Welcome to "On A Tangent", where we tell the stories behind the mathematics. In each episode, we meet a different early career mathematician from Mathematics Münster, and learn about their research, their path towards mathematics, and their hopes for the future. We explore the many different shapes that mathematical research can take, the early memories that led people towards the subject, and try to understand a little bit better the voices that make the mathematical community of today.

The podcast episodes are available on this webpage, on Podigee and on popular podcast platforms such as Spotify, Apple Podcast or Deezer. A total of seven episodes were produced.

The Hosts

In the second season, Carlo Kaul invites mathematicians to talk about their paths into mathematics. He is currently pursuing a PhD in arithmetic geometry and has already gained some experience in science communication. 

The first season was conceived and produced by Simone Ramello. He completed his PhD in model theory in Münster, undertook a postgraduate certificate course in science communication, and now works full-time as a science communicator specialising in mathematics and the quantitative sciences.

In the third episode von "On a Tangent", Carlo speaks with Nataniel Marquis, a postdoctoral researcher working on p-adic geometry in Eugen Hellmann’s research group, about their enthusiasm for mathematics, Batman’s influence on it, and the French classe préparatoire system, as well as how dancing and music help to clear the mind.

Send your feedback to mail@carlo.info.

• Transcript

  Carlo: From Mathematics Münster. This is On a Tangent, the podcast where we tell the stories behind the mathematics and explore the fascinating paths that lead early career researchers to Münster. My name is Carlo Kaul. I'm a PhD student in arithmetic geometry, and I have the privilege to bring new stories to you in the months to come. For the third episode of the season, I am joined by Nataniel Marquis. Nataniel pursued their mathematical training in Paris at the Ecole Normale Supérieure and Sorbonne University, where they completed their undergraduate, master's and doctoral degrees. The PhD research, supervised by Pierre Colmez and Antoine Ducros, focussed on multivariable Phi Gamma modules. In 2025, they joined the University of Münster as a postdoctoral fellow in Eugen Hellmann’s research group. It's very great to have you, Natalia.

  Nataniel: Great to be here.

  Carlo: As always, I want to first ask you about your first mathematical memory.

  Nataniel: I think that the first mental mathematical memory was rather in high school. Okay, I think I maybe I don't know what the equivalence in German of like, the end of the end of secondary school. I think it was mainly because I was very specifically interested in my math in primary school. Yeah. I just end up with a very good teacher at the beginning of my secondary school, which, I have to confess, I had, like, a detention because I was talking too much, and she just made me like a detention. That was like a geometrical thing. And at the end, it was Batman. And I think that was kind of like a trigger that math can be fun, I guess.

  Carlo: Like a graph, which is Batman.

  Nataniel: Yes. It was like some geometry exercises. You should draw something according to some instructions. And at the end it was Batman. I think it was a trigger. And then like in secondary school, I was mainly interested, but I didn't do some math outside of school. I think it was mainly interested, like what was happening during lessons. Yeah. My mind was not very filled with math. It wasn't the case until after high school. That's my first very precise memory of math.

  Carlo: So that was your math teacher?

  Nataniel: Yes, definitely. I think that in primary school it was not very notable for me. Like, it was nice and I was good at it, but it was very notable.

  Carlo: So you went to school in Paris, right?

  Nataniel: After my high school.

  Carlo: But your high school, where was it?

  Nataniel: It was in the south of France. Just in Cannes, to be precise.

  Carlo: So, what made you move to Paris?

  Nataniel: Okay. I think that in France, there is this very specific system, Grand École.

  Carlo: Yes.

  Nataniel: You might want to go to “classe préparatoire”, which is kind of like two years or three in a high school environment, but with very intensive teaching. Basically you cover both math, physics, informatics, maybe not all at the level of bachelor degree, but kind of. I had not the idea at this point to do some research. I just knew that I wanted to do some math. Actually, it was not sure when I was at the end of high school because I had published some novels. And I was also doing a lot of harp and mathematics, and like all three were options that I considered at some point.

  Carlo: So, harp the music instrument?

  Nataniel: Yes, the music instrument, some writing and some math. Math was at this point always enjoyable to do while for instance, writing sometimes was very valuable and refreshing, and sometimes it was like kind of a heavy effort. And for music … I think I wasn't good enough to do that even though. And also the security of math, I guess, which is kind of a bummer at this point.

  Carlo: Before you get into these classe préparatoire, you have already decided that you want to go into the math direction.

  Nataniel: Yes, but I didn't decide for research, for instance. So after this classe préperatoire, I went to the ENS in Paris. I learned about the ENS in classe préparatoire. So it was not at all planned all the way along.

  Carlo: So in your high school, what was your mathematical life like? So did you read a lot of math or was it just in school? You enjoyed the subject?

  Nataniel: I think it was in school. And I always did like the hard exercise and the books, but it was very limited to school because I was doing a lot of other activities where my mind was full of other things when I went back home.

  Carlo: So you didn't realise that yet that you have some exceptional talent?

  Nataniel: I don’t know what this means, first, it's weird. I knew that I was good at doing math in high school, but I did not investigate … I did not make it like, fill the other parts of my life.

  Carlo: Yes, okay.

  Nataniel: It was an option and it was something I was good at high school. But it didn't become part of my life at this point. Dance music and writing were more a part of my life out of school rather than math.

  Carlo: So you had very little free time there.

  Nataniel: I had no free time, basically. But it hasn't changed. 

  Carlo: That's right. You do much more math. So, then you went to classe préparatoire in Paris, and there you learned about the ENS.

  Nataniel: Yes. For the first time, I think in high school, I never really stumbled on concepts. And at classe préparatoire, I first saw, the things that like. Some exercises were really hard. I had a very good teacher in my first year. That was a long push. And he just was able to convey the love that math and all the funny algebraic stuff. Let's say he was more oriented I think in algebra even though his analysis lessons were very good. But he talked about a little bit of set theory, a little bit of patterns at least about categories. So that was the time when I was very interested in math and it made me more curious than average. I guess I was already curious about math, but that's the thing that really loved it. In the second year of classe préparatoire I also had another teacher that was very good. It was the year of my life when I worked the most, to be fair. So, I really went deep into hard exercises. I also went little bit deeper into number theory. I was really interested in, I first opened my books on my own and things like that, and I did a lot of exams to prepare the competition. But most of them were number theory.

  Carlo: This is something really special about the French system that you can already do real math at this point. For example, in Germany right before you start at university, you only did calculations your whole life and algorithms.

  Nataniel: I do think that you reach more abstract things faster in Germany. But maybe you reach harder things,… , There's really a culture in classe préparatoire to give you very basic notions like groups or things like that, but go with very deep, and hard exercises. I think, that's a little bit different. I do think that, like in Germany, from what I've seen at least (maybe I'm to biast about student in Münster), but I do think that you reached higher level of abstraction faster.

  Carlo: Totally, at the university. But classe préparatoire is not really university, but usually more. In Germany at high schools, you really don't do any concepts or any proofs or whatsoever. But then you start at university and there's this cut where you have this school math you never really use again. It's more calculations and algorithms you learned and then you start from completely scratch. Maybe the French system is a bit smoother in this regard.

  Nataniel: A little bit. I think the level of abstraction is really gap at classe préparatoire. I mean, you did some proof before. The big difference is not the proof because you did some, but the definition of the objects and the fact that the definitions are really like rigorous. It's really something that happens only in classe préparatoire.

  Carlo: Did you immediately live in Paris?

  Nataniel: Yes. I think that in classe préparatoire it was hard to tell because I was mainly working. I did all the things, I kept on with the harp and things like that, but I was still a little bit attached to some peoples in the South. So I was calling people a lot and things like that. So, I think that I would say yes, but I wasn't fully involved as I was when I was in the ESA.

  Carlo: I'm also asking you because you were quite young.

  Nataniel: Yes, I was 15 when I moved to Paris.

  Carlo: And you moved alone.

  Nataniel: I was in a boarding school. I mean, it was kind of a boarding school. So this was a really good thing because it meant that like for instance, in the in the weekends there was no Mensa. But we usually cooked together on Sunday evenings or things like that. So, it was not alone. It wasn’t a lone life at all. It doesn't feel like it.

  Carlo: That's very nice. And then you apply to ENS because this is the most exciting thing?

  Nataniel: Yes. At this point, it seemed to be the thing that did the most of number theorists. 

  Carlo: So you were already into number theory?

  Nataniel: Yes, I mean, there is a specific exercise for the expansion, where you have to present a kind of a mini bachelor thesis. I chose a prime number theorem, using analysis. Not with holomorphic functions. I totally forgot what the prove was.

  Carlo: It's actually very funny because I did something similar in Germany. When you do your A-levels, you can do a fifth grade. Normally you have four subjects, but then you can add a fifth A-level subject, a final examination subject, where you can do some mini thesis. And I did this and I did also the prime number theorem. But with the holomorphic method.

  Nataniel: We are complementary. I knew from classe préparatoire that I was more into algebra. I went to this partly for that. That's not one of the reasons, but it's the main advantages that it's because you're paid by the ENS. You're an employee of the state. And also it seems more open because in the same building you have people doing science, literature, and there's a lot of associative life around it.

  Carlo: So, in your classe préparatoire and also at the ENS was math always a sole focus. You also have to take other subjects?

  Nataniel: So I thought mainly about math that I definitely worked around the others.

  Carlo: But you had to do the others or did you really enjoy the others as well?

  Nataniel: I think that French was nice. It was very interesting. In physics I had an excellent teacher in the second year. So this was really interesting due to the professor, uh, and a little bit of informatics. But at ENS it's very different because it's obviously math I did mostly, but I also grappled 1 or 2 lessons from other subjects. But it was just for my culture or to dive deep into one thing. I did a literature lesson. I followed, let's say ecological development.

  Carlo: Did you have time in your classe préparatoire to do some literature or was this literature thing abandoned after you started doing math seriously? I mean, writing.

  Nataniel: No, it's a question of time. I did write a little. But like, slowly as my social life, let's say, grew in the ENS. I think that is something I placed my big projects on ups. But I have some in mind that I might put myself onto paper at some point.

  Carlo: How did you path to p-adic Hodge theory work? So you started with interest in number theory with the prime number theorem. What happened then?

  Nataniel: I think that it was thanks to two professors I had in algebra, Algebra one and Algebra two. So in algebra I had Antoine Ducros, which is one of my PhD supervisors. And in algebra II , I had Ariane Mézard. Her aim with this lesson was also to provide us with deep glimpses, let's say, of different areas. And then I had to do a master one internship, which was during Covid. So it didn't happen, but I was supposed to do it. At this point, she oriented me towards periodic number theory. And I was already like quite diverse debate. So then it just became worse and wors. Then she did a little seminar about perfect wages. So I did that. And then I had this internship with Arthur-César Le Bras, and then I ended up with [Pierre] Colmez as my master's supervisor. So, I went deeper and deeper and deeper.

  Carlo: So was it a the straight path for you, or where you're also in the middle of the thinking, should I really do this or maybe should I do other thing. Was there another thing?

  Nataniel: There was one point. When I was looking for a supervisor, it was also the second lockdown for Covid, I had emailed Colmez and he just said, “yeah, look at these and we'll see when you come back to Paris”, because I was away for the lockdown. I also received an email from another researcher who just said, I think you're good. So just don't waste your skills into p-adic number theory because it's just too busy. At this point, I was kind of hesitating. It was also not a good time for doing math, because it was the first time in master-two that I stumbled upon the fact that literature is not well written. I think, that until the master-two-year, everything was properly proved in the lessons I followed in the things I read. And then I stumbled upon the fact that, like some of the teachers were not doing the full proofs and that the literature could be, like, kind of sketchy. And so this Post-Covid was kind of a harsh time for the math, I guess. So at this point, I wondered about doing some C*-algebras. But afterwards, Colmez just said, “we will do your master's thesis together, and I was back on the path.

  Carlo: Did you like writing your master's thesis? Was it like the first research text you wrote, or did you do some research before that?

  Nataniel: What do you mean by research text?

  Carlo: I mean, probably for your master's thesis, it was the first time you really thought about very recent and new mathematics?

  Nataniel: For recent it's true because I did my master thesis is mainly about Zábrádi’s article. I already wrote some bachelor thesis about class number theory and finiteness of class number and things like that. So, it was already like a bit of an exercise of writing, but I agree that like, it was kind of the first time when I read current articles.

  Carlo: And even with a view towards something new, right? I looked into your master's thesis. 

  Nataniel: Yes, that's very true. I think it didn't make that much of a difference at this point. I think that, while writing my master's thesis, I didn't realise that it was like opening to some things. At the end of my thesis, that's when I realised that some of the article, I could generalise it and that become part of my PhD. I think, it was the interesting part. One thing I can say is that at some point I struggled for two weeks around the understanding of a specific proof in the article. So that was the first experience of really a little bit of research, I guess. And together with Colmez we thought a little bit at this point. So that's shift. But I also think that it was an area of my life when it was not like math as learning was not really interesting for me. Also because it was also the third kind of lockdown in France. I was all the time alone in my room trying to understand. So the meetings with Colmez were really good at this point. And it really put the first little pieces to like, bring me back to math. But it was a slow pace. I think the master's thesis is really the parts of my life were the work was hard and the motivation was also hard. I had some very huge personal issues at this point, too. So it was kind of all at the same time. It was slowly getting down. And the year after that, before my PhD was like the most personal problem. So it was mainly a year about regaining my link to math. It worked. But it wasn't so easy. 

  Carlo: But given your master's thesis, it was natural to pursue a PhD in this direction. And it was probably offered to you or how did this process go?

  Nataniel: I don't remember precisely. I think that we just talked about it with Colmez at the end of my master's thesis. He offered me three articles at the beginning of my master's thesis, and I only looked at parts of two of them. So, I guess that was in July/in summer I looked at Carter-Kedlaya-Zábrádi’s article and I just thought, I might just have an not that, not that complicated generalisation. So he was just okay, we can just continue to think about it. And it became my PhD thesis. So I think it was kind of natural.
  It was a rather smooth process. It was a rather smooth path from Master’s to PhD thesis.

  Carlo: At the end of the PhD thesis, you probably thought about, what doing now? Was it like clear to you that this topic is still very fun to you and you want to continue in this direction? Or have you also had some doubts during your PhD?

  Nataniel: Doubts about the thesis - No. I think that in the middle of my PhD thesis, in the second year it's the time where all the arguments about my thesis were kind of in place. I had to write them down and like, a lot of things happen in the middle or obviously, but it was the time the subject feels very overwhelming. And to see if I had skills enough to like properly say interesting things and was really kind of harsh at this point. And slowly in the end of my thesis I can maybe see where I'm going. The subject still feels very overwhelming. I think like very full of things you should know and all the people seems to know and you don't. And you don't feel that secure about it.

  Carlo: But how did you come up with going to Münster after Paris?

  Nataniel: I think that I just applied to a few places, and it's the first to answer. It's as simple as it.

  Carlo: Did you hear about Münster before?

  Nataniel: No, I heard about the people that interviewed me, Hellman and Schneider. I heard about them. I think it's more like the persons that were here that I knew that rather than in place. I didn't know how big was this university. 

  Carlo: Now you're here. What does a typical day of research look like for you? Do you have a typical day?

  Nataniel: I don't think so. I usually don't work after eight. I usually don't work before ten. But no, I don't think there is a precise schedule about what I'm doing. Because there are different things like writing articles, thinking by yourselves, having. I don't think I can describe a proper typical day of work.

  Carlo: How do you like to do research. Do you like to just think, or do you like to write things down all the time or just a lot of reading? Your research process maybe. 

  Nataniel: I think that's what I'm doing the most is reading and thinking. But I think that it's not the most efficient. I think that the most efficient is definitely, when I'm writing down things for instance for collaborators. If I'm writing down, I think the best thing to learn things is to have to do a talk about it. And that's really like the part when you read, but you know that you must extract something that is very suitable and shapes in a way that is very suitable for some people.

  Carlo: What do you do when you want to relax a bit and not think about work? What's your favourite activity in Münster?

  Nataniel: In Münster, there are three things I'd say. The first one is talk to people. A lot of the people that I bay or have a relation with. The second thing is just listen to music. The best thing is to go back at home, blast music and cook. Just being in the air with music. It's really shifts the mind from work to not work. And than it’s obviously sports. I do a lot of pole dance. That's also something that washes my mind.

  Carlo: What genre of music is best for coming down?

  Nataniel: It's not a genre. It's about lyrics for me. It’s something that I am obsessed about. If I am obsessed about the song, the lyrics. I have queer songs, rap, rock songs. I’ve got a lot of different things, they are mainly in French. 

  Carlo: Now that we've gotten to know Nathaniel, we will come to the second part of this podcast is the so-called A-or-B-game. For this game, which will consist of roughly 30 A-or-B-questions. I will ask Nataniel to answer as quickly as possible in order to get an idea about their intuitive thoughts. 
  0 or 1? – 0.
  Analysis or algebra? – Algebra.
  Coffee or tea? – Coffee.
  Beach or mountains? - That's harsh. Because I'm originated from beach. So I think I would take mountains because it's less usual.
  Snooze or getting up? - Getting up.
  Chaos or order? – Order.
  Summer or winter? - Winter, but none of them are the good seasons. A good season is spring and autumn. Autumn looks very fine. It's beautiful. Both of them are pretty nice. So the answer is neither. 
  Abstract nonsense or crazy calculations. - Crazy calculations.
  The automorphic side and the spectral side. – Spectral side.
  Smooth or locally analytic. – Smooth analytic.
  Socle or cosocle. – Socle.
  In French or in English? - Depends what? No, I cannot choose.
  Sorbonne or ENS? – ENS.
  Paris or Münster? – Paris, there's too much people there. 
  Fifth arrondissement or outskirts. - I spent more time in the fifth.
  Luminy or Oberwolfach? - I have been only to Luminy, so I take Luminy for now.
  Blackboard or slides? – Blackboards.
  PDF or physical copy? – Physical copy.
  Writing math or writing fiction? - I think writing math because thinking about fiction is best.
  Karaoke or salsa night? – Karaoke, I cannot dance salsa.
  Reading or writing. – Reading.
  Pole dance or hip hop? - Pole dance. I mean, I did a little of both, but I did more pole.
  Rock or rap? – Rap.
  Biking or walking? – Walking.
  Mensa am Ring or home cooking? - Mensa am Ring. For lunch it’s Mensa.
  Clapping or knocking? - Clapping.
  To be or not to be. – Not to be.

  Carlo: All right. Thank you very much.

  Carlo: So you were hesitating with abstract nonsense or crazy calculations? Why is that? How would you describe your research more?

  Nataniel: My research is more crazy computation. I'm just thinking that I like them both. So it's hard to choose. I do more crazy calculations, but I also think that most of my research is getting down to one crazy calculation.

  Carlo: So using abstract nonsense to get down 

  Nataniel: Yeah. But I'm not using that much abstract nonsense in my thesis for instance.

  Carlo: I asked you about fifth arrondissement or outskirts because, when you were in Paris and you were not doing math, which was probably not much time, but did you more like to wander around Paris where the ENS is or did you more like go into nature?

  Nataniel: I think I did both, but I spent a lot of time for the cultural life of Paris and also like in the associative life of the ENS. So I have to say fifth because a lot of my activities in Paris were linked to the ENS from one way to another, if I was not studying. So all my dance that I did was in the ENS hence in the fifth.

  Carlo: And for Mensa am Ring or doing home cooking. You said for lunch Mensa, I agree. I also like it very much. Not because of the Mensa but because of social aspects.

  Nataniel: I have this debate at lunch that I think the German people are thinking too low of the Mensa. Really. It's one of the best. 

  Carlo: Nice to hear for the Mensa. But you also do home cooking. You also like to cook a lot? What's your favourite dish to cook?

  Nataniel: I think it's like vegan Bologna I guess.

  Carlo: Oh, I can totally agree. I love it as well.

  Nataniel: You have to put some soy sauce in it and some red wine. Just a little bit of both to have the taste.

  Carlo: I don't know if the Italians approve the soy sauce.

  Nataniel: I know they don’t.

  Carlo: But I would try nevertheless.

  Nataniel: I was about to say, I don't know if Italian proves vegan, but I think that friendship was vegan. Okay, I will not shoot this arrow. 

  Carlo: For the end of this podcast, I want to ask you five questions and you can just answer. They are kind of difficult question, but try to answer in one sentence. So the first one is what is your favourite mathematical theorem?

  Nataniel: The first that come to my mind is the four square theorem.

  Carlo: Right. Then which of your results or achievements are you most proud of?

  Nataniel: I think it's the gaps that I field rather than the results.

  Carlo: Can you explain that?

  Nataniel: No, I just have the sensation that a lot of the results I use in my research, they may be something that is just not explain or that's lacking in the original article or the original reference. I think I'm more proud about, my way of filling the gaps or expanding things rather than the results themselves.

  Carlo: I just want to mention that when googling your name, there are a lot of achievements literature wise. So about the books you wrote. So I think very impressive

  Nataniel: I was really young.

