An approximation to the research of the 2024 Leibniz Prize winner

Prof. Dr. Eva Viehmann, investigator at Mathematics Münster, has been awarded the Gottfried Wilhelm Leibniz Prize 2024, the most highly endowed German research funding prize. The German Research Foundation (DFG) particularly highlighted Eva Viehmann's influential work on arithmetic algebraic geometry in the context of the Langlands programme.

On this website, we would like to present this research field in more detail. We have asked mathematicians from Eva Viehmann's environment for contributions. In order to make these insights accessible to as many interested persons as possible, we are approaching the topic in three levels:

Level 2 - for maths students "Affine Deligne-Lusztig varieties"by Prof. Dr. Urs Hartl who has published several papers together with Eva Viehmann over the past 15 years

The mathematical landscape of the Langlands Programme

An approach for mathematical laypersons by Dr. Stefania Trentin

In theoretical Mathematics we have three major areas of study: Algebra, Geometry and Analysis. At a first glance these may seem quite different and far from each other, as they deal with different objects and try to answer different questions. It is indeed a very ambitious program, the one set by Langland, which aims to find a unifying structure behind these three disciplines. In order to understand its deepness, we first try to give some intuition on the questions on which Algebra, Geometry and Analysis lie their focus.

One can think of Algebra as a generalization of the arithmetic we learn in school: instead of adding or multiplying numbers one can define new operations involving also more general expression, like polynomials. An every-day example of what we mean by defining new operations is the time of day: if it is now 22 o’clock and I want to sleep for 10 hours I’ll have to wake up at 8 o’clock. This means that in the algebraic system of the clock 22 + 10 = 8 and not 32! Of course, we do not think about the algebra lurking behind these computations when we set our alarm, but this is a good example of what algebraist like to do, that is defining new objects and operations and studying their sometimes surprising properties.

Geometry is probably the one of the three fields we have more visual intuition about. It is one of the oldest branches of mathematics and it originates from the very concrete need of measuring and modeling the physical world. Modern geometry is today however more abstract even though it still has many applications to other sciences.

Last, Analysis is the branch of mathematics which studies functions. We can think of a function in terms of its graph. An example of graph is the curve representing the evolution in time of the position of a particle in space or of its velocity. One can ask question such as will its velocity stabilize to zero, that is will the particle eventually stop?

From these first examples we can get the impression that Algebra, Geometry and Analysis are three separate disciplines, which never interact with each other. This is however not the case, and the borders between them are much more fluid and allow for common areas of interplay. For example, the region at the intersection between Algebra and Geometry is called algebraic geometry and studies curves and surfaces defined by polynomial equations. One particularly interesting family of such curves are elliptic curves, which nowadays play also a fundamental role in cryptography, the discipline concerned with security and privacy of communication. Elliptic curves have a double nature: on one hand they are geometric objects, as they are curves, on the other hand they are algebraic objects as one can define an operation similar to the usual sum of two numbers but between two points on a elliptic curve. This double nature is particularly fascinating for mathematicians as it blends two different disciplines in one object.

There are many more examples of mathematical objects that can be studied from more than one point of view and that provide a common ground for the interaction of Algebra and Geometry or for example of Geometry and Analysis. As mathematicians like to find abstract structures, seeing such double-natured objects raises in them the question whether there is a unifying structure or philosophy able to explain all three, Algebra, Geometry and Analysis. This is a very deep and fascinating question that is the core of the Langlands program.

We can put this into more mundane terms: image you have two good friends, which you think would make such a great couple and so you want to introduce them to each other. Well, what you could do is for example to throw a party and invite both of them. This is somehow what the Langlands program aims to do: finding a common ground where Algebra, Geometry and Analysis can meet and interact with each other in order to study their interplay and look for common or unifying structures. In this metaphor the two friends are Algebra and Geometry and the party are certain special curves, called Shimura Varieties, which are a generalization of the fascinating elliptic curves we have mentioned above.

Text written by Dr. Stefania Trentin, former doctoral candidate of Eva Viehmann at TU München and University of Münster.

An approach for mathematical laypersons by Dr. João Lourenço

The German mathematician Gauss once described number theory as the queen of all sciences. This is because, while we encounter during our school years many important sets of numbers such as the rationals $\mathbb{Q}$, the reals $\mathbb{R}$, or the complex numbers $\mathbb{C}$, or study graphs of continuous real-valued functions $f\colon \mathbb{R}\to \mathbb{R}$, number theory focuses on the most foundational part of mathematics, namely the integers $\mathbb{Z}$ themselves.

However, you should not think that those more advanced concepts are suddenly useless when it comes to number theory. On the contrary, mathematicians have often learned that one can only actually answer a question or parts of it by significantly enlarging the scope of the theory, i.e., by looking at new objects we might be able to say something about the old ones. In other words, what distinguishes number theory from other areas of mathematics is often not so much its means, but rather its final destination.