  Carlo: Still it's very impressive to see your achievement there. Just want to mention. So which mathematical field might you have chosen if not your current one?

  Nataniel: I think it would have been like C*-algebras applied to group dynamics. 

  Carlo: Is there something in your professional life you would have done differently?

  Nataniel: If I take my experience, style and I applied to myself. Then I would have found faster ways to pull out the pressure. I'm still not totally out of it. I'm more out of it than when I was like eight months ago or four years ago. And I could use this experience.

  Carlo: I think, That's a really important experience. And also it links very nicely to my last question. So now I'm curious, what advice would you give a student considering mathematical research today. Maybe you can also say what do you think is the best way to pull out the pressure? Is there like one advice or is it just like learning that?

  Nataniel: I think that I will only give one sentence of myself and one sentence of Ariane Mézard. Okay, so from myself is try to always at the most you can enjoy your math, even if it does not go in the way that like the system has an incentive for you. Yeah. And the the sentence of Ariane Mézard is “Don’t damage yourself.”

  Carlo: Thank you very much. I think that's a great way to end.

  Nataniel: Thanks for having me.

  Carlo: If you enjoyed the conversation with Nataniel as much as I did, please consider sharing this episode with a friend or colleague. In the show notes you can also find a link to the video part of my conversation with Nathaniel, where they explained how the notion of space and complex geometry can be adapted to a p-adic framework. I'm Carlo and I will catch you on the next tangent. 

In this video Nataniel explains how the notion of space and complex geometry can be adapted to p-adic framework.

Link to Nataniel's website

In the second episode of the second series of On A Tangent, Carlo speaks to Elisabetta Mazzolari. She is a doctoral researcher at the Institute of Mathematical Stochastics and works in the field of probability theory. In the podcast, she explains what led her from Pavia via Delft to Münster, and what she does to clear her head.

You can send feedback directly to mail@carlo.info.

• Transcript

  Carlo: From Mathematics Minster. This is ‘On a Tangent’, the podcast where we tell the stories behind the mathematics and explore the fascinating paths that lead early career researchers to Münster. My name is Carlo Kaul. I'm a PhD student in arithmetic geometry, and I have the privilege to bring new stories to you in the months to come. For the second episode of this new season, I am joined by Elisabetta Mazzolari. Elisabetta's academic journey began in Italy at the University of Pavia, where she completed both a bachelor's and master's degrees. During her master's studies, a research study took her to Delft in the Netherlands, where she finished her thesis on a large deviations and the revised model. In October 2025, she moved to Münster to work as a doctoral student and the group of Matthias Löwe. So it's great to have you, Elisabetta.

  Elisabetta: Thank you for having me.

  Carlo: The first question in this podcast is, what is your first childhood mathematical memory?

  Elisabetta: I don't know if this is my first mathematical memory, but I have this picture of myself, at the kitchen table at my parents house, while I was doing one of my first mathematics homework, I guess, under the supervision of my mother. And I guess the problem was something like, if I have three apples and I…

  Carlo: … give you …

  Elisabetta: … one apple, then how many apples do I have left? And I remember that I was liking a lot this type of problem. Because once you understand the mechanism there behind everything is very easy and is easier than it looks.

  Carlo: Yes.

  Elisabetta: I guess this is my first memory.

  Carlo: So. your mother was a mathematically involved person?

  Elisabetta: No, actually, but she liked a lot mathematics. She still likes a lot of mathematics. So I think, she kind of influenced us a bit.

  Carlo: Great. And this was in Pavia or where did you grow up?

  Elisabetta: No. I grew up, in a small city in the north of Italy. It's very close to Pavia. The city is called Cremona. It’s a very pretty city.

  Carlo: Very nice. So, you went to school and high school. At what point did you realise that it was more than like a kitchen-table hobby for you and really something which you have a lot of interest in doing and want to maybe also study later?
  Elisabetta: So, while I was in the high school, I was very indecisive between a lot of options. At first I was very oriented towards physics. Then at some point I wanted to do medicine and then even philosophy … and at the end, mathematics of course. One day I was speaking with one of my team-mates. I was doing basketball at that time, and she told me that she was studying mathematics at the university, and I found that very interesting. And I don't know why, but from that moment, doing mathematics at university started to become a concrete option. Also at that time, this was I think my last year or my second-to-last year of high school. And at that time I had a very good teacher in…

  Carlo: Mathematics.

  Elisabetta: And a not so good teacher in physics, if I can say.

  Carlo: Of course.

  Elisabetta: I think also those factors, they led me to choose mathematics at the end.

  Carlo: But did you have any concept of what studying math would be like?

  Elisabetta: No, actually, it was just I was attracted to it. I knew in the high school that I was good at it, but not just because I was particularly talented. And I think it was because I like to learn and to study new things. But then it was I didn't have a lot of idea about it. And at the end it was the right choice.

  Carlo: And then you decided at some point that you want to study mathematics. How did you choose to go to Pavia? And what was that like for you this time?

  Elisabetta: I don't know. I feel like a lot of things in my academic path were led by chance. I don't want to be misunderstood by saying this, but really. I wanted to go to university in Pavia because also some of my friends were studying there. It is a very good university in Italy and is also kind of close to my hometown. So it was the perfect solution. And then I decided both for my bachelor and my master to do a very general curriculum because I didn't want to specialise in only one topic in mathematics. So for instance, I did my bachelor thesis in algebra, and it was about the construction with straightedge and compass and the connection with the Galois theory. I liked it a lot. Then during my master I kind of shifted my interest towards probability. But I didn't see it as a cut. It was just a more natural changing of my interest. Then I decided to write my master's thesis in probability. But I actually was also very fed up with the city, because I studied in the same city, both for the bachelor and my master and also a lot of my friends left at some point. So I applied for the Erasmus traineeship, and then I had the possibility to go to Delft in the Netherlands to write my master's thesis.

  Carlo: What was for you the strongest difference between studying in Italy and studying in the Netherlands?

  Elisabetta: In the Netherlands I did the Erasmus. That was called the Erasmus traineeship. So I didn't have to attend the lessons of or give exam. It was just my master thesis. I felt like I was more part of the academic environment and not just like a student studying, doing exams and attending lessons.

  Carlo: Would you say, like the hierarchies were flatter in the Netherlands or what was the main reason?

  Elisabetta: Yes, exactly. For sure the relationship was very less hierarchical than the one that I had in Italy with my professor. And also I felt like the time that I spent in Netherlands was my first contact with the academic environment. Because I had weekly meetings with my supervisor and where I was discussing the doubts that I have, the problems that I have while writing my master's thesis. Also I was showing my progress and in the meanwhile I could speak the whole day with her PhD student about what was fascinating for me in probability and our research field. So yes, it was my first contact with the academic environment.

  Carlo: So that's very interesting because you said you did your bachelor thesis in algebra. And of course, I'm very attracted to algebra, naturally from what I'm interested in. Do you remember what the reasons were for the shift to probability? Was it, like more appealing that it was applied or more applicable, much more than algebra, for example?

  Elisabetta: Yes. Maybe I needed something that could be more applicable in some sense. But also, I think it was very tailored by the professor that I had. So I had very good professors in probability, and they stimulated my interest a lot. A real part of my interest in mathematics was due to the professors.

  Carlo: How much students were you in your master's classes in Pavia?

  Elisabetta: Very few, like around ten. Yes, we were super few.

  Carlo: Yes, the same thing, where I studied for a master's. I had also very close relationships to the professor because the courses were so small. So you just had a good mentor, you would say?

  Elisabetta: Yes, exactly. I mean, in Italy, the relationship at least in the bachelor was super cold with my professor. But then in the master, we were few. So we get to know the professor a bit better. 

  Carlo: How did you come up with Delft specifically? Why not another university?

  Elisabetta: So, I asked the thesis at first to a professor in Pavia, the city where I studied. And he had contact with this professor in Delft in the Netherlands. And it was very interesting for me. The topic there was statistical mechanics. So I decided to go there. I could have done everything for like living somewhere else.

  Carlo: I get it. Yeah. So you didn't know your Delft master’s advisor before?

  Elisabetta: No.

  Carlo: It was just a match.

  Elisabetta: Yes, exactly. It was a match. And she was also Italian.

  Carlo: Ah, okay. That's funny.

  Elisabetta: I liked her supervision a lot. And she also had a very nice group with two PhDs, and they helped me a lot.

  Carlo: And how did this work in your meetings with her? Did you talk in Italian or in English?

  Elisabetta: No, we were talking in English because those meetings were attended by two PhD students…

  Carlo: Okay.

  Elisabetta: And sometimes we had like a zoom meeting connection also with my other supervisor in Italy. But those meetings were very interesting, and also super intense. We spent like the whole afternoon together in a room with a blackboard.

  Carlo: Really?

  Elisabetta: It was it was very interesting.

  Carlo: So you just discussed everything that came to mind.

  Elisabetta: Exactly. So the progress that I made in one week, of course, the topic was also kind of easy. So not a big deal.

  Carlo: Very cool. Then you were in Delft. And how did it happen that you came to Münster after all?

  Elisabetta: So basically, that was my first encounter with the academic world. Because before, when I was in Italy, I didn't really know, how it was doing a PhD. But in the Netherlands, I really was much more attracted to this kind of environment. So, I decided to apply for a PhD, and my supervisor in the Netherlands suggested to apply for a position here in Münster, because she visited some weeks before the university. And she had a very good impression. And also, there was this perfect match. Because one of the main papers that I studied for my master's thesis was written by my current supervisor. So it was a very perfect match.

  Carlo: So you meet your heroes after all. I can also very much relate to that. To me it was really cool to finally meet the person who read the papers from all the time. You always see this name and then you see there's a person behind this. I really like that as well coming to Münster. So, my office is right across from yours through the window. And I see you studying a lot. What does this process do? I see you a lot at the white board. So, I have a black board in my office, but you have a white board. 

  Elisabetta: I prefer the white board.

  Carlo: I think this is like a different culture between different fields of math. I feel like in my field, people hate white boards, but maybe it's awesome. And everyone only wants a black board. But I also get why it's annoying because my blackboard is quite full. And when I also write always white things, I always have to wash my hands because my clothes...

  Elisabetta: The problem is also, when you cancel things… when you try to erase, then like the chalk still is on the board. It's not very practical. I prefer practical things.

  Carlo: But other than writing on whiteboards. How does researching look for you now?

  Elisabetta: I don't know. I have just started a couple of months ago. So, I'm still in this grey area where I don't really know what I'm doing. But I hope that this will lead some time to something.

  Carlo: Yes, I relate.

  Elisabetta: But now I'm studying a lot of literature. I'm reading some papers, and my supervisor just proposed me this first project that I'm working on. So, either I have meetings with him, where I discuss mainly the things that I cannot understand and the problems, that I have. Sometimes I also show my progress. I study these papers, and I started also to borrow some books from the library. They are super interesting.

  Carlo: So you are someone who uses the library. I know very few people who do that. That's great.

  Elisabetta: It's very nice. Basically, I try to learn as much as I can. This is the goal for this period.

  Carlo: I think it's a great idea I also tell myself. Because I think I'm in a very similar period of my PhD now. Other than being at the university or being in the common room… what do you like most about the city?

  Elisabetta: My favourite part of the city is that it is a very bike friendly city. I think I'm very emotionally attached to a bike because also in the city, where I was born and where I lived, that was Cremona. It is a very small town and we go everywhere by bike. 

  Carlo: It is also as flat as Münster.

  Elisabetta: Yes, it is flat. We are in this flat area called Pianura padana [Po Valley]. Basically there are zero mountains.

  Carlo: That's good.

  Elisabetta: It's my favourite means of transportation and the only transport that I use.

  Carlo: And you have a beautiful bike. I have to say.

  Elisabetta: Thanks, that you think that, because I think too it's a super pretty bike. But the place that I like a lot in the city is also the Saturday market on the Domplatz. One of my favourite activities is going there and go for a walk and then buy some fresh vegetables there.

  Carlo: You really buy vegetables? I feel like most people I know go there to grab something to eat.

  Elisabetta: That is the second part. I go there to eat, too. There is this stand, it’s incredible, it’s Greek stand. I like it a lot. Then they give you this so called Studentenbox.

  Carlo: It's great. Amazing. I didn't know that. I have to look for it next time. And if you like not biking to work or at the market, what do you like to do for coming down? Like relaxing after a full day of thinking about, I don't know, probability.

  Elisabetta: So these last two years I developed this very strong passion in running. It is an activity that often clears my mind. But it's also just fun. I mean, I go for a run and maybe the sun is shining and there is a very nice wind. I mean, after that, life feels perfect. And, you know, there is not any more chaos. I like running a lot. I started almost two years ago. In Germany, during the winter, it is hard to run, but now there is starting the spring season to be back on track.

  Carlo: The day of recording the podcast is like the first very beautiful day again after the winter. I think, we are almost through it. What do you recommend, where to go running best in Münster.

  Elisabetta: I live in a neighbourhood that is called Gievenbeck. It is beautiful because you have a lot of fields. I guess it's going to be very interesting running the Promenade, too. I was doing this calculation at the end of the day… we are mathematicians: The promenade plus the other should be like around ten kilometres. So, I would like to prepare a half marathon. And if I follow this route around the promenade plus the other, there should be half marathon.

  Carlo: Great, very impressive. 
  Now that we've gotten to know Elisa better, we will come to the second part of this podcast, the so-called A or B game. For this game, which will consist of roughly 30 A or B questions, I will ask Elisabetta to answer as quickly as possible in order to get an idea about her intuitive thoughts. 
  0 or 1.
  Analysis or algebra.
  Coffee or tea?
  Beach or mountains. 
  Snooze are getting up.
  Order or chaos.
  Summer or winter.
  Conceptual proofs or technical proofs.
  James Blunt or One Republic.
  i.i.d or not. 
  Random walk or Brownian Motion.
  Cycling in Delft or cycling in Münster.
  The law of large numbers or the central limit theorem. 
  blackboard or slides.
  PDF or physical copy - I'm sorry for the trees.
  Delft or The Hague.
  Italy or Netherlands.
  Pavia or Milano.
  Pavia or Cremona.
  The Ligurian Sea or the Adriatic Sea.
  The north or the south.
  Paris or Rome.
  Microscopic or macroscopic?
  Qualitative and quantitative.
  Biking or running.
  Matlab or R.
  Ruler or compass.
  Discrete or continuous.
  Wave or particle.
  Paris or Normandy.
  Sicily or Sardinia.
  Probability theory or statistical mechanics.
  Exact or asymptotic.
  Asymptotic calculation or proof.
  Physics or chemistry.
  Constant external fields or random external fields.
  Clapping or knocking.
  To be or not to be. 

  Carlo: Great. That's it. So you didn't need any skills. 

  Elisabetta: But actually, I'm not very sure about some answers that I give. I was thinking fast because of the pressure.

  Carlo: Well, that's great, because you should always just say the first thing that comes to mind. So was Pavia or Cremona hard for you to decide?

  Elisabetta: Yes. It was very hard for me to decide, because I guess that some years ago, I would have said Pavia. I kind of escaped from Cremona to study in Pavia. Also I lived in Pavia with two of my best friends. So it was the ideal setting. But I guess now that I'm abroad, I feel like that Cremona is my true home. Because also, my parents are here, and now some of my friends are there. 

  Carlo: So you said you were living with two of your friends. So do you like to live with other people? 

  Elisabetta: I think I really enjoy my self presence. So the presence of myself, but I guess it's very nice when you share your flat with flatmates.

  Carlo: You said you like technical proofs more, but you don't like calculations as much, right?

  Elisabetta: Yes, exactly. This is the point. I mean, I really like the rigour there is behind technical proof, but really, I hate calculation. Maybe it's because, I have trauma now because I'm getting through a lot of calculation and I'm having nightmares with them.

  Carlo: So you have to solve big integrals.

  Elisabetta: Exactly, this is a nightmare. But I guess, I have to go through them.

  Carlo: You also don't like as much playing around with the computer and programming things.

  Elisabetta: No, zero. I'm also pretty terrible like we do with technology in general. I'm not very much Gen-Z.

  Carlo: You never need to use a computer for any calculations or something?

  Elisabetta: No, zero. Thanksfully.

  Carlo: You would also say your research or your topic is more pure math.

  Elisabetta: Actually, it is not very applied. It's basically probability theory that you can apply.

  Carlo: You don't want to be the person applying it, right? I like it. What I'm doing is really not applied. It's really pure. But sometimes I feel like if you could apply it, it would be nice. So, you can tell people, this is why I'm doing this.

  Elisabetta: In the talk, then you can say, ‘we can apply this to this very nice setting. You don't do it, of course, like, you just study very few concepts. But it's nice saying that you can apply this.

  Carlo: Another thing I was curious about. You talked about German winters. Would you say that you miss the weather in Italy a lot? Have you settled in already, because you were in The Hague before and Delft?

  Elisabetta: I think in general, I miss Italy a lot because there is where I have my friends and my family. This is the main reason. I think I easily adapt to every situation. So I feel like this is now my home, Münster. And actually, I come from the northern part of Italy. So, the weather in winter is not very nice and is almost like this. But now my mother is sending me a lot of pictures of the sun that is now in Cremona with 20 degrees. The other day that here it was raining. I was super jealous.

  Carlo: So for the final part, I always want to ask five questions. Try to answer like in 1 or 2 sentences. There are five things I want to know of everyone I'm interviewing. So first thing is, what is your favourite theorem?

  Elisabetta: I don't know if this is my favourite theorem, but we mentioned it also in you’re A and B game. I think, I'm attached to the central limit theorem. Because it's one of the first theorem that you study in the first probability course that you take at university. And also it's extremely recurring because I'm also now studying in different settings. But yes, I would say the central limit theorem.

  Carlo: Which of your results are you most proud of?

  Elisabetta: I don't know. But so far, I would say my master thesis like having the opportunity to do my master thesis in that. I applied for the traineeship project. Then I got the scholarship. And then I also think having the position here, is the thing that I should not take for granted.

  Carlo: Certainly. It's a very tough, I agree. What mathematical field would you have gone into? This was also very interesting to me in the early A or B game: You said algebra and not analysis. How can you say this?

  Elisabetta: I'm very interested in algebra.

  Carlo: So would it have been algebra?

  Elisabetta: I guess so. And I also know that you can work in the intersection between probability and algebra. Some persons in Münster are doing that. And I think it's very interesting and could be the perfect mix. 

  Carlo: What have you done differently with the experience of today during all the time until you came here?

  Elisabetta: I don't have a lot of regrets. But maybe I would have enjoyed doing the whole master abroad because I really enjoyed my time in Delft and it was very little. I only stayed there for some months. But maybe doing the whole masters abroad, it would have been a nice experience, both professionally but also personally.

  Carlo: Finally, what advice would you give a high school student who's now thinking about if she should go into math or what she should pursue? What would be your one advice with all your experience you have today? Very tough question, I know. 

  Elisabetta: I think, don't be afraid of math. I know that math could sound intimidating, and it actually is. But, like often, with hard work and true commitment it really pays off. This is a very meritocratic world. Yes. Don't be afraid of it.

  Carlo: Do it and work hard. Thank you very much for coming here today. And it was very nice talking to you. And we would see each other in the video part of the interview. 
  If you enjoyed the conversation with Elisabetta as much as I did, please consider sharing this episode with a friend or a colleague. In the show notes you can also find a link to the video part of my conversation with Elisabetta, where she shows how probability theory can be used to explain why matter suddenly changes behaviour. I'm Carlo and I will catch you on the next tangent.

In this video, Elisabetta explains how large deviation theory provides the stochastic foundation for physical phase transitions in the Curie–Weiss model.

In the first episode of the new season of On A Tangent, Carlo talks to Maxime Ramzi, a postdoctoral researcher specialising in algebraic K-theory and homotopical algebra. Maxime talks about how inspiring maths teachers influenced him, how he came to Münster via Copenhagen, and where you might run into him outside of mathematics.

You can send feedback directly to mail@carlo.info.

• Transcript

  Carlo: From Mathematics Münster, this is On a Tangent, the podcast where we tell the stories behind the mathematics and explore the fascinating paths that lead early career researchers to Münster. My name is Carlo Kaul. I'm a PhD student in arithmetic geometry, and I have the privilege of following in Simone's footsteps to bring new stories to you in the months to come. For the first episode of this new season, I am joined by Maxime Ramzi, a rising star in the field of algebraic K-theory and homotopical algebra. Maxime's academic journey began in Paris, where he studied at Sorbonne University and the Ecole Normale Supérieure. From there, he moved north to the University of Copenhagen to pursue his PhD in algebraic topology under Jesper Grodal, a period during which he was also awarded a prestigious travel grant from the Danish Ministry of Education and Research. Since 2024, he has made Münster his home, working as a postdoc in Thomas Nikolaus's group. And so it's great to have you here, Maxime.

  Maxime: Thanks for having me.

  Carlo: The first thing I want to ask you is what is your first experience with mathematics? How did you get into mathematics? What's your first memory?

  Maxime: I think it's difficult to answer. I think I have one memory that it strikes me as the first. I guess I went to see this exhibit at some point when I was like eight. And I discovered that mathematician was a was a job that you could have. And I remember being kind of confused because when you're eight, you don't really know what math is, you know, about numbers and everything. So it was somewhere in Paris, but I forgot where it was in Paris. My mom took me to this exhibit, and there was like a video of [Cédric] Villani talking. He didn't have the fields medal at that point. So he was just kind of there. So that was kind of nice.

  Carlo: Cool. And so this was a math exhibit?

  Maxime: It was a math exhibit. There were all these shapes, all these figures. And, I remember triangles or something, but I guess I always was somewhat good at math at school, but for a long time it wasn't really a thing that I considered as a thing.

  Carlo: So you also grew up in the Paris area?

  Maxime: I grew up in the suburbs of Paris, southwest of Paris.

  Carlo: And how did a typical day look for you when you were like in middle school or high school?

  Maxime: I think pretty standard, I went to school. Usually most days. We woke up around seven or something, and then we'd go to school. I guess it was a bit longer in high school than in middle school. And then I would come home and sort of do my homework and maybe play video games, play Warhammer with some friends, watch some series.

  Carlo: And when did it start to become serious? Your interest in mathematics?

  Maxime: I would say the last two years of high school. Probably. Even specifically the last year of high school is when I started thinking seriously about doing research in math or physics. I wasn’t sure at that point and I had a really good friend with whom we spoke about physics and math all the time. We were doing fencing together and our fencing teacher hated us for like talking about math during warm ups. And I think it really started when my dad told me about this video about ordinals. So I didn't know about ordinals at the time. It came from a weird conversation we had where we're arguing about ∞ versus ∞^2. And my dad had explained to me that they were actually the same and linked me to this video. At this point I started being interested in set theory. And I started doing research with big quote marks. But like I would spend some of my breaks in high school where you're supposed to think about school. But I was starting to think about math at the time. So I would say last year of high school.

  Carlo: So it was more like a personal thing that you became interested or was it also a teacher or how did this start for you, this interest?

  Maxime: I think it was mostly personal. So as I said, like I think with mostly with my dad. I had really a huge amount of luck with my teachers in high school. I had the same teacher in first year of high school and last year of high school, and he was really an amazing teacher. It really helped to have a teacher like this, but he's not really the one who made me want to go into it, I think. Even though he really, sort of influenced it in that sense.

  Carlo: Okay, cool. And then, so you still didn’t know between math and physics. So what was the next step for you after that?

  Maxime: So after that I went to this... I don't know if you know about Prépas. So this is like a sort of weird university thing in France. So I went there and it was pretty heavy curriculum with some math, physics, a bit of computer science, a bit of French, a bit of English. But the most two things were like math and physics. And again, I had a really excellent math teacher, like top notch math teacher that point. On the other hand, I had a not so great physics teacher, and I hope she will not listen to this podcast. I had a really sort of way worse physics teacher. And so at that point, it sort of became clear to me that I wouldn't get a lot of physics out of her. And in this sort of intense curriculum, you don't really have time. Well, you do have time if you don't do all the other subjects, but if you do all subjects, you don't really have time to actually sort of do stuff on your own. So I realised I would have to do mostly math and I really liked it there specifically because of my teacher and I really didn't like physics anymore. So the choice was made for me at that point.

  Carlo: Okay. So how old were you then?

  Maxime: I think I was about 17 at that point. And I gave up on physics. Even though I still have non research interest in it, but it was… at that point.

  Carlo: And then after this Prépas is there's like a canonical choice which school to go to or how does it go on?

  Maxime: There's a bunch of different schools. But this thing is you have some competitive exams at the end. So, you know, even though you have maybe schools you want to go to you can't necessarily get into them. So the canonical school for research was ENS. And I was very lucky to be able to get there. Otherwise it's unclear what I would have done.

  Carlo: Okay. But you did two universities at the same time right. How do you imagine that?

  Maxime: It's not really the case. So the ENS is kind of this thing besides the university. So you have all these universities which have a standard curriculum and it's the same thing all across EU, but then you have these sort of side thing, which is kind of French. And they don't give you a master's diploma. So you have to register at a different place to get a master's diploma. And what they do is they also outsource a bunch of their classes. So I had in total not so many classes in ENS, mostly the first year and a bit the second year, but most of my time was actually at university. I mean, most of my school time was at university. ENS was mostly social.

  Carlo: All right. Sounds good. So how did you become then interested in your field of research, in topology or algebraic topology or categories? How did this start for you?

  Maxime: That was a very long process. So I've always been more of a algebra oriented person. Like, I've never liked analysis or anything like this. And I got my worst grades from probability theory and analysis when I was at ENS. So I was always sort of bound to go to one of these more algebraic abstract fields. And there was a lot of hesitation when I was in my second year; I was hesitant between algebraic topology – I had a really good teacher in algebraic topology as well – representation theory, which I really liked because it involved a lot of linear algebra, and logic and set theory. I think that was also the main contenders were logic and set theory as well. Because I had a really strong relationship with some set theorists on an online math forum, where we talked a lot about math. And I think ultimately, it's kind of random luck and circumstances that made me go for algebraic topology rather than logic.In fact, I almost gave up on algebraic topology at some point, because my teacher in the first class I took on the subject was really good—except for one specific notion that he introduced in a way I simply couldn’t grasp.  So if you know a bit about this: CW complexes, a specific type of topological space. And the way he introduced them… there are several perspectives and the one he chose made no sense at all to me. And it seems to be a pretty important part of algebraic topology. So, If I don't understand this, I can't do this field at all. Luckily afterwards, I understood them better and so I got back on the track. But the main reason I ended up doing algebraic topology is that in the second year of the ENS, we have to do this abroad sort of studies for a few months. And when I asked around the main people who gave me advice were the algebraic topologists specifically Muriel Livernet in Paris, told me if you want to do algebraic topology, you should go to Copenhagen. So I went there for a few months and there I still had a lot of like hesitations between different fields. But the environment was really nice and I started learning a bit more of this algebraic topology stuff. And so when I came back and did more classes than I got really into it at the end of the day. But I still was hesitant. And ultimately what happened is that I did my masters with with an algebraic topologist because, again, he was the main person who answered, but at this point it would have been more difficult, but I would still possibly have done maybe logic or set theory or something. But because of this circumstances I ended up doing algebraic topology. And I don't regret it at all. I'm very happy I chose this path at the end of the day.

  Carlo: Very nice. So Copenhagen was then a natural choice to go on for a PhD.

  Maxime: Exactly. So I had gone once and I had met already my future advisor Jesper. During this semester abroad and we had really gotten along. So at the end of my master's, I asked my master's advisor like, what should I do? And he said that I could definitely do a thesis with him if I want it, but that he recommended going to Copenhagen instead if I could. And so I emailed Jesper and then from there, like I applied and everything and sort of you know.

  Carlo: So this kind of algebraic topology you did was a really big thing in Copenhagen. Did you have a big group?

  Maxime: Exactly. I think so. The first time I went, because I was sort of just a weird master's student coming around and no one knew really why, because it's a kind of weird thing to do. So I didn't interact with the with the whole group, but I think it was already a pretty big group. But when I was there for my for my thesis, then it was really a huge group. I mean, they did an amazing job hiring people. So there's like really amazing post-docs and PhD students and professors. So a lot of opportunities to speak and to interact with people. So that was a really strong place for that field.

  Carlo: And it was already in your masters that this was kind of this categorical algebraic topology working a lot with these infinity categories and higher algebra?

  Maxime: Yes, that in a way developed because at the beginning, one other field that I really liked when I was hesitant between many things was category theory, but I was told by various people that you shouldn't really just do research in category theory because it's a bit sort of too abstract and not a lot of like, real things. But I knew about it. And so when I did my masters in algebraic topology, my advisor really taught me infinity categories along the way. So it was at masters where he told me we're going to try and prove a result. It's not a very difficult result, but along the way, if we get there, you'll have learned this stuff. And this is exactly what happened. So along the way, I really learned a bunch of stuff. It was during Covid, so I had literally nothing else to do than read a bunch of stuff about this and email him a million questions. And he was very present to help to answer these questions.

  Carlo: Very nice. And your PhD thesis topic, did you decide that for yourself or how did it go on then with your interests?

  Maxime: Yeah, I think Jesper sort of has a bunch of papers that he recommends to students, to all his incoming students. And this is a really good thing to do because you kind of feel free to do something. To do what you like. But you still have a really strong goal to understand these papers in the end. Along the way, I forged my own sort of tastes and questions that I cared about and everything. Under his guidance and my other advisor that I met a bit later, Markus Land, we discussed a lot and somehow, more than being given problems, I was really sort of advised about the problems I cared about. And so I didn't really have one specific thesis project, ultimately. I had a bunch of different projects that I thought about a lot. And at the end towards the last year, I had to decide what would actually be the thesis. But that came towards the end really, that it didn't have sort of a project that worked for three years.

  Carlo: So can you give us the title of your PhD thesis?

  Maxime:  It's called “Separability in homotopy theory and topological Hochschild homology”. I think because it's basically two independent parts that have basically nothing to do with one another. I really decided at the end I need a thesis. How should I do this?

  Carlo: And whoever's in Münster knows this topological Hochschild homology maybe as a kind of keyword for the research of Thomas Nikolaus probably.

  Maxime: Exactly, he's one of the big players in this kind of field.

  Carlo: So how did you come up with going to Münster? What was that like then?

  Maxime: So my master's advisor actually suggested two places for me as opposed to France. He said, you should go to Copenhagen or to Münster if you can. I ended up going to Copenhagen mostly because of the time I had already been there, so I was already familiar with the place and so time wise it matched better. But then during my thesis, I interacted a lot with Thomas and with Achim, who was here at the time. And somehow we get along really well. And Thomas is one of the leaders in our field. So I think there's not a lot of people who would get the opportunity to work here and wouldn't take it. And so when I came to visit him, mentioned that I was looking for jobs. And then we started talking about this, and I applied and ultimately, came here. But I, of course, knew about this way beforehand. I knew about Thomas and Münster way before, because he and Achim are very influential in our field.

  Carlo: Okay, now you’re in Münster. And besides maybe being a very good place for research, at least in my opinion, it's also a great city. What was your first place outside the campus where you really enjoyed going, or do you now have maybe a favourite place inside the city?

  Maxime: So I was a bit stupid in my first year coming here. I travelled a really huge amount and so I was not very present here and I didn't get in the first year to enjoy the city so much. I'm only now really getting to grips with the city. I think one of the places I go to often is the climbing gym. It’s not very original and it's not really a place to go to, to hang out, but I really like the Aasee. I take regular walks around the Aasee. That's really nice. I also like the botanical garden. And there's like a playground next to my house where I go when there is no one around…

  Carlo: And maybe now after two years as a postdoc. So you have a lot of research experience already, also doing your PhD, and it's maybe for someone who didn't yet do any research in mathematics or also for people who did, it's always hard to imagine how other people research. So what does research really look like for you on a daily basis. How do you do your research?

  Maxime: It's a funny question. It's the trick I use when people ask me what I do for work. Because then they ask me like, 'oh, what does it mean to do math research?' They often ask what I research and I often like bend the question to what do you do during your daily life. Which I think is a better question for people.

  Carlo: I totally agree.

  Maxime: So, I think it varies a lot, but most of the days, when I don't have to teach or prepare a teaching or grade or whatever, most of the days look like I have 1 or 2 meetings where I'll talk to someone about either a project that's ongoing or a project that we're starting or finishing. And that's a lot of discussion and collaboration. That part I really enjoy. I enjoy it much better when it's in person. But of course, given like how international research is, you kind of have to also often sort of be on zoom. That's a bit less fun, but it's still very fun. And the other parts of the day are spent either writing stuff, which is kind of boring and annoying, and it's the part that I like the least. Or thinking about stuff, and that can look weird because you're on your desk just staring out into the air, and after two hours you feel like you've done something productive.

  Carlo: So you're really someone who just thinks and doesn't write anything down for, like, a whole hour and just think ...?

  Maxime: No, it depends. I start writing when I think I have a place to go. But when I don't have a place to, I try to do stuff in my, I mean I don't try to, but I end up doing stuff in my head and I only write when I think I have a clue of what to do. But I do use a lot of paper. You've been in my office a couple of minutes ago. You've seen the mass of papers that was there, so I do write also. I also have a black…, a whiteboard, but I use this mostly when I talk to people. Alone I don't use that so much.

  Carlo: No, but that really speaks to the conceptuality of your research, if you can do things in your head, then it's really most about concept and less about calculation.

  Maxime: Yes. I'm very afraid of calculations. I have respect for people who are able to do calculations, because I'm always super scared of them. I'm both super scared of them, because they're difficult, but also when you do one yourself, I'm always very afraid of checking that I didn't make a mistake. It's much easier for me to be convinced that I didn't make a mistake in conceptual arguments, the ones that you can actually hold in your head.

  Carlo: Now that we've gotten to know Maxime a bit better, we will come to the second part of this podcast. The so-called “A or B”-game. For this game, which will consist of 30 A or B questions, I will ask Maxime to answer as quickly as possible in order to get an idea about his intuitive thoughts on topics like music, Münster or mathematics. 
  Okay, so let's start. My first question is 
  0 or 1
  Analysis or algebra
  Coffee or tea
  Beach or mountains
  Bouldering or hiking
  Snooze or getting up
  The morning or the night
  Summer or winter
  Order or chaos
  cooking or ordering food
  Mensa or no mensa
  By bike or by foot
  Slides or blackboard
  Technical proofs or conceptual proofs
  PDFs or physical copies.
  Condensed Anima or Infinity Topoi
  Quasi categories or complete Segal spaces, Maxime: Neither.
  Higher Topos Theory or Higher Algebra
  Model theory or categorical logic
  Invented or discovered
  Conjecture or counterexample
  Copenhagen or Paris
  Scandinavia or France. 
  Sorbonne or ENS
  Math Stack Exchange or Math Overflow.
  Clapping or knocking.
  Ratatat or Supertramp. (That's a difficult one.)
  Etymology or semantics
  Ballroom or hip hop.
  To be or not to be.

  Carlo: You've already done it. Great. Well, that was fast.

  Maxime: It was.

  Carlo: That was already 30. Maybe the next time I need to do more question. But not everyone is as decisive as you.

  Maxime: The thing is, I'm usually very indecisive, so I tried my best. Sometimes I answered, even though that was not what I meant.

  Carlo: So with complete Segal spaces or quasi categories, you say 'neither' because?

  Maxime: Because there's this idea of model independence. And it's not a very well defined thing, but it's the idea that infinity categories are a concept that's like implemented by either quasi categories or complete Segal space, but is neither. And I like arguments that don't really specify which one you're working in, that are still precise, but that use abstract features, that aren't specific to either model. And I think specifying the model usually makes the arguments less conceptual and clear.

  Carlo: Very good. And another thing, I was wondering about. So when deciding between bouldering and hiking. You said hiking, right?

  Maxime: I think I said bouldering.

  Carlo: Ah, you said bouldering.

  Maxime: But that was one of the ones where I was hesitant because I obviously boulder a lot more like on the day to day, because I do it like once or twice a week. But I actually really like hiking and I think if you want to hike here, you have to go out of the city and so on. So it takes more work. But maybe if it was more accessible, I would maybe answer something different, I’m not sure.

  Carlo: Okay. Because there's also maybe a middle ground which is climbing like these climbing gyms, they also exist. But you're like on a daily basis you go bouldering.

  Maxime: I actually haven't learned to climb yet. I mean, mostly you need to learn how to belay and everything. And I think you need a class for this, and it's cost some money. And you’ll need to be at least two people and so on. So I decided… I didn't do it yet.

  Carlo: And what was also really interesting to me was, how decisive you were when deciding between Copenhagen and Paris and also between Scandinavia and France because you grew up in Paris, right?

  Maxime: I did, yes. It's been a long process, to actually decide because it's going to matter at some point and I think ultimately there's a couple of different things. One is from a purely professional perspective, research wise, Scandinavia and Copenhagen are just much better places than Paris and France, because as a researcher, you're not trying to be rich, but there's this comfort aspect to your life that is much more present if you're in Scandinavia. Personal wise, the most difficult thing was to decide like is this professional advantage sufficient to make me want to move to Scandinavia or something. But personal wise, my best friend / partner is from Denmark. And so that already hinges you towards a place. But also, I really got to enjoy the Danish way of life. And so ultimately, I aim towards getting back to Denmark.

  Carlo: Right. So for the final part of the podcast, I want to ask you five questions, which are just things I'm curious about. So let's start with the first question: What is your favourite theorem?

  Maxime: Wow, what is my favourite theorem? I think, it's a very basic one, but it comes back to one of your earlier questions about how did you get into math? I think I really loved the proof of Bezout's theorem the first time I saw it, because it was such a clean argument. When you were in high school, in math, you don't do a lot of actual proofs. And this was one of the first, real proofs that I saw. And so it kind of stuck to me. And I think this might be for nostalgic reasons, basically my favourite theorem.

  Carlo: I can totally relate to that. I also liked my first algebra course a lot. Which of your results are you most proud of?

  Maxime: This is difficult question as well. I think it's in between something I've already done and something I'm currently writing and soon done with. But I think currently it's probably a result with my colleagues and friends Vladimir Sosnilo and Cristoph Winges, which is about this category of localising motives, which is, something very abstract and very difficult to explain in the podcast. But basically there was this construction that was very nice and got a lot of interest. And we showed that there's a different perspective one can have on this construction, that sort of makes an analogy with different field of math, algebraic k-theory. And it makes it much closer to that other field of math. And it sort of clarifies a bunch of things about, in my opinion, about this category.

  Carlo: If you if you can relate to fields of math, this is always very powerful. You see this all the time now. So which mathematical field would you have gone into if it wasn't higher algebra and what are you doing now? Maybe this is an easier one for you to get one.

  Maxime: It probably would have been set theory or logic. I think I still really enjoy thinking about and reading a little bit about cardinal arithmetic and stuff like this, whenever I get the chance and it does come up a little bit in algebraic topology. So I'm happy I have this background also.

  Carlo: With condensed anima, for example.

  Maxime: For example.

  Carlo: So you like to choose a κ.

  Maxime: Exactly. I like to choose a κ.

  Carlo: Okay. So maybe a harder question again. What would be your biggest professional regret? So what would you have maybe done differently during this whole time? It sounds like you had really the perfect... you pass through everything. But do you have anything which you would have changed in retrospect? The answer can also be no.

  Maxime: No, I think there are two and I think they are of the same nature but applied to slightly different thing. I think maybe I would have thought a bit harder about physics, because now I feel that I really don't understand a lot about. Of course I know all the popsciency things about physics that one can know when one is interested in things, but I don't know anything about serious physics, quote unquote. And I think that would have been nice to know a bit more about. And in a similar note, there's other fields of math that I didn't focus on as much as I would have wanted to. Now for example, I think I took a few classes in algebraic geometry, but I think I could have easily done more in that direction. And I think, at the expense of some other things, probably I would have broadened my scope a little bit more.

  Carlo: And the final question which is maybe even the hardest. So what advice would you give master students, someone who's a master student right now and who's maybe looking to go into research in your field or maybe a related field.

  Maxime: I think it's related to what I just said. I think just learn a bunch of different things, even if you have an area of predilection that you think you definitely going to go into, and that may well be true. Do try to learn other areas that will necessarily help no matter what. I think that would be the most important advice.

  Carlo: Very nice. So now we've gotten to know you a bit. So it was really nice talking to you about your past and your present and your thoughts on mathematics. So now, of course, we would like to get to know your research a bit more. So we will now switch to a video format for the second part of the podcast. And we will talk with each other in a minute again. 

  Thank you for listening to the first episode of this new season of On a Tangent. Since we're launching a new format, I would love to hear your thoughts. You can send feedback directly to mail@carlo.info. If you enjoyed the conversation with Maxime as much as I did, please consider sharing this episode with a friend or colleague. In the show notes, you can also find a link to the video part of my conversation with Maxime, where he explains how to do algebra on a doughnut. I'm Carlo and I will catch you on the next tangent. 

Want to learn more? In this video, Maxime explains how to do algebra on a doughnut.

Link to Maxime's website

In this episode of On A Tangent, Simone is joined by Edith Hübner, a doctoral research in topology. We learn about her journey from physics to mathematics, what headbutting gnomes can teach young children, and the mysteries of infinity categories.

Link to Edith's website

• Transcript

  Simone: Hello and welcome to On a Tangent, the podcast where the main characters are the stories behind the mathematics. My name is Simone. In each episode, I meet a different early career researcher from Münster to learn about their paths towards mathematics and their hopes for the future. In this episode, I interview Edith, a PhD student in topology. Learn about her journey from physics to mathematics and what the future holds for infinity categories. I hope you enjoy the episode.