The ancient Greek mathematician Diophantos initiated the study of so-called Diophantine equations, i.e., the search of integer zeroes $(a_1,\dots, a_n) \in \mathbb{Z}^n$ of polynomials $P(x_1,\dots, x_n) \in \mathbb{Z}[x_1,\dots,x_n]$. The famous example $x^n+y^n=z^n$ underlies Fermat's last theorem proved 30 years ago by the British mathematician Andrew Wiles. Already for one variable, the question is sufficiently enticing. In school, we are taught how to solve quadratic equations $ax^2+bx+c=0$ via the closed formula
\[ x=-\frac{b\pm \sqrt{\Delta}}{2a}\]
with $\Delta=4ac-b^2$ being the discriminant. Here, one already sees how allowing for irrational numbers such as square roots can help determine the existence of integer solutions. For the entirety of the classical and medieval period, cubic and quartic equations remained out of reach, until the Italian mathematicians Cardano, Ferrari and Tartaglia published closed formulas by radicals for their zeroes in the XVIth century: these developments led to the discovery of the complex numbers $\mathbb{C}$ as square roots of negative numbers appear as an intermediary step. Starting with the quintic however, closed formulas can no longer be found by a theorem of Abel and Ruffini.

At that point, the French mathematician Galois operated a transformation in the field. He realized that rather than caring so much about explicit solutions, one should consider the field extension $F/\mathbb{Q}$ spanned by the solutions of a given rational polynomial and consider its Galois group $\mathrm{Gal}(F/\mathbb{Q})$ of symmetries, i.e., automorphisms $\sigma \colon F \to F$ that respect addition and multiplication. This could be used to explain the quintic impossibility above in terms of simple group theory and it forever turned number theorist's attention to Galois groups. A pivotal development in our understanding of them was made by Artin, who gave an explicit description of {\it commutative} Galois groups in terms of arithmetic invariants of number fields, thereby generalizing earlier work of Gauss on quadratic extensions.

But mathematicians did not want to quit their search just yet. Namely, it remained the problem of understanding non-commutative Galois groups, of which there are many. The guiding principle for this is that one should start looking for vector spaces where these groups act. This is called a Galois {\it representation} and a lemma of Schur says that the commutative case is covered by $1$-dimensional vector spaces. The $2$-dimensional case was therefore the next step. In order to arrive at Galois representations, we need certain inputs from geometry. The key objects are certain planar curves given by cubic equations
\[ y^2=x^3+ax+b\]
called elliptic curves. If we look at the geometric locus consisting of the complex points $E(\mathbb{C})$ of an elliptic curve $E$, we get a donut shaped space (note that the real dimension is twice the complex dimension, so there is no contradiction in our terminology). However, the set $E(\mathbb{Q})$ of rational points of $E$ is rather discrete in nature, and it has key arithmetic significance. For instance, it can carry a lot of torsion, especially as we enlarge our search to algebraic points $E(\bar{\mathbb{Q}})$ and these torsion subgroups are used to realize a Galois action on a $2$-dimensional vector space.

To continue even further, we need another idea from geometry. (At this point, it should become clear to the reader that number theory really draws a lot from many different areas in mathematics; its recent debt to algebraic geometry is so humongous that the practitioners in this field started naming themselves arithmetic geometers.) Suppose there is a certain class of special algebraic varieties that you wish to study. Then under favorable conditions, it should be possible to regard this class as an algebraic variety itself, and we call it a {\it moduli space}. One of the first examples ever studied were the moduli spaces $X_1(N)$ of elliptic curves equipped with $N$-torsion points, also called {\it modular curves}. By looking at spaces of differential forms on $X_1(N)$, one obtains {\it modular forms}. Their Fourier series have algebraic integer coefficients (up to renormalization) and lead to wonderful arithmetic properties studied by Jacobi and Ramanujan. Again, modular forms contribute to the study of Galois groups via associated Galois representations by work of Deligne. The Shimura--Taniyama conjecture proved by Wiles (and which finally furnished a proof of Fermat's last theorem) stated that every (semi-stable) elliptic curve over $\mathbb{Q}$ relates to a modular form in terms of their Galois representations.

In higher dimensions, modular curves are replaced by certain real manifolds (which unfortunately do not always have a natural complex structure) and one gets a corresponding notion of {\it automorphic forms}. The Galois groups act on the vector spaces of automorphic forms, giving us {\it automorphic representations}. Langlands' vision, derived from important calculations orbital integrals of automorphic forms, was that there should be some mysterious correspondence between automorphic representations and Galois representations. Constructing a bridge between these two world is a vast program that has occupied number theorists for the last half century.