  Alright. Hi, welcome to the podcast. It’s a pleasure to have you. So, as you know, in each episode, we sort of explore the stories that mathematicians have gone through here in Münster. So, we always start in the past. What is your earliest memory of mathematics? What’s the first time you think maths entered your life?

  Edith: Yes, first of all, thanks for inviting me. That’s actually a funny question because, in essence, I don’t really remember a particular event that marked my interest in mathematics. It’s more like I was exposed to it constantly, roughly speaking. What I do remember is that, as a child—and I guess this must have been during primary school—I entertained myself with mathematical puzzle books. They were definitely about calculating things. There were also some computer games, like one based on a Swedish story called Petterson and Findus. Maybe you’ve heard of it? It’s about an old man with a cat, and they live on a farmhouse in Sweden, having various adventures. There are computer games based on this, and in one of the games, there were little gnomes also living in the farmhouse. They wore bike helmets.

  Simone: … because they go biking?

  Edith: No, they not go biking. They were running and …

  Simone: This is safety first.

  Edith: From safety first actually. The idea in the game was that these gnomes would run different distances at different speeds, and their biggest wish was to collide in the middle.

  Simone: … Ah, so that’s why they were wearing helmets.

  Edith: Helmets. That was why they wear helmets. Yes, of course. Your task as the player was to calculate the appropriate starting times for each of the gnomes so they would actually collide. I definitely had lots of fun with it.

  Simone: I feel like that’s a dangerous game to give to a child. Like, you’re teaching the child, “If you want to bang heads with your friends, at least compute the trajectories correctly.”

  Edith: Yes, definitely. Well, I mean, I don’t think anyone really got hurt.

  Simone: That's good.

  Edith: And you should still remember it was a children’s game.

  Simone: Right. I don’t know if there was a spike in incidents of children banging heads after the game came out. They all learned maths very well but also how to hit each other!

  Edith: Yes, probably. I don’t know. I never tried that in real life, I have to admit. So, yes, this was one instance. Another one—maybe the most abstract math instance —was when my mother showed me some identities for binomial coefficients. I must have been around 15. I appreciated that she took the time to show me this, but I wasn’t too enthusiastic because, during my time at school, I never really encountered mathematics as it is taught at university. My main interest was actually in physics.

  Simone: Yeah, somehow I feel like that’s a common phenomenon. It’s hard to get enthusiastic about maths in school because the maths you’re taught is just like, “Here are the top ten rules to compute this integral.” But surely there must be a more fun way to do maths, right? Nothing against integrals, of course.

  Edith: No, nothing against integrals. I mean, I kind of appreciated maths as a tool to compute things in other natural sciences, like physics. But I didn’t really consider it as a subject in its own right because I was never really taught what the essence of mathematics is.

  Simone: You’re taught maths is a tool, exactly as you say.

  Edith: Yes, and I think physics is much more accessible to the general public, partly due to things like astronomy and the nice pictures.

  Simone: And physics and all. Of course.

  Edith: Yes, exactly. There’s something about spotting constellations in the night sky that you can directly experience. I think it has a much bigger fascination.

  Simone: And when you went into physics, were you sure which kind of physics you wanted to do? Like more experimental, more theoretical?

  Edith: Well, I guess more on the theoretical side. But I belonged to the group of people who were very enthusiastic about astrophysics. I wanted to learn about the universe: how it came into existence, where everything is going—these big, almost philosophical questions. In the end, though, I have to admit, I never actually took an astrophysics lecture.

  Simone: But maybe that would ruin the magic a bit, no?

  Edith: I mean, maybe, I don't.

  Simone: I remember pondering whether to study astronomy. In Italy, there’s a bachelor’s degree in astronomy somewhere—I’ve forgotten where—and I wanted to apply. But when I looked at the courses, there was a lot of statistics and data analysis, and I thought, “Hmm, maybe that’s not what I want to do.” I felt it might ruin the magic because, in the end, you’re looking at data from a satellite. Of course, it’s incredible that you can get this data, but it’s very different from the romanticised view of astrophysics.

  Edith: Exactly. After finishing school, I did an internship at the Max Planck Institute for Astrophysics in Munich. The task I was given wasn’t very advanced, of course—I had just graduated from school, so I wasn’t particularly helpful. They asked me to transcribe a programme written in an older programming language into a newer one called ROOT, which is based on C++. The programme was intended to analyse data from a satellite they had launched ten years earlier. The data was still there, waiting to be processed. So, yes, it was exactly what you mentioned—programming in order to analyse collected data. I also learned that most of these projects are designed for the scientific lifetime of one person. You have to build the equipment, have engineers construct it, launch it into orbit, and then collect and process the data. That takes an incredibly long time. If the project fails, it’s a huge setback.

  Simone: Yes, of course. I remember doing something similar in high school. They gave us data. I don't remember what it was probably like the light of a certain star fluctuating over time. It was fantastic and brilliant that we had this data, but it took away some of the magic of astrophysics. It felt like a lot of number-crunching, which is great, but somehow it depends on what the taste that brought you into that direction is. Right?

  Edith: Yes.

  Simone: So, you said you always had a taste for the theoretical. Is that what ultimately brought you towards mathematics?

  Edith: Not directly. When I started studying physics, I was pretty sure that’s what I wanted to do I never really gave mathematics a serious thought as a subject of study. Then, I was confronted with mathematics during a linear algebra lecture that we all had to take. It was quite horrible for me. We dealt with a lot of abstract stuff, like the axiom of choice to prove that any infinite-dimensional vector space has a basis, or showing that every surjective map has a right adjoint. That was the first exercise on the very first exercise sheet. It was brutal. It was definitely brutal. And I definitely did not enjoy it because it was like it left me in some kind of state of shock. I did not enjoy this, but I kind of had enough perseverance in order to say, “well, I don't want to let this stop me.” So I kind of finish that lecture and pass the exam and.

  Simone: And then.

  Edith: Well, the lecture itself definitely did not spark my interest in mathematics, but I kind of was eager enough to sit down with a book, I think a general calculus book and read through it myself, because I had the feeling that during that lecture there was like too much input and too short of time. And I said, “I do want to go over all of this again in my own speed and have a look at this and see whether I like it or not.” And when I was doing it by myself, that was much more fun, of course. I started understanding things, which was also great. And then I had a calculus course, like calculus for physicists, that was taught by a mathematics professor. He provided lecture notes beforehand. The lecture itself was not great because, in general, mathematicians don't like teaching courses for non-mathematicians.

  Simone: In my former university, we used to call them “service courses”. Everybody had to pay this price of getting at least one service course per year. I think the physicist ones were actually the ones that people were okay with. But it could be worse. You could have teach calculus for biotechnologies.

  Edith: Yes, I know that's what I mean.

  Simone: Not enough to say, I feel like in the hierarchy of which courses people do not want to teach, statistics for biology is like top ten things people don’t like.

  Edith: I am not totally sure.

  Simone: I want to.

  Edith: Teach this, I totally agree. But this professor did not want to teach the physicists and the physicists also did not want to be taught this kind of. It was a mutual relationship. He would project his lecture notes on the wall and write on them with a marker. That was the entire lecture. People not really hated it, but because I was reading the lecture notes beforehand and I was just trying to really understand each detail in the proof. And at some point I was convinced that a proof was like already finished and that the other things that were written later in the script were not needed. And I kind of mustered the courage to tell the professor during the lecture. And it was a kind of a shocking experience because nobody normally talk during the lecture.

  Simone: Right? Everybody just sort of sat through the pain and were like, okay, it's over, I think.

  Edith: Yes. And then somebody's telling you, “I think your proof is already done; you don’t need to write more.” He came and looked at me and said ”: are you very sure about this?” And I was very much intimidated. It kind of sparked some enthusiasm in myself and also the professor. He became more enthusiastic himself noticing that there's a student who's interested in what he's got to say.

  Simone: At least who's paying attention.

  Edith: … And is paying attention as well. I think it was kind of this professor, who got me into mathematics, because I also felt I wanted to show him that physicists can understand pure math.

  Simone: And a few years later. Yes I'm, I crossed the bridge.

  Edith: Yeah. I think, that was really kind of the event that let me consider mathematics as the thing I wanted to do. I think the more I learned about Mathematics, the more difficult it became for me to understand physics lectures, because I really wanted to know about the precise mathematical requirements that they were like asking for in their theories.

  Simone: And then, which famously bit sloppy somehow.

  Edith: Yes. And then it became very frustrating and led me to the decision to do math rigorously because I had that feeling that only a very few people manage to be both a physicist and a mathematician. It seemed to me like the way of thinking about mathematics was kind of fundamentally different between these two groups of people.

  Simone: And would you say, even for very theoretical physicists, because somehow sometimes you read these sort of inflammatory claims, which are like a string theorist fundamentally is just a mathematician.

  Edith: Yes, I mean, I have to say, I don't know any string theorist personally, so I cannot answer on that. But still, I think that physicists have a different approach to mathematics. Like more regarding it as a tool and being more willing to do things that are obviously forbidden by mathematical, I mean things that are like forbidden operations in a mathematical sense, just to try out and see what, what kind of results they would get in a calculation. Yes. And maybe that would actually help mathematicians to be a bit faster doing stuff, but, at least I kind of like, I had this. And also like, I don't know whether you would with a theoretical physicist say that he or she appreciates mathematics for the sake of itself. Like without any.

  Simone: Yeah. It's not art for the sake of, I mean, it's not for the sake of mathematics, right. It's a means to an end, of course, for the physicist. Whereas for the mathematicians it's somehow different.

  Edith: Yes.

  Simone: And so this is your story to maths. And so, what brought you to what you do nowadays and what is that you do nowadays?

  Edith: During the course of my bachelor's studies and especially at my bachelor thesis, I started learning about category theory. I became very interested in that. And then, I learned that there was something called infinity category theory.

  Simone: And this was in your bachelor's?

  Edith: Yeah. That was in my bachelor's. Yes.

  Simone: Was the bachelor's supervisor a physicist?

  Edith: It was a joint supervisor supervision. And one of them was a theoretical physicist, and the other one was a mathematician. But in the end, like the topic was provided by the mathematician and the physicist agreed for me to work on it and said ”well, do as you please.”

  Simone: You can do infinity category.

  Edith: I am very, very grateful and indebted to him that he allowed me to do this. I have to say, because there is no immediate connection to physics.

  Simone: I see.

  Edith: So I started becoming interested in this. I mean, I did not learn about infinity categories during my bachelor's, but I decided to go to university for my masters, where there is a professor working, in infinity category theory. And yes, I first went to Regensburg and then learned about Thomas Nikolaus being in Münster and decided, okay, I'll try this. And that's kind of how I ended up working in homotopy theory and infinity categories.

  Simone: Right. So that's what you do nowadays.

  Edith: Yes. I mean, I am well currently working on a joint project with Thomas on a question that is. Well, in essence, we are trying to assess whether there exists a theory of something that you might call non additive stabilisation for infinity categories. This a very general question intrinsic to infinity. One category theory which is motivated by certain examples. And I might tell you about some examples perhaps?

  Simone: Yes. I feel like occasionally, you know, I don't want to say they're a bit taboo, but I feel like occasionally when you feel the categories are mentioned, people have a moment of like, “okay, yes, yes, I'm ready. But like, give me a moment.”

  Edith: I agree that infinity categories are scary at first sight, and maybe they remain scary at a second glance. I think this is mainly due to unfamiliarity with their inherent concept, namely specifying relations between mathematical objects or operations only up to coherent homotopy, as opposed to, for example, isomorphism. Doing this involves remembering a lot of data, and, of course, this data has to be made precise. Thus, there is rather heavy machinery underlying the theory of infinity categories, which you have to become familiar with in order to work with them.
  But I should also say that the theory itself—at least the model that is most commonly used, developed by Joel and especially Lurie—is a very elegant one. Let me give you an example of an unfamiliar feature of infinity categories: If you think of an ordinary category, then whenever you're given a pair of morphisms with matching source and target, there is a unique composite map. This is essentially by definition. In an infinity category, however, a composite still exists, but it is no longer unique. In general, there will be several different composites, and the uniqueness statement translates into the statement that the space of all composites is contractible.

  Simone: You've talked about this project. What is one question or problem you would like to see solved in the next ten years? Of course, it could be the same project you just mentioned. But you could also allow yourself to be a bit more ambitious, if you wish. As I said before, it could be a problem you have an emotional connection with—maybe a question that frustrates you because there’s no answer yet.

  Edith: My wish for the future—that’s a very difficult question. Homotopy theory is a relatively young and rapidly evolving field of research, so there are lots of things yet to be uncovered and plenty already being uncovered. There’s a lot of progress being made. I think one of the big goals is to achieve a better understanding of the basic objects in this field, particularly the infinity category of spectra. That’s certainly very important. Another challenge is that a lot of knowledge already exists but hasn’t been written up. It would be great to see a collective effort from researchers in the field to document and share their insights to make this knowledge more accessible. But, of course, writing things up always takes much longer than expected—this isn’t unique to homotopy theory. As for my personal goal, I’d say it’s to figure out what interests me the most. Homotopy theory has many different subfields—for example, equivariant homotopy theory, chromatic homotopy theory, infinity categories, algebraic theories, and generalizations of categories. I haven’t yet decided where I’d like to focus. That’s probably my biggest personal goal right now. The most selfish wish I can share is that I’d like to see lambda rings in action someday. Lambda rings are algebraic objects that appear, for instance, in K-theory. One specific question is whether they can be used in a way similar to delta rings in prismatic homology to define integral cohomology theories. I have no idea whether this question will be answered in the near future, but honestly, if it isn’t, that’s fine. There are so many interesting things to explore that I doubt I’ll ever get bored.

  Simone: Well, thank you very much for the for the very nice insights.

  Edith: Thank you very much.

In this episode of On A Tangent, Simone is joined by Marjory Mwanza, a Young African Mathematicians Fellow working in group theory. We learn about how one goes from statistics to group theory, and where to find deep structure in seemingly simple objects.

Link to Young African Mathematicians (YAM) Fellowship Programme

• Transcript

  Simone: Hello, and welcome to "On a Tangent" the podcast where the main characters are the stories behind the mathematics. My name is Simone, and in each episode, I meet a different early career researcher from Münster to learn about their paths towards mathematics and their hopes for the future. In this episode, I interview Marjory, a young African fellow at the Centre of Excellence, to learn about her journey from statistics to group theory and how to find deep structure in groups. I hope you enjoy this episode.

  Okay, so. Hi. Welcome. Thank you for joining us.

  Marjory: I'm happy to be here. Thank you.

  Simone: On this podcast, we try to understand the stories of the people in the cluster, the mathematicians in the cluster. So I’d like to ask you, what is your earliest memory of mathematics? When did you first encounter mathematics in your life?

  Marjory: Well, I would say my memory of mathematics goes back to primary school, actually. I’ve always been interested in mathematics as a subject since then, and growing up, I had very good mathematics teachers who motivated my interest in the subject. In high school, we had career days where they’d ask what you want to do in the future. I had no idea that someone could pursue mathematics as a career.

  Simone: Yes, you don't really see many mathematicians in everyday life, right?

  Marjory: Yes, exactly. So I wanted to find out which careers involved mathematics. I was told engineering involved maths and sciences, so I used to say I wanted to be an engineer — not necessarily because I wanted to be one, but because I wanted to study maths.

  Simone: I understand that completely.

  Marjory: When I finished high school, I applied for a bachelor’s programme at a university in Zambia. The universities there have a general first year where you study natural sciences if you’re aiming for sciences or engineering. I applied for natural sciences, intending to go into engineering. However, while I was there, I researched the programmes offered and discovered that mathematics was one of the options. That was intriguing to me.

  Simone: So you realised you could actually pursue what you wanted to do.

  Marjory: Yes, I found out that mathematics could be a career. So I started looking into it — what it would involve, what options there would be after completing a mathematics programme. I wanted to study maths passionately, but I also wanted it to lead somewhere practical.

  Simone: Yes, that’s what my parents used to say. They’d ask, "What are the jobs afterwards?" because, while passion is good, you still need to make a living.

  Marjory: Precisely. And as the firstborn in my family, I felt I needed to set an example for my siblings by doing something worthwhile. I did some background research on the mathematics programme, asking about people who had graduated from it. I spoke to someone who told me about a fantastic opportunity for top students — a sponsored master’s programme in various centres across Africa, called the African Institute for Mathematical Sciences. There are centres in South Africa, Cameroon, Rwanda, and Senegal. The person I spoke with had studied there and was doing well, which motivated me to pursue this path.

  So, I entered my second year, specialising in mathematics, with the goal of performing well to qualify for this programme. It was a full scholarship covering everything, including transport. As I was progressing through my bachelor’s degree, my understanding of mathematics deepened. In high school, it was just maths, but during my bachelor’s, I started seeing it as something real. We had both applied and pure courses.

  Simone: So a comprehensive overview.

  Marjory: Yes. In research, you can then specialise. I was particularly drawn to statistics as an applied course and algebra. I liked the applied aspect of statistics, but choosing between the two was tough.

  Simone: That’s challenging. In some programmes, you have to choose a theoretical or applied curriculum from the start, which is tricky since you barely know what mathematics is about in the beginning. I initially wanted to do physics, but my physics professor suggested I was more suited to mathematics.

  Marjory: Yes, I understand. My journey wasn’t straightforward either. I liked group theory. We studied linear algebra, then group theory and other topics, and I was drawn to the challenging nature of group theory. However, I didn’t get to choose my final-year project; they assigned us topics for ease since our class was large. I was given a project in statistics.

  Simone: So someone else made the decision for you?

  Marjory: Yes, unfortunately. I liked statistics and numerical analysis, but deep down, I knew I wanted to delve into group theory. I completed my bachelor’s degree and applied for the master’s scholarship in South Africa. While waiting for acceptance, I worked as a tutor because I love teaching. Mathematics is something I enjoy sharing with others.

  Simone: So, did you get accepted?

  Marjory: Yes, I was thrilled when I received the acceptance email. I was accepted to pursue my master’s in Cape Town, South Africa.

  Simone: And what was it like when you arrived there?

  Marjory: It was fantastic. At the African Institute for Mathematical Sciences, you’re exposed to various topics in mathematics before choosing your research focus. I even took a course in group theory.

  Simone: So you could explore different areas before specialising.

  Marjory: Yes. We were introduced to applied courses, machine learning, coding, and pure courses. I mostly chose pure courses because I knew I wanted to do a pure mathematics project. I took modules in quantum theory, logic, and group theory. The best part was the diversity—students came from across Africa, and we had lecturers from around the world, not just South Africa. We had lecturers from Germany, the UK, and other places.

  Simone: You were exposed to different approaches to mathematics from various countries.

  Marjory: Exactly. At the end, we got to choose our own research topic and supervisor. I had a very enthusiastic group theory lecturer, which helped solidify my decision to pursue group theory.

  Simone: It's fascinating how having a motivating teacher can really make a difference. It can turn the subject into something exciting and alive.

  Marjory: Yes, it really gave me a boost. My lecturer during my bachelor’s wasn’t so enthusiastic about group theory; I developed the interest largely on my own. But what truly made me certain this was what I wanted to do was my lecturer in my master’s. He was so passionate—one of those people who talk with such love for what they’re doing.

  Simone: Yes, the passion really comes through, doesn’t it?

  Marjory: Absolutely. I was motivated, and I spoke to him after class. At first, he had his doubts because my bachelor’s project wasn’t in a pure field; I’d done statistics. But I convinced him, and he agreed. So I began my research with him, and it was incredible. I learnt so much during my first pure project, although it wasn’t easy.

  Simone: Right, especially since your background was more, well, applied. I mean, your previous project was in statistics, wasn’t it? So, what was the project about?

  Marjory: I studied the structure of groups that satisfy the Glauberman-Thompson theorem. This theorem is one of the most important in finite group theory. However, not all groups satisfy it, but some do, and my focus was on the structure of those groups. That was the main subject of my master’s project.

  Simone: And now that you’re here as a fellow, what has your project been about?

  Marjory: Here in Münster, I actually shifted fields slightly. During my master’s, I developed a love for graph theory through a course I took on it. It’s fascinating because it allows you to represent mathematics visually, which I really enjoy. When I came here, my supervisor specialised in geometry, topology, and group theory, specifically in geometric group theory. So I thought, why not switch to geometric group theory?

  Simone: That seems like a perfect blend — you’ve got the groups, but also the visual element through graphs.

  Marjory: Exactly. You get the groups, and you can also have all the visual aspects you enjoy in one place. It’s great. At first, I was a bit anxious because I hadn’t done much geometry before, but I thought, there’s always room to learn.

  Simone: I think that’s something people fear quite a lot. I remember when I started my PhD, there was this constant feeling of not knowing enough, but eventually, you realise that there’s always room to learn.