Finally, we would like to explain one last ingredient behind recent progress in the Langlands program. We have seen how important it can be to enlarge our set of numbers from $\mathbb{Z}$ to $\mathbb{Q}$ or even $\bar{\mathbb{Q}}$, $\mathbb{R}$, and $\mathbb{C}$. But sometimes it can also be important to reduce them, by considering the rings $\mathbb{Z}/n\mathbb{Z}$ whose elements are integers up to congruence modulo $n$, i.e. we declare the integers $a,b$ to have the same image in $\mathbb{Z}/n\mathbb{Z}$ if they are divisible by $n$. If $n=p$ is prime, then $\mathbb{Z}/p\mathbb{Z}=:\mathbb{F}_p$ is actually a {\it field}, meaning every non-zero element has an inverse. Congruences have been used in number theory for centuries, as they help in excluding certain types of solutions to Diophantine equations. Hensel took the idea further by considering the limit $\mathbb{Z}_p:=\mathrm{lim}_n \mathbb{Z}/p^n\mathbb{Z}$ of all congruences modulo powers of primes. This construction is similar to the construction of $\mathbb{R}$ as the completion of $\mathbb{Q}$ for the usual norm, in terms of equivalence classes of Cauchy sequences, except now $p^n \to 0$ as $n \to \infty$ for this $p$-adic norm. We similarly get a field $\mathbb{Q}_p$ of $p$-adic numbers by completing $\mathbb{Q}$ with respect to the $p$-adic norm. The advantage of the $p$-adics is being now able to perform limiting arguments as in real analysis, even if they behave in rather different ways. These fields are understood as the {\it local} avatars of $\mathbb{Q}$ for every prime $p$, and together with the reals $\mathbb{R}$, they have become an integral part of number theory in the XXth century.

For a long time, it remained however difficult to understand how to perform some type of algebraic or analytic geometry over the $p$-adics in a way suitable to number theory. The important thing is that one needs to be able to work in {\it families}, to construct new $p$-adic moduli spaces. Many important discoveries were made by older mathematicians such as Bosch, Lütkebohmert, Kiehl, Raynaud, Tate in the field of so-called rigid-analytic geometry, a part of what today is called $p$-adic geometry, but unfortunately they had to work under many finiteness assumptions that made it impossible to reach the key moduli spaces. Advances were made recently by Bhatt, Fargues, Fontaine, Scholze, Zhu in the foundations of this $p$-adic geometry that have now opened the doors to number theorists to exploit the many possibilities of the $p$-adic world. It is within this setting that Eva Viehmann has made significant contributions to arithmetic geometry in roughly two decades of activity. Her research has focused on understanding the geometry of these $p$-adic moduli spaces and certain types of numerical invariants called {\it cohomology groups} associated with them, and her results are inextricably interwoven in the development of the area.

Text written by Dr. João Lourenço. João Lourenço is a postdoc ("Akademischer Rat auf Zeit") in Eva Viehmann's research group at the University of Münster.

An approach for maths students by Prof. Dr. Urs Hartl

Eva Viehmann works in Arithmetic Algebraic Geometry on Shimura-Varieties and moduli varieties of $G$-Shtukas. Both are much studied objects in algebraic geometry. They have a rich structure given by the intertwined actions of various groups and are of great importance for the Langlands program. To study their reduction over finite fields one uses affine Deligne-Lusztig varieties, which are one of the central research topics of Eva Viehmann. Let $k$ be an algebraically closed field of positive characteristic $p$ and let $L = k((z))$ be the formal power series field. Let $G = GL_n$ over $L$ or a subgroup thereof, such as $SL_n$ or $SO_n$. Let $b$ be an element of $G(L)$ and $\sigma$ be the Frobenius automorphism of $L$ and $G(L)$ with $\sigma(x) = x^p$ for $x \in k$ and $\sigma(z) = z$. An affine Deligne-Lusztig variety is defined as
\[
X_\mu(b)(k) := \bigl\{\, g \in G(L)/G(k[[z]]) : \tau = g^{-1} b \sigma(g) \mbox{ is bounded by }\mu \, \bigr\}\,.
\]
Here, the boundedness by $\mu$ for $G = GL_n$ means, for example, that the matrix $\tau$ has entries in $k[[z]]$. For general groups $G$, $\mu$ is a cocharacter of $G$.

Eva Viehmann proved that affine Deligne-Lusztig varieties are algebraic varieties over $k$ and determined their dimension, connectedness and irreducible components. These varieties are so-called Rapoport-Zink spaces for $p$-divisible groups in mixed characteristic and for local $G$-Shtukas in equal characteristic. In mixed characteristic, these spaces were constructed and studied like Shimura varieties in the 20th century. In contrast, Eva Viehmann is the founder of the theory of local $G$-Shtukas and the originator of Rapoport-Zink spaces in equal characteristic. Over the last fifteen years, this has led to the development of arithmetic geometry with objects, phenomena and questions on a par with the arithmetic of Shimura varieties. And it also served as the foundation from which Peter Scholze drew the inspiration for his epochal new developments in mixed characteristics.

Text written by Prof. Dr. Urs Hartl.

Urs Hartl is Professor of Pure Mathematics at the University of Münster. He conducts research in arithmetic algebraic geometry, including on moduli spaces of G-Shtukas, and has published several papers together with Eva Viehmann over the past 15 years.

Geometric representation theory and p-adic geometry

An approach for the mathematical community by Dr. João Lourenço

Abstract: We discuss the number theoretic origins of the Langlands program, its geometrization and categorification over function fields, and more recently over $p$-adic fields by Fargues-Scholze. We conclude by describing some of our own contributions to the emerging field and possible future directions.