  Marjory: Yes, there’s always room to learn. So, I took a course in geometric group theory to understand how it connects with the group theory I already knew. It was fascinating—like finding a missing piece. I really appreciated diving into geometric group theory. When it came to choosing a project for my Young African Mathematician programme, I wanted something that combined groups and graphs, and my supervisor and I found a perfect topic. It’s on constructing graph products of groups from finite groups. My master’s focused on finite groups, and now I’m working on constructing graph products of groups from these finite groups.

  Simone: The puzzle is really coming together, isn’t it?

  Marjory: Yes, it feels like the puzzle is complete.

  Simone: It’s wonderful. How would you explain the topic of your project to someone without a mathematics background, perhaps at a party? Although people don’t often ask about our work at parties!

  Marjory: True! Well, if I were to explain it to someone unfamiliar with mathematics, I’d say, since I’m working on graph products of groups, I’d start with the concept of a graph. Imagine connections between people — say, your family and friends. You’re connected to certain people, and those people are connected to others. In a graph, people are represented as vertices, and their relationships as edges. So, when I work on graph products of groups, I’m talking about how these connections can be represented in a mathematical structure. Cayley graphs, for example, are another interesting area — they allow us to represent groups in terms of graphs. If I were explaining it at a party, I’d talk about relationships and connections, which is how I link the concept of graph products of groups.

  Simone: Graphs are everywhere! They’re so versatile, and people can easily picture them. It’s a great subject because it’s so visual.

  Marjory: Yes, exactly. And with Cayley graphs, you don’t have to represent every element of the group — just a generating set. A group can be generated by different sets, so you can have various graphs representing the same group depending on the generating set you choose. It’s really quite interesting.

  Simone: Yes, I think it’s fascinating. Personally, groups are a bit daunting for me because they have less structure compared to fields, which have more operations. I find groups intimidating because they only have one operation; it feels limiting.

  Marjory: That’s true! Groups on their own don’t have much structure. My master’s project didn’t have a lot of structure, and I almost forced one into it! Groups themselves might not seem structured, but when you start relating them to geometric objects, that’s when you start to see structure.

  Simone: Yes, that’s reassuring! So, we’ve talked a bit about your past and present. Now, let’s look to the future. I always ask our guests: what’s one big question or problem you’d like to see answered, maybe by yourself or someone else?

  Marjory: That’s a great question. I’ve recently come across something that has been unsolved for a long time, and I’d love to see it solved. It’s connected to my current project on graph products of groups. There’s a specific type of group called the right-angled Artin group, which is a type of graph product of groups. Right-angled Artin groups are simpler in structure due to commutativity, but in general, Artin groups are quite diverse and complex. One major open question is determining the quasi-isometry of these groups. Yeah. So, this is one issue I would like to see resolved because it's challenging. When you want to determine the quasi-isometry, which is like a map between certain groups, you need to understand the large-scale geometric structure. This understanding enables you to compare spaces effectively. These groups act on spaces and complexes, making it difficult for people to determine the quasi-isometry without a deep understanding of the large-scale geometric structure of these groups. I would really like to see this issue resolved in the coming years.

  Simone: At some point, yes.

  Marjory: Yes, at some point. I think solving this could actually have a significant impact in the field of geometric group theory. Since groups combine both algebraic and geometric properties, it seems that one way to approach a solution might involve collaboration between mathematicians from different fields. Bringing together combinatorial group theory, geometric group theory, and topology, for example, could lead to a breakthrough. I look forward to seeing this issue resolved.

  Simone: Well, thank you very much for the insightful thoughts. See you around in the corridors.

  Marjory: All right. Thank you.

In this episode of On A Tangent, Simone is joined by Rin Ray, a postdoc in Topology. We learn about why an inventor becomes a mathematician, and how to build bridges between deep questions in different areas.

Link to Rin's website

• Transcript

  Simone: Moin! Welcome to On a Tangent, the podcast where the main characters are the voices behind the mathematics. My name is Simone and in each episode I will interview a different early career mathematician from Münster to discover their paths towards mathematics and their hopes for the future. In this episode, I meet Rin, a postdoc from topology, and learn about how an inventor turns into a mathematician. I hope you enjoy the episode. (...) Thank you for joining us. Welcome.

  Rin: Hello.

  Simone: So, let's start from the past. Let's start from your earliest memory of maths. What is the earliest time maths entered your life?

  Rin: I think the first time… I have kind of three candidates for this. The first one was that my mom had a lot of jewellery, and one of the things she found that kept me occupied was undoing the knots in the jewellery. So probably that's my earliest. Just like I'm doing knots in, like, shoestrings and in jewellery and such. Also, my dad is a professional antique repairman and has a full woodworking studio, and often people would come with just totally broken and splintered pieces of furniture that we would carefully glue back together. And I also remember that. But probably my first like maths memory, where I was like, ooh, there's something here, and I knew that that was called maths was when I took calculus. I remember like learning about Riemann sums. This sounds really silly, but I remember when I was learning about Riemann sums, I felt like this thing that I had done when I was younger and bored in class is that I would sometimes kind of draw lines in the air and then mentally, like stack things from around the room under the lines. And I was like, oh my God. They packaged this thing that I do in my head in like one tiny package. And however else anyone else thinks about it, they can unpack it in that way. And I was like, that's really cool that you can really like package it because it's a huge amount of like tools and information in this, like really small symbolic framework. And that's like the earliest time I was like, ooh.

  Simone: Is this something I want to see more of in my life?

  Rin: Yeah, exactly. I enjoyed this ability of talking about one thing many different ways in like a unifying language, I guess.

  Simone: And do you feel like these three experiences you talked about somehow were already precognition of what later you would do in maths or not really?

  Rin: No. I mean, I do think that I'm a pretty visually thinking person, but I don't think they necessarily pointed to the area of maths that I ended up working in.

  Simone: Which, now that this comes up, what is it that you do in maths?

  Rin: Yeah. So, um, I work on kind of a combination of a lot of different fields, but mostly I do kind of arithmetic geometry, homotopy theory, representation theory, number theory, I guess.

  Simone: And how do you explain this to people at parties? I mean, if they ask.

  Rin: Of course. So I mean, I guess, well, the first thing that I say, if someone asks me what I do, I say I like thinking about shapes and numbers. Um, and if they ask more than that, then I say, well, I study the way that maps between higher dimensional spheres are related to counting points on curves.

  Simone: So spheres we can imagine, curves we can more or less imagine.

  Rin: Yeah. And counting. People know what that is roughly.

  Simone: Yes.

  Rin: Right. And that's like, yeah I struggle sometimes, but so do we all.

  Simone: It's always a humbling experience, right? When you're the mathematician in the room and they expect you to sort of… can you please compute the bill at the restaurant and. Well, no, I can't. I mean, I could write I mean, with some time, but not this. Definitely not what I consider within my skill set.

  Rin: Ah, yes. It's very different from, I think, the kind of everyday type of maths that we do for sure. Although I have had a few of these moments over the years where sometimes I'll be doing math and I'm like doing math, you know, computing a five by five matrix or like counting, uh, you know, comparing a bunch of numbers to each other and trying to find a common factor. And I'm like, this is what people think mathematicians do. And right now I'm actually doing this.

  Simone: Exactly. And this is what mathematicians often struggle at doing. I mean, the high dimensional spheres are fine. It's the matrix of integers which we sort of not necessarily grasp.

  Rin: Right, I mean, this is somehow more of a trained skill in like the ICM crowd or sorry, ICM is the wrong word. What's the word for the competition?

  Simone: You mean the IMO? Were you a competitor?

  Rin: I was not an IMO competitor, but I do think that IMO competitors tend to be better at these sorts of kind of quick calculation skills than the average mathematician is.

  Simone: Yeah, there's often, I think the belief that if you're a good IMO competitor, then then this means you're going to enjoy a lot of of mathematical research, which I don't think there is necessarily a big correlation there.

  Rin: No, it burns a lot of mathematicians out. But I do think that the IMO is pretty good at effectively teaching proof methods. In fact, when I eventually kind of decided to professionally enter mathematics, the thing I struggled with is that I had never written a proof before, and I used a book for IMO students to learn how to do basics of proof writing.

  Simone: So how did you get to the maths? And how did you get to what you do nowadays?

  Rin: I have a somewhat convoluted path towards the way I got into math. So I was working in robotics and engineering… Actually, it kind of started even earlier. When I was at university, I really, kind of didn't really quite know what I wanted to do. And then eventually I took this calculus class. I got really excited about that. And I wanted to do pure math. And at the time, the person who is mentoring me was a burnt out postdoc and he was like, Rin, you don't want to do mathematics, you can actually help people. And I was like, okay, so then I did like robotics and plasma physics for a while, and then I got a grant to be an inventor, and I was doing that for a while, and I was doing maths…

  Simone: I need to backtrack on this a bit. An inventor, as in, because somehow, maybe to me, the image of an inventor is like Archimedes in the Mickey Mouse comics, like somebody who is in the lab the whole day and actually build things. Is that actually…?

  Rin: That was what I was doing. I mean, the grant that I got is called the Teal Fellowship, and it's given to 20 people under 20 years old who kind of have a project in mind that they want to do. And it's for $100,000 and it's given to the person, not the project. So I mean, what my project was, was doing medical technology, improving wheelchairs in various ways and neuroprosthetics and stuff like that and improving pre-clinical trials. So being able to detect stuff that we weren't previously able to detect in pre-clinical trials until they reached human stage earlier using technology. And um, so I was doing that and I was doing math as a hobby. And then I just like really reflected on what aspects of my job I liked and I didn't like. And I realized the part I liked the most was the math part. And I decided that I would, you know, try that, just wrap up the projects I had at the moment, give them to other people and try just doing math for a month and see how I felt. And the month never ended.

  Simone: Yeah. So how did you go from somehow medical technology to arithmetic geometry? I mean, it's quite a jump.

  Rin: Yeah. Um, so I guess the way that I kind of went through it was, I was at this like party where the person who is behind the Teal Fellowship, Peter Teal, has meetings with the fellows. And we had this kind of questionnaire that was open and kind of we could all ask him questions. And at the time, I had a friend who was a pure scientist who was struggling in various ways, and someone who was then a hedge fund manager, but had been a theoretical physicist before, Eric Weinstein, he came up to me and he was like, oh, you're a pure science Teal fellow, why haven't I met you before? And I was like, well, because I'm a mechanical engineering fellow. And he was like, well, tell me about like some of the things that you're interested in. And this was the day before I was going to start studying math for a month.

  Simone: “A month.”

  Rin: Yeah, a month… eight years.

  Simone: Uh, and counting.

  Rin: And counting. Yeah, so I kind of asked him various things, and he was like, you know, have you heard of the Atiyah-Singer Index Theorem? And I was like, no, what a good place to start. Let's look at it. And I was like, damn, this is really cool, you know? And that was one of my entering points into maths. And one of the ways that I still learn and the way that I learned then, was I would print out a paper that I found interesting, I would outline all the words I didn't know, which were most of them, and then I would look up John Baez and that word, and I just read every post he wrote on the topic.

  Simone: And on most words you would find something.

  Rin: So I actually was able to learn enough from doing this, just locking myself in a room for like three months and doing this, enough to start attending classes at Berkeley, which was nearby because I was living in San Francisco at the time. And so I started sitting in the back of lectures, and at some point, I kind of reached a point in Eric Weinstein's mentorship where he realized I needed to actually be talking to a real mathematician. And Edward Frenkel was also affiliated to the Teal Fellowship, who's a professor at Berkeley. And he started meeting with me weekly. And at some point, I reached a point in his mentorship of me where he was like, the questions you're asking me are a little too topological for me to understand. And he started to introduce me to people in the mathematical world. And also I met them through just showing up in classes and asking questions.

  Simone: So somehow by this sort of repeated process of reaching the limits of the expertise of somebody and going to the next one, is how you eventually ended up in your field.

  Rin: Yeah. And I think that's still how I do math. I mean, something I have really enjoyed about my Uni Münster postdoc is that I think the way that I best learn is by doing this. And so if I want to learn a topic, I’m able to fly to wherever that is, take a train to wherever it is, talk to that person for a week, a month, however long, then go to the next place, or write up what I learned and then go to the next place.

  Simone: And somehow do you think there was any connection between your exam or your past as an inventor, and your daily life as a mathematician? I don't want to get into the topic of is math created or discovered…

  Rin: No, I don't care about that.

  Simone: And it would take, you know, infinitely many hours. So let's not get into that. But somehow this idea of letting ideas flow and coming up with something new. Do you think somehow this was always in, in at least hidden away somewhere?

  Rin: Yeah. So I think I've always known that I wanted to be a researcher of some kind, or maybe a creator, and preferably both. And so I think that has continued to influence my development as a scientist and artist, as a person, like in all areas of my life.

  Simone: So what would you like to see happen in this somehow, remainder of this “one month in maths” that you've been doing for eight years and a bit more? Uh, what is some what is an answer to some question that you would like to see?

  Rin: I've always been really fascinated about the interplay between manifolds and topology and number theory and arithmetic geometry. And this has been evolving quickly and like beautifully in the past couple of years. I mean, I feel like a paper comes out almost every month now that just puts together another piece of this puzzle. And it's kind of a broad picture that I'm devoting myself to, but let me try to express it. So there's somehow a connection between L-functions, topological genuses, and like Euler characteristics and boundary conditions on topological quantum field theories or kind of Riemann-Roch type statements, the Atiyah-Singer index theorem, etc.. And I mean, it's really just amazing how these things are starting to feed into each other. I mean, for example, this kind of started with me looking and seeing that the Bernoulli numbers were showing up all over homotopy theory and in homotopy groups of spheres. And my goal is to kind of sit down and be like, okay, so we know that really numbers show up, but what other arithmetic invariants show up? What other L functions, what other generalizations of the Bernoulli numbers show up? And as I sat down to do that, I realized that we in fact don't understand why the Bernoulli numbers show up.

  Simone: They just happen to show up.

  Rin: Yes. And right now it's just kind of a set of numerical coincidences. And together with my collaborators Noah Riggenbach and Andreas Mejia, we put together this picture that kind of categorizes all of the different instances that the Bernoulli numbers show up in the homotopy groups of spheres, and seems to imply that they come from kind of this greater structure. And so that's nice and in some sense a key part of like the perspective that we developed is that the Bernoulli numbers are capturing the lattice of Z sitting inside of the real numbers. And you can ask… If you generalize to other lattices, you know SL2(Z), or like the symplectic group on Z or any other group you'd like, that's discrete, living inside of a topological space. You can also ask for a generalization of the zeta function there. You can ask what sorts of Gauss theorems do you get? Do you get a generalization of this beautiful statement that haunts me to this day, that the zeta function of one minus 2g is equal to the Euler characteristic of the moduli stack of genius g curves with one marked point. So you have this like topological invariant of a moduli space being equal to the zeta function of the dimension of the moduli space. And you can ask, well, what if you put something other than the mapping class group there? What if you put the symplectic group. What if you put SLn. Do you get on the other side the generalization of the zeta function to that lattice? And this kind of fits into a different picture. Do you get that the orbifold Euler characteristic of like SLn for example, or Spn? And to the symplectic group is equal to a product of zeta functions. But there's no known connection between the mapping class group case, which gives you one zeta function like a zeta function of one minus two n, and the symplectic group case, which gives you a product of the zeta function minus one all the way to zeta function of one minus two n. And like those, there's a map from one of these groups to the other, and there should be a connection, but we don't know what it is. And a lot of this amazing recent work. So for example, this recent work of Amina Abdurrahman and Akshay Venkatesh, where they were really able to use Reidemeister torsion and the topology of knots to prove things about L functions of symplectic representations. And I feel like in recent years we've really been able to have these two fields fully talk to each other. These analogies are becoming actual proof techniques that are emerging into like a bridge in a language.

  Simone: No more heuristics, but but actual frameworks to understand things.

  Rin: But for now, the frameworks still need more work. I mean, somehow they're just they're hinting at this very simple, beautiful global structure that specializes to these two worlds. It specialized to the topological case and specialized to the zeta function case. But right now, getting in between those two worlds is still unbelievably difficult. I mean, their paper is really, really hard work. And again, it comes down to a sort of numerical coincidence which would be really interesting to try and see if we can specialize. And it's also kind of cool because you can see things like, the Euler characteristic could be written as a type of duality statement, like the Euler characteristic of a manifold is equal to the Chern character of the manifold times the Chern character of the dual of the manifold evaluated on fundamental class, and in fact, in a lot of path integrals that show up in field theories, you get these bosonic fermionic cancellations that make your path integral converge. And it seems like there's a connection between some of the path integrals that show up in string theory, where you're not just mapping, say, S1 into your manifold, but you're mapping some other world sheet, you know, genus g curve into your manifold. It seems that those path integrals in the statements, for example, for genus one curves, you get that the Riemann zeta function of minus one shows up as a correcting factor, which is the Euler characteristic of the moduli of genus one curves of one marked point. And right now there's no framework that really explains why that happens. I'll just like reiterate this, somehow this triangle of like L functions and zeta functions, they're generalizations to other lattices, topological genuses or Euler characteristics. And these like Riemann-Roch duality statements which show up in boundary conditions of invertible topological quantum field theories and invertible is important here, because somehow the way that it's connected is showing up as something and its dualising object. So a lot of the arguments that show up there end up being some sort of Fourier transform that picks up a term, a measure on a space that allows it to converge. And this re normalizing factor seems to be the Euler characteristic I see in like elliptic curves. So I'm really, really curious to see, first of all if we can really make this picture that I've been developing work that really shows that the Bernoulli number instances that we know specialize. That this categorification really does give you an explanation of where all the Bernoulli numbers show up in the homotopy groups of spheres, and also in general, this kind of a categorical framework which allows us to talk in the same breath about symplectic groups and the mapping class group, um, and their measures and their Euler characteristic just living in the same world.

  Simone: That’s really fascinating. Thank you very much for this.

In this episode of On A Tangent, Simone is joined by Rob Sullivan, a postdoc in Combinatorics and Model Theory. We learn about what children do to entertain themselves at boring weddings, and why ultrahomogeneous structures are hard to understand.

Link to Rob's website

• Transcript

  Simone: Welcome to On a Tangent, the podcast where the main characters are the stories behind the mathematics. My name is Simon and in each episode I am joined by a different early career mathematician from Muenster to learn about their paths towards mathematics and their hopes for the future. In this episode, I am joined by Rob, a postdoc in combinatorics and model theory, to learn about what children do when they're very bored at weddings and why ultra homogeneous structures are quite hard to understand. I hope you enjoy the episode. (...) All right. Hi, Rob. Welcome. Thank you for joining us. So let's start a bit with who you are and what kind of mathematics you're interested in. How do you explain this to your family, to friends, to people at parties?
  
  Rob: Okay. So, yeah. So the general I think generally I've always been interested in combinatorics, I'd say it is the general theme. And then how do I explain this to people? I generally lie, first of all, and sort of give them a sort of mild misrepresentation of what I'm doing. And generally I think it's better... I often find it's better to give someone a problem to play with because, you know, I can tell them all kinds of terminology and whatever. But I think the best thing to do is to give people like, you know, a case. So what I often do is someone says to me, what do you do? And I say to them, okay, let's prove that the Ramsey number three is equal to six, right. And you know, obviously I don't say it like that. I start off with, you know, particularly if you're speaking to someone who's got, you know, very little mathematical background, I'll say, so you've got six people at a party and then, you know, I claim and it's supposed to be this sort of, you know, confounding fact that, you know, so you've got six people at a party and you can choose any six people you like, and there's always going to be three of them who know each other, or three of them who don't know each other. Right. And so then people sort of say to me, so some people say, why do you care? Which is a very interesting question.
  
  Simone: That is not an easy one.
  
  Rob: Yeah, yeah. But I don't necessarily have a good answer to. And but generally what people do is particularly if they've not got a mathematical background, is they'll start off and they'll try and define the problem. They might say, oh, but hang on. So, you know, Annabel is Dan's mate, but then Bella is Dan's wife. So does Bella know Dan more than Annabel does?
  
  Simone: Right. They sort of focus on the irrelevant part of the problem.
  
  Rob: Yeah, exactly. And so it's really interesting because you see, you see people trying to go through this process of abstraction and... Yeah, so we sort of essentially play a little mini tutorial. And so I say, okay, so points and lines and then, you know, colourings and I don't know, you know, blue means know and red means don't know and so on. And then we start talking about the pigeonhole principle. And we do small little cases. And so this is, you know, and I think that's more, that's much more instructive because I think that much more closely represents what I actually do on a daily basis rather than, you know, theorem statement or this is my field or whatever, you know.
  
  Simone: It gives them a taste of what actually, I mean, what your work actually sort of feels like or what the concepts are behind.
  
  Rob: Yeah. And more than that, it tells them why I might care as well, because the goal is to essentially, for me, is to convince people that these questions that we're asking are natural, because a lot of people have this preconception that maths somehow exists in this separate platonic, existential realm. And, you know, I just want to convince people that, you know, everyone's got a certain set of problems that they're trying to solve, and we're not doing kind of any magic. It's just often the problems have a certain flavour.
  
  Simone: And people prefer certain problems over others because somehow they feel like that line of reasoning is more natural to them.
  
  Rob: Yeah, or because they've been educated on that, exactly.
  
  Simone: And so, so how did you come to combinatorics? What brought you to the topic? I mean, somehow you've sort of already implied that the taste of doing it, this idea of concrete problems that you can do, maybe you can draw the things and this on is something you like about your subject.
  
  Rob: How. Yeah. And I think it's something that you can you can teach children or teenagers in a very attractive way, because you don't have to lead them through a whole series of prerequisites. You can literally and, you know, people who enjoy problem solving, you can just sort of get them going, right. So for me, …, so I did maths Olympiads, and so I was sort of involved in this stuff in the UK. So I went to a school that as a matter of course, enrolled everyone in. So there was something I think it's called the kangaroo or something. There's some sort of very Junior maths Olympiad. And then there was the British. I'm probably getting all the names wrong, but there was the British Junior Maths Olympiad as well. So if you've got a good score in that then you've got a magical Hogwarts letter in the post, and you got invited to go to a summer camp. And so this is sort of, where I had my first sort of taste of combinatorics, I suppose, because I don't think I was really at that time capable of differentiating different styles of mathematical thinking, and so I just naturally gravitated toward whatever seemed the most interesting, you know, without any necessarily having any preconceptions. And I remember at this summer camp, there was... they had some, you know, professional mathematicians there as well. And one of them was Imre Leader, who's at Cambridge, and I think it was after dinner, maybe he gave a little proof of Van der Waerden's theorem. Shall I state Van der Waerden's theorem?
  
  Simone: Why not?
  
  Rob: Why not. Okay, so it's to do with colourings of the natural numbers. So let's say that we colour the natural numbers in two colours, red and blue. Then Van der Waerden's theorem states that one of the colours contains arbitrarily long arithmetic progressions. So if I want an arithmetic progression of length five where you know all the points are of the same colour, right?
  
  Simone: Okay.
  
  Rob: Um, and so. It's a great theorem to sort of present at a summer camp, because what do you do? It's basically a clever induction. And you start off with small cases and so you say, okay, so you start talking about the pigeonhole principle and you say, okay, well, obviously if we colour three integers into colours, then we're always going to get two integers of the same colour, right. Yes. You know, and obviously it forms an arithmetic progression because it's trivial. And so you start off with that as the base case, and then you sort of build it up and you say, okay, so what if we want an arithmetic progression of length three where everything's sort of the same colour and so on. And so he literally did this. So he sort of had a series of slides and he had a blackboard. And he got us to prove it collectively as a group. And I don't think we proved the full theorem, but, you know, I think we did enough of the sort of low value cases to sort of work out how it worked in general. And this sense of like play. I think something that I found very intimidating as a child was this sort of 19th century romantic idea of a mathematician, which is often what you see in films where, you know, I'm sitting on top of a mountain and, you know, a lightning bolt strikes and suddenly I prove the Riemann hypothesis.
  
  Simone: Possibly in some awfully little office and, you know, sort of closed there for eight years, and then you come up with the...
  
  Rob: Exactly. You know. And this gets glamorised. You know, if you look at Hollywood films, I don't know. People always love the story of Andrew Wiles, where no one knew what he was doing for seven years and so on. And I was always very frightened by that because there was always this question of, well, I'm probably not good enough to do that. Right. And so I think that's a very frightening stereotype.
  
  Simone: I think we went through this already in a previous episode. It's somehow in a way, it's not a very standard way for people to live, and even you think, okay. I mean, I'm not going to do that. Like that's not natural for me.
  
  Rob: Yeah.
  
  Simone: And then you think, okay, well, does this mean I cannot be a mathematician?
  
  Rob: Yeah, and it intersects with lots of really troubling notions, I think, of personal worth and things like authorship, where, you know, you have to be able to put your flag down and say, I did this, right. And so what am I trying to say? You know, having this theorem being presented to me and, you know, it's a theorem with a capital T, and it's got a guy's name, Van der Waerden and, you know, 1929. And I say, oh, well, he must have been some genius guy. And then, you know, it's presented to us as like, you play around with some small cases, you spot some patterns and you generalise. And I thought, well, this is great. You know, this is what I wanted to do. I mean, it's a little paradoxical because often combinatorial proofs involve having, you know, one of these lightning bolt style ideas, maybe, or at least they're often presented as that. Right?
  
  Simone: Yeah. Somehow, I've always, or well, I've often had that feeling when some combinatorics entered the picture. So doing model theory myself, there is always sort of a moment where you have to do a little bit of combinatorics, even if you do very algebraic model theory. And then people sort of drop from the sky as if, you know, here is here's what you should do, here's the set you should consider, here's the, you know, the bound that actually happens. And you're like, okay, how did you figure this out? I mean, it's not clear, but I mean, I guess if you do a lot of small cases, right, then heuristically you can sort of start guessing what the general picture might be. And that's one of the good things you can do in combinatorics that you maybe cannot do elsewhere, because maybe concrete examples don't really exist in other, when other questions are considered.
  
  Rob: Yeah. And no, I mean, there is there is a paradox there in the sense that I like combinatorics because it's accessible and you can play around with it. But at the same time, I think a lot of the culture around it is this kind of...
  
  Simone: Miraculous.
  
  Rob: Yeah, miraculous. I had some amazing idea. And then I publish a paper and it's one page long and it solves this, this, you know, this long standing problem.
  
  Simone: And it's sort of saw in a dream that the bound was supposed to be log, log, log log ten times and then but not 11. Yeah, ten is enough.
  
  Rob: And so I think when you're much younger, this combination of the fact that you can play with it, but also, you know, because people are very contradictory. Right. And, you know, combinatorics has got these big personalities like Erdos with these very colourful biographies. And so I think, it's naturally something that, you know, if you're exposed to this, you can feel very attracted to. And the environment I grew up in said that combinatorics was important because it was hard. And, I think there was this idea that it really tested you out because you couldn't come along having read a textbook, and then do any better on the problem than anyone else necessarily. I'm not sure whether that's actually true.
  
  Simone: Yeah, somehow it feels a bit like a stereotype, but it does seem true, right? That in a certain stereotypical way that what's been tested is here. And again, this is a problematic notion. Right. But sort of your cleverness because there's not much theory that goes into that. So it's not like you can actually know better than others.
  
  Rob: I mean, it's not true, right?
  
  Simone: It's not true, of course.
  
  Rob: But if you've been thinking about this style of problem for a long time, you build up a little compendium of problem solving, and it's clear that experience goes into it and knowledge goes into it.
  
  Simone: Yeah, but somehow I think it's true that stereotypically, combinatorics is seen as the field where the actual clever people can maybe just go in, dip in, fix a problem, go out. Yeah, because you don't actually need anything else rather than some clever intuition. And of course this is false, right? I mean, but there is a bit of a stereotype there, and I can understand its attractiveness because it really tests. It's a bit like street smart versus book smart. In a sense.
  
  Rob: Yeah. I mean, and this is that's actually a very good way of putting it. I've always liked this idea of, um, you know, I'm sitting there having a coffee and another mathematician comes and says, what are you working on? And I say, oh, well, you know, and I just sort of sketch it out for them because I don't need to say to them, ah, so do you know what an abelian variety is? Yeah. Right. Yes. I do think there's a fundamental paradox there in terms of my attraction to the subject and in terms of how the subject is generally perceived as well. And so then, you know, after the Olympiad background, I went on to Cambridge and this was a place where, again, there were lots of. I don't know. It's difficult because no one ever actually sits down and says to you, oh, so this subject is important here, and this subject is more important elsewhere. And no one, because, you know, it can be incredibly rude to the people working in these fields to do that. But yeah, there was a sense of... combinatorics is exciting and fun.
  
  Simone: And if anything, because probably there were a lot of combinatorialists, or still are a lot of combinatorialists in Cambridge.
  
  Rob: I mean, a lot of them have moved to Oxford now, but certainly, at the time I was there, they were big names in arithmetic combinatorics who were doing lots of exciting things, you know, so this is sort of late 2000, early 20 tens. And so they're teaching courses on graph theory, they're teaching courses on functional analysis. And then at the master's level combining things, and often these people tend to be very good lecturers and presenters as well.
  
  Simone: Yeah. Of course this is sort of these naturally builds an exciting fields. Not just cool in the sense of, you know, popular to do this subject, but also you're naturally drawn into it. Because if, of course, if the lecturer is are very good in it and there's a lot of exciting courses and seminars, then I mean, you're going to be exposed to more, more results and more, more enthusiasm even.
  
  Rob: Yeah, exactly.
  
  Simone: Okay. So let's go a bit further back and further back than your Olympics. So what is sort of your early...
  
  Rob: Olympiacs? Olympiads.
  
  Simone: What did I say? The Olympics?
  
  Rob: Mathlete.
  
  Simone: Yeah. Mathletes. People call themselves mathletes, I think. So let's go back to maybe even further back. So what is your sort of earliest memory of doing something mathematical, which, of course, mathematical is defined as whatever you felt was mathematical at the time.
  
  Rob: So we should probably distinguish between my earliest memory and my mum's earliest perception of my mathematics.
  
  Simone: I don't know about you, but I have like zero memories from when I was born to like age ten or something. So I might as well have done infinitely many things back then.
  
  Rob: But I mean, so my mom always tells me that when I was in the the pram...
  
  Simone: The pram?
  
  Rob: Oh yeah, the pushchair, I guess people call it in the US. So I'm originally from Australia and obviously because a lot of the towns in Australia were built in this very sparse way. You have these sort of American style grid systems a lot of the time. And so it's very normal for your house number to be like 1152. Right. And so my mum always tells me, you know, when I was in the pram, I used to point at a number and say, mum, what's that number. And you know, then she'd say 217 and I'd say wow, you know.
  
  Simone: Yeah, I mean kids point at things and ask, but you had a particular fascination for numbers.
  
  Rob: I actually don't think my original interest in maths was... I think I was more into computers, really, and codes and things like that. So I remember I was always trying to code messages, you know, like these Caesar shift ciphers, you know, where instead of writing A, you write B and instead of B, you write C.
  
  Simone: So these are the ones that are sort of alleged to be the ones that Caesar actually used to move information.
  
  Rob: To send messages during times of war. Exactly. I was trying to invent my own alphabet and, you know, God knows why. There was something that my dad always did that I always found very, very impressive. But I don't think he did it deliberately. It was that he was very good at not knowing what was age appropriate.
  
  Simone: Let's see where this goes.
  
  Rob: Yeah. You know, and so we had a computer and, you know, a computer at home. And in the early 90s, that's like a non-obvious statement, obviously. My dad was an engineer, still is, I suppose. We had a computer at home and one day I can't remember where we were. I think we were at some kind of second-hand boot sale. You know, where people sell things out of the back of their cars and things they want to get rid of. And he bought a book, and he said, oh, doesn't this sound interesting, Robert, wouldn't you like to do this? And it was called QBasic for dummies, right?
  
  Simone: Okay. I mean, I sort of expected this to go into like, Marquis de Sade situation.
  
  Rob: You can edit that out. So, he bought this book for me, you know, QBasic for dummies. And what does it do? I have to program for a for loop. And, you know, the computer screen prints out all the numbers from 1 to 100. And then this was back in the day where not everything was done on windows. So this was mS-DOS. And so you could do things like you could change the terminal background to be pink, of course.
  
  Simone: Yeah.
  
  Rob: And this is amazing.
  
  Simone: Yeah. Of course, of course. I mean when all you have is just the terminal.
  
  Rob: Yeah. And so I loved it. So and the other thing was when I was a child, I used to, completely differently to how I am as an adult, I used to wake up very, very early, like 5 a.m. early and go and jump on my parents bed.
  
  Simone: Um, which I'm sure they loved.
  
  Rob: Yeah. And so we eventually reached a compromise, which was that I could go on the computer, if I got up early and, you know, unsupervised.
  
  Simone: Yeah. What age were you?
  
  Rob: Oh, like 4 or 5.
  
  Simone: I see. Okay. You could barely read.
  
  Rob: I don't know. I mean, I definitely feel like, I learned to read via the computer or via programming rather than... I mean, I read lots of books as well. And so I used to sit there, and so the family cat used to come up and curl up next to me next to the computer, and I used to sit there when I was five, and I used to try out these programming exercises. And, you know, I used to play computer games as well, but this is how I initially got into it. Oh, I've got a much better answer. I've got a much better answer. I've just suddenly remembered. (...) So I was at a wedding when I was four. I'm at this wedding, and I'm absolutely bored out of my skull. Right? Because I'm a four year old. I'm at a wedding. You know, what's a wedding?
  
  Simone: You're not allowed to run around.
  
  Rob: Yeah, exactly.
  
  Simone: Sit quiet.
  
  Rob: There's no children at a wedding, usually, because people don't bring children to a wedding because they make too much noise, right? So my dad was an engineer, and his idea to entertain me because we were both very bored was to teach me how to count in binary.
  
  Simone: Right. Okay.
  
  Rob: And so, you know, I already knew how to count. I think I could already count, I don't know, over ten at least. And he said, oh, well, there's another way of counting. And then, you know, we were sort of putting each finger up and down. And then can you convert between them? And I remember it just seemed like impossibly difficult and I didn't understand anything, but like, uh, it seemed very interesting.
  
  Simone: Yeah. So enough about the past. What about the future? So what is a big problem in your area you would like to see solved? Of course, not necessarily by yourself, although one can, of course hope, but unlikely. But no, what is a theorem you would like to see proven, or a question you would like to see answered, or really, maybe even more. I mean, a concept that you think should be explored more and so on.
  
  Rob: Yeah, I think I just tend to think more in terms of... I often feel like we're just sort of swimming in a sea of complete ignorance.
  
  Simone: Sort of in the deep sea. We can't see anything.
  
  Rob: Yeah. I just think I have this feeling that we're only really scratching the surface. And so what do I do? A lot of the stuff that I do is to do with ultrahomogeneous structures. So, there is this sort of generic limit objects, these sort of generic countable objects that result as limits, of course, of finite structures. And the point is they kind of, from the finite perspective, they look the same everywhere. And so if you want to do arguments, you know, about, I don't know, finite graphs, you can look at this limit object called the random graph or the Rado graph. And the kind of the interplay between them is very interesting. I think... we don't know very much about ultrahomogeneous structures in general, ultrahomogeneous structures where all the relations are binary. We know some stuff, but I think in general we just don't know that much. And a lot of the intuition goes out the window when you start looking at ternary things or, you know, more complicated things in general, really.
  
  Simone: Also, it feels like in mathematics there's a lot of emphasis on what is the next big problem, what is the big problem. But of course, most of the work, which is maybe sort of... submerged. Yeah, I mean, the sort of submerged part of the iceberg is just understanding the things we don't know. I mean, which is of course, much less popular, much less fancy to say, oh, I want to expand knowledge on this class of objects. It's less fancy than saying, I want to prove this big conjecture.
  
  Rob: But I think this big conjecture style thinking feels a bit again. It goes back to what I was saying before. It feels a bit Hollywood. It feels a bit. If you look at if, you look at Grigory Perelman, who, you know, who declined, the Fields Medal and you know, declined the Millennium Prize as well. I say this because, Simone and I have just paused the interview to check exactly what prizes he declined because I think he declined about five of them. And, you speak to people about this, and I think there's this sort of general idea that, you know, this is a bit of an odd guy, right?
  
  Simone: In a sense it is an extreme view of the problem.
  
  Rob: Yeah. I mean, I don't know how many people would decline a Fields medal, but you know, if you look at what he says, he says, well, you know, a lot of this stuff was building on previous people's stuff and some of it was done collaboratively. And he doesn't like this idea of the money and fame and the, you know, the sort of the focus just being put on him and saying he solved it, you know.
  
  Simone: Yeah. It is this submerged iceberg idea that, of course, in the end, most of us will not prove a big theorem or a big conjecture in their lifetime. And the idea to me is more like every one of us gives a bit of a push, and then there's a sort of, a critical mass is reached and the big thing can be proved. Yeah, but all of this is just the sum of all the little pushes of people.
  
  Rob: Yeah. And you do need the genius people, of course. But if you start to think to yourself, oh, well, why don't we just take the genius people and...
  
  Simone: Lock them in a room?
  
  Rob: Yeah. And just we don't need any of the rest of them. They're all kind of not really doing anything particularly substantial. I think that's a completely wrong way of seeing it.
  
  Simone: It's actually probably factually false. I mean, somehow someone will have to do a maybe have some big intuition. But yeah, even that big intuition will be nothing without all the work of the previous people behind them. I mean, even Newton, who was not the most agreeable fellow on earth, used to say this about being on the shoulders of giants, right? I mean, yeah, it was Newton, right? Maybe we have to pause again.
  
  Rob: It was Newton. And this impetus towards ownership, sometimes it's forced on people externally and it's not something that they would necessarily want. I mean, we don't want to and you have to sort of ask yourself, how much is it got to do with the scarcity of academic positions?
  
  Simone: The job market being what it is. A stronger focus on, you know, who proved what and what is the exact contribution.
  
  Rob: What's the more important thing? The what or the person who proofed it? I think it's obviously the theorem that's the more important thing.
  
  Simone: And of course you want to give recognition to the person. But this doesn't mean that it's a scarce substance that you cannot give to more people. I mean, I always feel like you're not going to run out of acknowledgements.
  
  Rob: No, certainly.
  
  Simone: And somehow it's always... it's how I feel about giving a single name to a theorem, because of course, this person proved the theorem, and it's a useful mnemonic to know who proved a theorem.
  
  Rob: Yeah. And often it's just a little placeholder that you put in. Right. Just because you need a name for it, there's no better way and so on.
  
  Rob: But yeah, but this whole concept of ownership I find very... I mean, look, you know, when you're selecting people for a position, you've got to have a rough idea of what they've proved.
  
  Simone: But I think that this is really just because the market has become so competitive. To the point where you actually have to sort of surgically compare the contributions to choose which one is better according to some metrics.
  
  Rob: Right. I mean, you can't give everyone a job.
  
  Simone: I mean, I agree, but I'm not sure that pushing so hard on, you know, how many papers you have and what percentage of that paper is your contribution is actually a healthy thing.
  
  Rob: Yeah. And it's often very murky.
  
  Simone: I mean, I wouldn't know what percentage of a certain paper I have.
  
  Rob: But certainly during my PhD. The first time I gave a talk on the results from my PhD, my supervisor had to come up to me afterwards, and he said to me, okay, so two comments. He said, number one, you didn't have any theorems in that talk. You just labelled everything as a proposition.
  
  Simone: Yeah, I've done this. I've done the same in my first talk actually. It felt too much to just call them theorems.
  
  Rob: Yeah. Yeah, exactly. And then secondly, he said to me, actually, you forgot to attribute some of these theorems to yourself. And you wrote my name. And I said to him, you know, because I don't feel like I proved them. And he said, well, you did, and I really struggled with it because, it's still not clear to me now, you know.
  
  Simone: But I think this is common, right? I mean, it's definitely not that someone comes out of nowhere with a single author paper, and they've come up with every single idea in the paper. It's just not how it works.
  
  Rob: Yeah. I mean, sometimes it does.
  
  Simone: Sometimes. Occasionally. Yeah. But it's not how the standard thing goes.
  
  Rob: And you can wrap yourself, you can tie yourself up in knots with this stuff.
  
  Simone: Definitely.
  
  Rob: The solution is okay. Did the theorem get out there? You know? Yes. Am I going to be able to get a job? Yes. And then if those two preconditions are met, then nothing else really matters, you know?
  
  Simone: All right. So thank you very much for the lovely conversation.

In this episode of On A Tangent, Simone is joined by Azul Fatalini, a doctoral researcher in Set Theory. We learn about how the Axiom of Choice transforms the universes it holds in, how logic is the mathematics of mathematics, and what’s the quickest path to ice cream.

Link to Azul's website

• Transcript

  Simone: Welcome to On a Tangent, the podcast, where the main characters are the stories behind the mathematics. My name is Simon and in each episode I am joined by a different early career mathematician from Munster to learn about their paths towards mathematics and the hopes for the future. This episode, coming out during Pride Month, I am joined by Azul, a doctoral researcher in set theory. We learn about how the axiom of choice transforms the universes it holds in, how set theory is the mathematics of mathematics, and what's the quickest path to ice cream. I hope you enjoy this episode. (...) So, Azul… Welcome to the podcast.

  Azul: Hi. It's nice to be here.

  Simone: Yeah. Thank you for joining us. So as you know, in the podcast, we sort of explore the stories of our guests. And so I would like to start from your past. So what is your earliest memory of mathematics?

  Azul: Actually I love that question. So when you told me you were going to do this podcast, I ask you what type of questions you would ask. And you told me this example, and I was so glad because I have this story and nobody ever asked me this. And I didn't even know this until you ask me them. So I'm super happy to tell this. So the thing is, when I was a kid, in my hometown, the map of the city is a grid. And there is this main square in the city centre that is a square. So it's 100m times 100m, but it also has a cross in the middle. And one of the things we did with my mom a lot, was walking from my house to the city centre and going to have an ice cream in the corner of the square. And to do that, the best way was to cross the square through the diagonal, right? And most of these, most of the blocks in my city don't have this, it's just that part that you can cross in diagonally. And when I was a kid, I was always thinking, so first, if it was really better, I think, that one I could see, like, I could just look around and see that it was shorter than doing two blocks of 200m. But I always wanted to know precisely…

  Simone: How much better.

  Azul: How much better it was, because it seemed to me that it was more than one block, but less than two.

  Simone: So better than two.

  Azul: But yeah. But not like so much better. So… how much. And I was obsessed with this and I was trying to estimate. But it was super hard.

  Simone: Getting faster to the ice cream.

  Azul: Of course. How much faster are we getting to the ice cream?

  Simone: Exactly.

  Azul: But there was always trying to do this, and in my mind estimation was like 1.5. I didn't even know decimals I think, at this point. But like half.

  Simone: Half. Yeah.

  Azul: But I never could answer if the diagonal was less or more than 1.5. And this was just something I had in my mind for a long time.

  Simone: Until many years later in school, you suddenly found out?

  Azul: Yeah. Yeah, definitely. It was much longer, and I was not even living there anymore. It came the answer ten years later.

  Simone: And do you somehow think that this way… Well, this is in a way, a very geometric problem then sort of connects to what you do nowadays in your research. Have you always been a very visual thinker or a geometric thinker?

  Azul: I would say so. When I was in a Math Olympiad in high school, my favourite branch of problems was geometry. So Euclidean geometry. So in that regard, yes. But I would say I'm terrible at visual thinking in the sense of… It's very hard for me to imagine figures. Okay. In that sense I would say no, but because I cannot imagine so much, then I would draw things. I mean, I like to compensate this lack of imagination with like doing diagrams and drawings to help myself.

  Simone: And so… I just sort of already went into this topic: what is it, that you do nowadays in research?

  Azul: So I'm studying set theory. That's the branch. It's inside logic, as you know.

  Simone: Of course. We finally have a logician on the podcast.

  Azul: Yes. I mean, it's easier to tell you, right?

  Simone: Of course. But for our audience…

  Azul: Yes. So I study problems that are related with the axiom of choice and the subsets of the real numbers. So somehow the objects are very much like normal math, let's say. And there are some geometrical objects that I study. But from the perspective of the theory, because the questions are related to how much axiom of choice do you need to construct these objects or not, which is essentially a set theoretic question.

  Simone: Yes. And is this how you explain it to non maths friends? If they ask, of course.

  Azul: Of course not. I mean they don't know what axiom of choice is.

  Simone: And why set theory is important for people I mean.

  Azul: I mean, I don't even know if a mathematician… I'm not sure mathematicians know why set theory is important in general, I think. Yeah.

  Simone: Except for the fact that, I mean, yes, this is the foundation.

  Azul: That's it. Not mathematicians. I usually say I do something in logic. Logic is a word that people know, right? And I mean, of course they don't know what logic, mathematical logic means, but there is a concept of logic, an idea. And then if they ask me more, I say, okay, there is a branch of logic that is set theory that studies more or less the foundations of math and the structure of how math works. So there are these meta mathematical questions that set theory takes care of.

  Simone: I think I usually tell people we do, like what linguists do to the real world, like, people study the language in which people communicate and logicians study the linguistics of mathematics.

  Azul: Yeah, it could be.

  Simone: In terms of funding, probably it's a similar parallel.

  Azul: In terms of funding?

  Simone: We get as much funding as the linguistics people.

  Azul: Yes. I'm not sure I don't have any idea of this.

  Simone: Yeah. But I imagined it’s not much, as all humanities people.

  Azul: I mean, I agree on what you're saying, but in particular set theory I like more to simplify as the math of math. I think because you're not using linguistics to study, you know, the language of real life, it's like a different thing. Right? Linguistics is different from real world or, or the language that is spoken. But the theory is itself a part of math. And I think that's one of the most beautiful things.

  Simone: And the most confusing.

  Azul: And it's super confusing. But this thing… I mean, math can study things that are things of math. And when you say this to people, even if they are not mathematicians, it is like, what? How do you do that?

  Simone: Yeah. And I do remember my first set theory class, there is all the sort of paradoxes which seem to stem from the fact that you actually talk about set theory when you do the set theory of math. I mean, when you do the maths of maths, you're in particular studying the maths. I mean, you know…

  Azul: Yeah, there's this self-reference. Self-reference at the beginning.

  Simone: Yes. I mean, it is confusing. I mean, it's also something to explore.

  Azul: Well it's definitely both for me. The fact that is so weird in a sense, in comparison to other branches that don't have this feature makes it more interesting.

  Simone: And so your research deals with sort of, let's say, normal mathematical objects, which if another mathematician said this to me, that it would be offensive, but we can say it. Yeah. I mean, you know, because if somebody told me, ah, yeah, we do the normal mathematics, you do the logic, I would be a bit offended. But I can say, you know, the natural mathematical objects that you find in nature and in nature, well, you know, in nature, like in, in mathematics, I mean, mathematical nature. In mathematical nature. Exactly. And you often hear about how the axiom of choice builds these sort of paradoxical objects. Is this the kind of things you're interested in?

  Azul: Yeah. So my PhD thesis is called paradoxical sets and the axiom of choice. That's definitely the object. And yeah, I agree this about the normal math and not normal math.

  Simone: But would paradoxical sets be considered a normal mathematical object.

  Azul: Yeah, I think so. I mean, for example, you have this example that I think is one of the most well known. I think that is the Vitali set. So the standard example of a non measurable set in the reals. And this is I mean this is part of the course of real analysis. This is just something most of people would have seen at some point or Banach-Tarski that is so famous. And, well some people don't like actually Banach-Tarski, like it's weird. Right. But for example, the Vitali set is really a part of understanding of measure theory.

  Simone: No, we don't work in the, … what is it? The Solovay paradise where all sets are measurable?

  Azul: Yeah, exactly. So, I mean, but of course, when you start asking a theoretical question, then the object kind of doesn't satisfy the same role as if it is because you're not studying the object itself, but rather also or in my case, which are the axioms you need to construct, or how consistent is this existence of this object with some other set of axioms or some other objects? So, even if the name is right the same, like Vitali set, then what you're doing with it is very… different, you know.

  Simone: You’re not studying the properties of it, but rather what, what its existence says about the universe you find it in.

  Azul: Yeah, somehow. Where does it lie? In the map of things.

  Simone: And I know you're not meant to play favourites, but what is your favourite paradoxical set?

  Azul: What is my favourite paradoxical set? Let's see. Well, I think if I have to pick one, it would be the one I thought more about. So there is this theorem in ZFC. So using the axiom of choice. That R3 can be partitioned. So the space can be partitioned in circles, so only with the border. And these circles can be taken to have radius one. So they are all the same radius. But still, you can cover R3.

  Simone: Which is, I think you the first time you hear it, like of course I can do this. And then you mentally start placing circles one into the other and then very quickly get to the point where you don't know where to place the next one. Right?

  Azul: Exactly. I mean, you can put more and more circles, right? Because each one is kind of small in inside the space. But how would you end up completing everything over there?

  Simone: Yes.

  Azul: Yeah. Like, imagine you put a lot of circles and like it's dense in R3. But there is some space that you have to fill. If you have like some isolated points, for example, then of course you cannot do it. So even if you have like a full circle, but a few points are not already taken, then you cannot put a circle again.

  Simone: So this means you did it wrongly from the start?

  Azul: Yes. I mean at some point there was something that you did that didn't allow to continue the procedure. Right. But there is a way to do this process, wisely so that this doesn't happen. And this is a theorem of ZFC. So you really need the axiom of choice. Well, less than that, but some form of choice. Some part of choice. Yeah. I think this is my favourite. It’s nice that you can tell people you are studying something like this.

  Simone: Because they can visualise it.

  Azul: They can visualise it is just Euclidean geometry. So I can even tell my family about this if they are patient enough to hear this.

  Simone: Yeah. Or to look at the pictures at least.

  Azul: Yes.

  Simone: And do you feel like because this feels a lot like one of those problems in number theory where the statement is very easy. But then the math behind it is very complicated. So I presume in a similar way the problem can be stated very easily. But then proving it is not Euclidean geometry is actually.

  Azul: Yes. So the proof of this theorem is not well, of course it has some Euclidean geometry because the object is. But the main tool is doing transfinite induction on the cardinality of the reals, whatever that is. But the point is you do this induction, but it's longer than the natural numbers, but even more. You keep going after you did countable many steps, and then you keep going. Yes. And then you keep going on and you keep going until you reach the cardinality of the reals, which is the same cardinality of all the points in the space that you have to cover. So this tool is not from Euclidean geometry. It's set theoretical. So the existence of this paradoxical set already has a proof that is set theoretical.

  Simone: So you said you need less than choice. So maybe something like a, well, ordering of the reals or dependent choice, whatever this means for the non logicians. But then maybe in the vein of what we said before, from the existence of this set, just so you look sort of look at some universe where the set exists. Can you then say anything about what axioms are true in this universe? If, for example, is some amount of choice still true?

  Azul: Right. So that's exactly the questions we were trying to think of while my PhD was happening and so basically the answer is no. There is I mean, you could have this set and basically no choice. I mean. This, you can formalise this. For example, there is this concept of countable choice. Doing the choice countable many times. And we got the result that there is a model of the theory in which you have this set, so this partition of the space in unit circles, but you still you don't even have countable choice.

  Simone: And I guess for the non logicians or maybe even for the non set theorists, the striking idea is building a model of set theory because it's kind of feels like building a universe of but we are in one. So it's a bit confusing.

  Azul: Of course.

  Simone: If you start thinking about this philosophically you can go on for hours, I guess. But yes. How does one build a like a universe of set theory or model of set theory?

  Azul: I mean, I think that's the full thing of what a theory does, building models of set theory that satisfy the things you want. So how do you build the model? Well, first you assume there is one.

  Simone: Which is already… Yeah.

  Azul: And then from there, I mean, there are many techniques. So one way to build a model is build a smaller model, inside the one you have. So from all the sets you have in your model, let's take only the definable sets which have some definition and then you'll get something in principle, smaller. It could be strictly smaller. It could be all of it, depending on what you're doing. But there is also a way to construct bigger models. Okay. So there is this technique that is called forcing that allows you from a model and some other elements, construct a bigger model than the one you started with. So those are the two ways people do it.

  Simone: I think I remember in my set theory class, it was described to me as… So in the model, there are people and they believe in some sort of like entity. They know sort of how this entity looks like. They know some things, you know. Like so in the first thing you have some properties encoded, but then you sort of look up at the sky and this entity doesn't exist. But then like in the level above, the entity is there and looks down on the people and sort of has an effect on them. So it's a bit like religion, which I don't know. I mean, you do have a lot of cardinals in set theory.

  Azul: I do think it's a bit like that. So I don't know if you know this. Is it a book or…, I think it's a book.

  Simone: The higher set theory?

  Azul: No, no. Nothing to do with this. Okay. Flat something.

  Simone: Flatlandia.

  Azul: Flatlandia. Yeah. Is it English?

  Simone: Flatland.

  Azul: Maybe. Flatland.

  Simone: Yeah.

  Azul: Yes, flatland. I didn't know it in English. Sorry. So in flatland, there is this people that cannot. Well, actually, did I read this book?

  Simone: I'm sure I read it in school, like, when I was very young.

  Azul: Yeah, but. So the point is, if you are in flatland, you cannot see three dimensional things, right? You're a circle. Yeah, or a square or something. But then a sphere in flatland is like a circle that moves and has different radius. Right. Because depending on the section of the sphere, the circle changes. So somehow from flatland you can see this three dimensional things as another thing which is not three dimensional, right. Because you cannot but you can guess what it is somehow and can imagine what three dimensionality is by all the sections, for example. And this is more or less what forcing is. Yeah, it's a simplification of course. But what forcing is about. So from the original model you can kind of imagine what the exterior model would be like.

  Simone: And so from your research you're doing nowadays, let's maybe look into the future a bit. What is something you would like to see prove in the next ten years by somebody? Not necessarily by you, but… some questions you would like to see answered?

  Azul: I don't know. I think I would like… so I have this maybe vision about what I want to do in. I don't know the future very abstractly in general at some point with math, and it is trying to connect a set theory with, yeah, like subset of the reals or things that are like normal math now.

  Simone: That you find in nature.

  Azul: Yes, I am not comfortable with it.

  Simone: I think I often say, you know, model theory does logic on objects found in nature. This is what I usually say.

  Azul: Well, yeah, I don't feel it is in nature. Right. So, I feel uncomfortable with that concept. In theory the real numbers, for example, are not the real line. If you tell any mathematician, imagine the real numbers, they probably imagine the real line or maybe they just say pi or something. But the set of reals, which is the set of reals? They will draw a line. And for the set theorists usually it is not. For example the power set of the natural numbers or all the functions from the natural numbers to the natural numbers or things like this. Like there are several sets that we consider them all as the real numbers. They are all bijection with each other. So for most of the questions, this is the same if you care, for example, only about the cardinality of that set, then they are all the same. But for the objects I'm looking at, for example circles in the space. You are thinking about the real line. I mean, the space is three times the real line. So you have to look at the reals as the real line to just approach these questions. So what I would like to be solved is the things I'm curious about, of course. Which is this abstract idea of some theoretical questions that are related to the reals as the reals of the mathematicians, which is the real line.

  Simone: Mhm. To do a more geometric version of the reals.

  Azul: So yeah, but not only, I mean, even if you consider it for example the reals as a field, it's also something that is not… I mean, the power set of the natural numbers is not a field.

  Simone: It’s just a set.

  Azul: Yes. So even if you ask more algebraic questions, this also has to do with this view that I have, that we have a gap there.

  Simone: Okay. Well thank you very much for the insights!

  Azul: Thank you so much.

  Simone: See you in the corridors.

In this episode of On A Tangent, Simone is joined by Mert Bastug, a doctoral researcher in PDEs and Calculus of Variations. We discuss Mert’s first meeting with mathematics, how we can understand materials with the help of PDEs, and where they might take us next.

Link to Mert's website

• Transcript

  Simone: Welcome to On A Tangent, the podcast where the main characters are the stories behind the mathematics. My name is Simone, and in each episode I am joined by a different early career mathematician from Muenster to learn about their paths towards mathematics, and their hopes for the future. In this episode, I am joined by Mert, a PhD student in PDEs and Calculus of Variations, to find out about what mathematics can say about memory alloys, why they're not the same as memory mattresses, and how mathematicians think about their work. I hope you enjoy the episode!

  Simone: Hi. Welcome. And thank you for joining us.

  Mert: Hello. Yeah, thank you for having me.

  Simone: So let's start a bit further in the past. And so let's start from your very first memory of maths. So what's the first time you've seen maths enter your life?

  Mert: Well, the first time. I mean, I guess I could surely mention all those times I would draw on numbers or learn how to count. But I think my first moment where I consciously thought about a math problem was when my dad asked me a simple counting problem. But to me, at the time, it didn't look simple at all because it involved very large numbers. Actually, I can talk about the problem. It was about some number of horses all needing some horse shoes and each horse shoe requiring some number of nails. And then one would need to tell the total number of nails one would need. And the total came up to something in the millions which, as a child in kindergarten or maybe elementary school, it was ... I didn't really grasp this.

  Simone: A lot of zeros!

  Mert:  Right. I mean, somehow… Yeah. Yeah, definitely. And I don't even think I attempted to solve the problem. It was just to me that defined what a difficult math problem was. So that's as far as I can remember, right.

  Simone: So from somehow seeing what a difficult math problem was when you were a child, now you come to dealing with difficult math problems now, in your life.

  Mert: Yes, totally.

  Simone: So what is it that you do? How do you explain it to your friends? How do you how would you have explained it to yourself as a child?

  Mert: Oh, myself as a child, that would certainly be difficult, but, I mean, I guess I would use some tricks. Now it is okay to lie to others when you say you're doing something in math when you're not actually doing it. For me personally, the kind of math that I'm interested in ultimately goes back to problems in physics. So if I want to talk about math, well, not want to, but maybe if people ask me to explain to them what I do, I will start with some physical motivation. In my case, it would be something related to understanding materials. More specifically, what happens to materials when you try to bend them or when you just in general try to deform them? What sort of shapes are possible or why do they bend the way they do? Or, if they retain their form after you've let go of them, why that happens? I mean, okay, these are certainly not all questions that I try to deal with, but yes, if I'm trying to give people some sense of what I do, then I will freely talk about other things as well.

  Simone: I mean, somehow it's always helpful, right, when you can appeal to some physical motivation or so. There are any particular materials, I mean, that you actually work with or…

  Mert: Me personally, no. But I know. Well, yeah. So different people, of course, maybe they try to understand. I think they're called memory shape alloys.

  Simone: Are these a bit like memory mattresses, like you know, the ones where you sort of you lay down and over time, they sort of take your shape?

  Mert: I guess not really. Yeah, I would say these are a little more different than memory mattresses. They actually retain their form again. Maybe if you heat them or you expose them to some other external condition.

  Simone: So it's like those videos on TikTok of these materials that you bend over in strange shapes, and then the moment you sort of heat them up, they go back into, I don't know, a line or…

  Mert: Right, right.

  Simone: Maybe this goes back to earlier than TikTok, I mean… on YouTube in the early 2010s or something. This is magic of these alloys. Right. So this is the physical motivation. Right. So this is what you tell people. But what is the maths that you actually do for this. Or I mean I'm assuming you don't do experiments.

  Mert: No. Well I don’t. I’m sure some people out there do. The part of mathematics that deals with these questions is related to calculus of variations and also partial differential equations. In calculus of variations, one looks at minimisation problems. Now the quantities that one tries to minimise typically come from geometrical considerations, but as in my case, also from physical problems.

  Simone: So these materials you were referring to at the beginning.

  Mert: Exactly. So if you look at a material, then the shape that material wants to assume, will actually be the shape that minimises some sort of quantity, and this can be some sort of elastic energy or some other type of energy that one might want to consider.

  Simone: So, the natural state of this material is the one where this energy is minimal. And I think if everybody has ever played with a spring, you know. You extend the spring somehow. This is, you feel that there's some tension and you let it go. There is none.

  Mert: Yes. Right. And also, when you apply force to these materials, they will still try to minimise the energy in the best way they can. But now there will be some external condition, which is the force you're applying to it. Right. And in partial differential equations one looks at again, maybe physical or geometrical models of something. And one tries to understand how that model changes over time or over space and these are then described in terms of some equations that involve derivatives, which is where the name differential equation comes from.

  Simone: So somehow these in this case you're again thinking about models of materials and how they change over time when you give them some input.

  Mert: Right. And actually the two areas are related because it turns out that minimising something, amounts to solving a particular equation. So in other words, to every minimisation problem you can associate well, in most cases at least a partial differential equation. Yes, but I mean, if I wanted to talk about math to my friends or my past self say, I don't think I would necessarily mention the actual work that I do, but I would just try to give them a feeling of what doing math is like. That's why I often also mention, um, number theory, because, well, naturally, numbers are the objects that everyone can relate to mathematically. And I might also mention famous open problems from number theory, because in the end, no matter what it is in math, you're doing, um, you're trying to solve a problem that hasn't been solved before. And it's just the nature of the problem differs from area to area. But the motivating force, which is trying to understand something that hasn't been done so far, is the same in each area, I would say.

  Simone: And somehow you write model theory. Sorry, number theory professional deformation, there's all these open problems which are somehow easy to state, but then require years and years of difficult or difficult maths to be solved. Right? I mean, like Fermat's Last Theorem, where you would say, okay, I want to solve this, this equation with integers, and then, okay, everyone can more or less think, okay, how do you solve this? And then it takes, I don't know, eight years of Andrew Wiles to do this, which is of course the striking fact.

  Mert: Um that's right.

  Simone: So so you use often number theory as an example. Do you think somehow number theory and what you do then actually feel the same when doing them. Do they, do they sort of. Let me let me try to explain. Do you when you do maths on your on a daily basis, what is your guiding intuition for example? Last time we had Alex, who had a very geometric intuition about things. Uh, in this case, do you also somehow imagine your materials actually somehow evolving in space visually? Because, for example, some mathematician I can give the example of myself, I'm not a visual mathematician. It also doesn't help that my field is not visual at all. Um, but in other areas. In GR we had the discussion last time. It's a very visual area. You can imagine the geometric configuration. Does this also help in your problems? And do you think it's somehow, um, do you also think in this way or is it or is there some alternative way of people think about problems in PDEs or in, in calculus of variations?

  Mert: Well, um, I would say that I am also quite a visual thinker. Um, now. For me, it's difficult to apply my visual thinking into problems directly, but it's always helpful to draw a sketch of what I'm doing. Um, just so I have something that I can work on. Uh, even though the methods that maybe I require in solving a problem don't necessarily appear in the picture that I'm drawing. Um, I mean. So to answer your question, whether geometric thinking helps in, um, say, solving a differential equation or maybe more broadly, and in the two areas that I mentioned. Um. I would. Well, I would first have to say that I don't think I'm qualified enough to answer it, but my impression is that most of the time in analysis the proofs come down to some smart estimates or some smart, um, identities that one should recognise or, somehow rather the than the intuition about geometry often is useful the intuition about the quantities that you actually computing with uh or some standard tricks.

  Simone: I mean, because I always have this, this clash when I talk to people who maybe do differential geometry or this or analysis where they there's a lot of estimations and computations going on. There is no computation going on in what I do on a daily basis. And somehow I always struggle a bit to imagine how that feels like, because there is hardly any computation you can do to solve a theorem or to prove a theorem in model theory, you usually have to maybe give some structural argument or appeal to some standard tricks. Um, so somehow what you're, you're saying is, is there is a more quantitative, maybe intuition that people have rather than geometric.

  Mert: Right. Um, I mean, I would imagine, I would imagine that the situation changed actually, as one, gains more experience, um, because as I've said, the, the problems that one deals with, um, well, at least in my case come from physics and there maybe not the geometrical intuition, but the physical intuition plays a greater role. Um, now, certainly if one has a mathematical proof of something, that proof doesn't have to relate to the physical world, but. It is usually the case that the proof is based upon some physical intuition that the author that the mathematician had. And um, for me so far, it has been more or less possible to work without the physical intuition. But I think for everyone wanting to work in an applied field that is, um, quite an important tool that one should have in their arsenal.

  Simone: And so I guess then the natural question is how did you come to an applied field? I mean, what brought you to working in analysis, but not just analysis, but applied to some actual concrete problems like materials.

  Mert: Right. Um, well, actually, analysis was the first math subject that I wanted to learn. Um, and this was back in high school when I was trying to think about what to study in university. And, um, somehow math, um, became, for me, the most reasonable choice. And then I wanted to, before even starting university, see whether I would like it. Um, and then I looked online. I searched what people recommended, and I found this one book that was, um, on analysis. And as I studied more and more from this book, I came to notice that analysis was, for me, full of, um, nice ideas and geometrical intuition, at least at the level I was learning. Um, and then when I got to university, I didn't want to immediately cling onto this one subject, but rather keep my options, um, open. However. Yeah. Due to having a sort of head start in analysis, I felt more comfortable working with it. And ultimately, in my case, it turned out that professors and analysis were more pleasant. Uh uh. Well, they gave more pleasant lectures to attend. Um, and around the same time, I was also getting, um, interested in the idea of seeing how math could be applied to real life, because at a certain point, the math curriculum at the university can get quite abstract. And I noticed that there was a sort of a certain rewarding sense I felt whenever I saw math being applied to a real world problem. Or at least I could find. I found it that for me, it was more interesting to feel attracted to the problem, knowing that it was coming from something physical.

  Simone: There is a natural satisfaction that comes from the fact that you are actually having an impact, which I think is something that maybe people who do very theoretical maths struggle a bit with. Certainly I can say, I mean, or, um, that your math does not have a direct impact. And it's very common to say, well, yes, but I don't know. Alan Turing developing the computer. His theoretical maths also did not have any impact on the world. And see now what happens. Well, okay. Yes. But I don't think we can use exceptions as our motivating examples. And I understand somehow it's, something that Alex was saying is, you can motivate yourself a lot through the fact that you're dealing with a concrete problem that actually has an impact.

  Mert: But I would actually like to argue in another way. For me, of course, it's helpful to know that what I do might be impactful in real life for the development of science or whatever you can call it. But I would say the, the selling point for me was that. I have a more immediate way of seeing why this problem is interesting. Now you have problems in, say, algebraic geometry that require a lot of background information just to be able to understand. And I'm sure if I could understand those problems, I would find most of them interesting as well. It was just the amount of sophistication necessary to appreciate problems from analysis was lower. And it is not to say that the problems have lower quality.

  Simone: Of course not. As we've said before, right? Is this number theory problems that you can state to a child, then they require all the work in the world to be solved. Right. And somehow, in the same way, just because you can appeal to the physical intuition, it doesn't mean that the problem is less interesting or less worthy or less hard, right? I mean, yes, sure. Of course we don't want to discriminate in any way. So what are your hopes for the future of your field now then somehow, what is something that you would really like to be to see solved or to see addressed?

  Mert: Well, I have to admit that I am still quite fresh in my field. So I don't feel qualified, to pick a problem that would be interesting. Now I can say, for example, that if one thinks of materials in a broader sense as, things that are made of continua, such as maybe fluids as well, then the biggest open problem at the moment would probably be the problem relating to Navier Stokes equation. But again, even though. This problem is quite popular. I don't feel quite invested in the problem yet. So it is difficult for me to pick a particular subject. But I've always had certain thoughts about the future of mathematics. You hear often that there isn't a unified theory for partial differential equations, and that each equation is by itself, very special and requires different methods to be dealt with. I mean, this is natural when you think that for every physical system that you can describe, there is a differential equation that would maybe associated with that system. So we cannot hope to maybe find a unified theory for the whole field of partial differential equations. But then I'm reminded of the history of math and how calculus came to develop at a time when there were lots of problems, all requiring a different methods to be solved and how calculus built the ground for, well, not necessarily a unified treatment for each of them, but at least a language to describe all those problems then. I always find it exciting to think what could happen in the future in the field of partial differential equations or more generally, in analysis. In general I would be curious to see something like a new revolution happening, a new kind of calculus appearing and forcing us to think about the things we already know in a different light, but in a way that opens up new avenues of research. Of course, I don't know what that would look like, but it's always, for me at least, interesting to ponder about the possibilities, because we tend to think that what we know is the edge of knowledge and things will remain the way they are. But often we're surprised by advancements in our knowledge.

  Simone: Well, thank you for the lovely insight. And we'll see you in the corridors.

  Mert: Thank you. See you.

In this episode of On A Tangent, Simone is joined by Alex Tullini, a doctoral researcher in General Relativity. We discuss Alex’s journey towards mathematics, going through oranges, eclipses, and the odd similarities between a career in mathematics and surgery.

Link to Alex's website

• Transcript

  Simone: Welcome to On A Tangent, the podcast where the main characters are the stories behind the mathematics. My name is Simone, and in each episode I will meet a different early career mathematician from Muenster to learn about their stories, their paths towards mathematics, and their hopes for the future. In this episode, I am joined by Alex, a PhD student in General Relativity, 'GR' for short, to learn about eclipses, cosmic censorship, and surgery. I hope you enjoy the episode!

  Simone: Hello, Alex and welcome. Thank you for joining us.

  Alex: Hello, Simone. It's really nice to be here.

  Simone: So you're a PhD student in general relativity. So when you go home and your friends ask you, what do you do? What do you tell them?

  Alex: So first, I take a very deep breath. Actually, I think that as a mathematician, being in general relativity makes this job of sharing what I do easier, because it's one thing to say, oh, I'm a model theorist, and I study this and that.

  Simone: This is a personal attack to myself.

  Alex: It is. I mean, it's another thing to mention, say, a black hole, which is much more part of pop culture. And that helps me in some sense. It also saddens me because I realise that I have to be imprecise in order for them to receive the message. But okay, what I typically tell them is that I try to study through math the stability of black hole solutions to Einsteins equations, and this is kind of easy to get through because then people imagine, oh, black holes, there's these things that are described using math and okay, these things are hard, but I can do something with this math. But it creates an idea in their mind typically. Then the next thing they ask me is, so do you make experiments, which I think is something unique to people in my field, I don't think people ask you about experiments.

  Simone: No, definitely not.

  Alex: Okay, then I have to explain to them that what I do cannot really be experimented, but it's more connected to questioning the consistency of a theory. And I do think that what I do, the message really arrives to say my family, my friends. So I think I feel like I'm in a privileged position.

  Simone: I was once reading a message, I think maybe on Twitter by a professor who was saying, you know, everybody learns what a black hole is in high school or what the DNA is, right. Nobody learns what a manifold is in high school, which might be trickier, but somehow it is a bit easier if you can explain it with the pretty pictures.

  Alex: Yeah, and the physical phenomenon maybe is maybe easier to relate to. I think the weird thing is that they don't realise that what I do is still very abstract, and they think of it as being, um, enjoyable in a way that is not really what I go through. Um, there's much more technicality involved, but there's no hope of passing it through, of really creating a picture of this type of technicality. So I just accept that I'm going to mention interstellar and they'll be happy and very excited.

  Simone: Having a mainstream reference for it is a very good attack point.

  Alex: Yeah. I'm almost as lucky as a physicist, let's say, even if I don't really do what they do. I mean, I don't have the abilities to do what they do.

  Simone: So this is you now. Let's take a step back. What is the earliest moment you can remember in your life where maths entered the picture?

  Alex: Um, so I'm afraid I might be unrelatable to the general public when I actually answer this question. But I have to be honest, math has always been with me, and that for whatever reason that I couldn't really understand, among all the things that I like, it was my greatest interest and the first memory of it I that I have, which might seem also unrelated, but to me it was a moment of mathematical interest and understanding was when at some point we were waiting for an eclipse to happen. I was home with my mom and dad, and they just told me, the eclipse is going to happen. And I had some trouble understanding what it what it meant, um, because I couldn't picture in my head the kind of the geometric configuration that would allow for the eclipse to happen. And so I asked my dad to explain to me, and we went, you know, over his big bed, and we took some balls, I think an orange and, like, maybe a lemon or something. And my dad, like, put them on the on the bed and said, okay, this is the solar system, this is the moon, this is the Earth. And what we're going to witness today happens because of this geometric configuration. Right? And for me, that was my first approach with math. And I think it also in some sense it is coherent with the sort of interests I've had later, which were more connected to geometry.

  Simone: I was going to ask, do you think somehow this is what brought you to a field where maybe geometric intuition or imagining visually what's going on is important?

  Alex: I don't know if it brought me. I mean, I don't see it as a cause, rather I see it as a maybe as a symptom of something already there. Maybe I don't have that story of a little kid thinking about numbers, but I was always intrigued by geometry, by a perception of distance, by just configurations of objects in space. And this definitely is something that I also witnessed in the choices I made later during university and uh, and everything that led me to where I am right now.

  Simone: So how did you arrive where you are? How did you choose what to do?

  Alex: Um, so as I said, I was always drawn to it. It almost felt like a choice that was made by someone else. And that for whatever reason, I always felt like, oh, this is actually what I like the most. Then I also liked other things very much. I liked philosophy very much. I really wanted to go into that. I liked ancient languages very much. Part of me wanted to study Latin and Greek.

  Simone: Did you study them in school?

  Alex: I only studied Latin in school, unfortunately. But I made this choice because I thought, okay, if I want to study math, and it is certainly what I want to know the most before anything else. wI also believe that there is a right time to do it, because I felt like my brain would be better off studying math at an earlier age, and then maybe other things at a later age, then the opposite. And so this made me confident that what I felt was right for me, which was to study math, was also objectively, in some sense, the most optimal choice\'85 in terms of long term plans, even if it meant sacrificing other interests that I maybe had. And sometimes I wondered, okay, could I try doing something else? And something which may seem unrelated, but that I was very interested in: I've always wanted to be a surgeon, just because how cool is it? Okay, this is going to sound weird, but to actually see the inside of a body.

  Simone: It's certainly something, right, you don't see every day. It's a bit of a mysterious thing.

  Alex: And also, okay, the idea of having that role where some problems that are very hard to solve, they maybe can be solved through surgery. And somebody that has, you know, studied for years has a special set of skills. They can do that for you. And, um, I try to imagine myself into that career. But then I realised I would have suffered too much the loss of the mathematical knowledge, whereas I could have, you know, accepted the loss of, say, the surgical knowledge, or the knowledge that I've had to sacrifice because of course, there is only one thing you you can reasonably master. I mean, unless you're specially gifted or something.

  Simone: Or you have a lot of free time, and somebody at home that takes care of everything. So this is how you came to math. But how did you come to general relativity then? I mean, you told me you've always been geometrically inclined.

  Alex: Yeah, but I would say I came to GR because at some point I really took into my own hands my journey. I studied in Italy, where, as you know, we tend to have less freedom than in other countries. Which okay it can be both good and bad. It certainly allowed me to see a lot of different math, which was good. But then I reached a point where I realised that even if I was very much interested in geometric problems and certain type of problems specifically, there was something about GR and black holes because of course, I live in the same world as anybody. And as we said before, GR is much more part of pop culture. And of course this has an effect on me too. And at some point I realised, okay, maybe I like it. This is kind of the base, you know, the bottom line, it's that I like it, but it's also something that it is probably easier to maintain interest in because of course, studying math is hard, right? Sometimes you're like, it's the equivalent of walking into a dark room. Um, and at least for me, I thought it was a good move to pursue an interest in something that, um, I can much more easily renew my interest for. Because in some sense, it's at least for me much easier to renew my excitement for black holes than it is to renew my excitement for the homology groups of positively curved manifolds.

  Simone: And also, how cool is it that you can go and tell people "I work on black holes", right?

  Alex: Yes, yes. I mean, let's not hide that. There is, at least for me, an influence in my choices given by the fact that I like to be the person that does this, other than I like to just do it. And of course it is. It is cool. I mean, I like to think of myself as a person that has this interest but at a certain point either you're lucky and the professors and the people around you in the university where you study actually share those interests, or you have to create a path for yourself. And this is what I did. I did it a specific time. I don't think you can do it at all times, like when you're much younger, of course, it's much harder. It certainly would have been too hard for me. But it also was somehow natural because I reached a point where I had a I felt like I had a decent understanding of all the tools needed to pursue this, this interest. And I realised, who are the people that could help me get into this field? And even if they were not in my home university, I looked for ways to get in touch with these people. And I don't know, for whatever reasons, the planets aligned again.

  Simone: Like in the eclipse.

  Alex: Like an eclipse.

  Simone: The orange, the lemon...

  Alex: And somehow I've ended up doing something that that I like very much, which was not the only option here, I want to say again, but it's the one on which I bet.

  Simone: So you moved to Switzerland first and then here.

  Alex: And then I came here, yes. Because there was some previous student of a professor that I had that was somehow between what I was doing before, which was a geometric analysis, and GR, which is what I'm doing right now. And this person was a bridge. But most importantly, moving away from my hometown, which was actually not my hometown, but my home university. It allowed me to meet other people that also made a bridge. And, you know, most of the times, unless you meet the people, you don't even realise that certain areas of research exist. This was my case. I didn't realise that it was possible to study black holes using this set of skills that a mathematician has, which is the set of skills that I have at my disposal. Until I went to Zurich and I met a person that was doing this. So I think overall, other than believing in yourself and kind of for once not doing exactly what you're supposed to do, more than that, you also need to be a bit lucky right, to meet the right person that tells you yes, you can do this. And then you realise, oh, I could do this, and then you do this. Um, so it's a series of factors and. Okay, luck is always a factor.

  Simone: So this is the now. This is why you do it. So what is the future ahead of you that you would like to see? What is a theorem that you would like to see proven well by yourself maybe, or others of course, in the next ten, 20, 30 years.

  Alex: Among all the things that I initially looked at when I started exploring this mathematical general relativity field, there is something specifically that really caught my attention, and that's something called strong cosmic censorship conjecture.

  Simone: Which is a very metal name.

  Alex: I mean, it's very cool. It's very easy to sell. And I mean, they sold it to me effectively. Yes. I mean, reasonably, this is something that is probably gonna take on a hundred years, maybe they won't even be enough because also my field is relatively young compared to other fields, but in essence, what I like about it, and I think this is like further proof of how much more easily I can communicate what I do if compared to other people. Basically, this conjecture says that the theory of general relativity is deterministic, and which you would say, oh, why? Why shouldn't it be deterministic? Well, it turns out that this is not something to take for granted because, okay, in very layman terms, there are some black hole solutions to Einstein equations that present something that's called Cauchy horizon, which I won't go into the the reason why it's called like that, but in essence, it's like a geometric locus where the theory at times appears not to be capable of predicting uniquely what happens to an observer or a person or an object passing this horizon. And this conjecture that I mentioned, the strong cosmic censorship conjecture, in essence, it says that although you can write down solutions of the Einstein's equations which display this behaviour, they are not generic in the sense that if you consider the initial data that give rise to the solution and you perturb them, then once you apply this perturbation, you no longer end up with a spacetime or a solution that that has this feature.

  Simone: So somehow the solutions that don't work like we expect, we expect physical theories to work deterministically, right? Yes, we put in the initial data and we know and we predict what's going to happen. Yes. So the solutions that don't behave like this are just an exception. So they just happen. And then but if you move a bit away from that then it works just fine.

  Alex: Yeah. Like trying to balance a picture upside down. Right. There is an equilibrium point. Ideally you could balance it and put it up on the wall. But you know the moment you perturb it a little bit, it's just going to go down and eventually it's going to stabilise along the other reasonable equilibrium point that you can use to put a picture up on the wall. And this is the same idea. And also I feel like this is the best representation of what it means to do general relativity. As a mathematician with the skills of a mathematician, I don't have the ability to do what the physicists do. But you know, with our set of skills, we can investigate questions such as this one, which they are really about the consistency of the theory. They are really giving meaning to the words determinism of general relativity. And they're doing it in a very reliable way, because I'm really, you know, taking the equations and rigorously trying to prove that the theory is deterministic.

  Simone: And actually predicts our reality as we expect it.

  Alex: Yes, yes. Um, so this would be very nice. Um, yeah. I think whoever puts the last brick on this collective effort is certainly going to be a famous man. Uh, but I don't know how long it will take. I would be happy if I gave my contribution. But not necessarily for the glory of it, which I confess for sure. And for the first years of my journey in math it was part of the motivation. But then I think, certainly once you entered the PhD, what I really want for myself is to have given some contribution and at the same time have a certain set of skills and of knowledge and of understanding that allows me to sit with my peers and give something to them in conversation, right? What I envy is not, say, the list of awards that professors have or may have. I mean, in part, we all envy that, right? But what I envy on a daily basis is the fact that they, they can navigate their department and talk to other people, and they have conversations where there's an actual like exchange between them. They don't just have to sit there and take as much as they can, which is what you just have to do at the beginning, right. Somehow it's also a necessary step, I mean, to have these conversations, right? Because nobody will prove the strong cosmic censorship conjecture or any other big conjecture on their own.

  Simone: Math is necessarily a collective effort that we all have to do together.

  Alex: And then we need to be able to interact and communicate. Yes. I feel like this is in the end, what once you're when you're inside, when you're doing your PhD or doing some research, or maybe if you're just a master's student, you realise that really what gives you the feeling of having owned your spot inside the community? It's really just this, you know, sitting down, having a coffee after lunch and being able to contribute to a conversation. It's not, I don't know, the list of pieces of paper that you've earned. And really it comes from the human interaction, which is something that I only discovered and touched with firsthand once I started doing my PhD. And maybe, maybe this is something that professors and researchers and everybody in charge should try to pass on to students while they're studying, maybe to boost motivation, because I think people think of math as a very lonely career. Unless maybe you're at the top of the top.

  Simone: Yes. Because we have these images, right? We see movies or in or in on also the stories we hear. I mean, of Wiles proving Fermat's Last Theorem alone for eight years.

  Alex: Yeah, and okay, this can happen. But actually the majority of the work is not like this, right? I mean, it's more about really sitting down, having a coffee, discussing what's going on. Slowly. Brick by brick is anybody would do in their generic work like office work. You have your colleagues, you work together on something and you collaborate. If anything, we're luckier because we are actually working for ourselves. And so yeah, I think that is the best part. That is what I also what reasonably I think keeps us motivated because one does not just blindly pursue this career in hope of one day maybe receiving an incredible piece of recognition. It really is a daily life to daily department life that fuels you and it allows you to move forward.
  
  Simone: Yeah. Also because right, you cannot force maths to work.
  
  Alex: No, sometimes it doesn't work.
  
  Simone: Right. Uh, and so sometimes you, you go and go and go at the same problem for, for infinite time and then you still don't get results because that's how it goes. So yeah, that's true.
  
  Alex: I mean, motivation can hardly come from the success because you cannot control it. You cannot predict it. You have to find a way to make the process, if not enjoyable, because maybe this is aiming too high, but sustainable. So a good balance of positive emotions and negative emotions otherwise I mean you're not going to go anywhere because I feel like it's just against the human nature. And it's weird that the narrative that that people receive from the outside is of people somehow going against the human nature, which is probably the reason why those that are depicted, maybe movies are this very, um, um, this genius people or this very, um, uncommon people in some sense. And I feel like this is what makes them uncommon in the eye of the, of whoever is watching. But really, this is far from the reality of the department, or at least of the departments that I've inhabited and navigated.
  
  Simone: Well, thank you very much for this lovely chat.
  
  Alex: See you soon around the corridor.