The recent computation of the K-theory of $\mathbb{Z}/n$ by Antieau, Krause and Nikolaus [1, 2, 3] is a major success. This problem, a well-known open question since the 1970s, was a goal in the research programme of the establishment proposal. This achievement was made possible by the introduction of higher categorical and arithmetic methods into the field of K-theory (see, e.g., [4, 5, 30]) thus showcasing the strength of our integrated approach.

Krause, a former postdoc, was appointed professor at the University of Oslo in August 2024.

Benjamin Antieau, Achim Krause, and Thomas Nikolaus. On the k-theory of $\mathbf{Z}/p^n$. arXiv e-prints, May 2024. arXiv:2405.04329.

Benjamin Antieau, Achim Krause, and Thomas Nikolaus. Prismatic cohomology relative to δ-rings. arXiv e-prints, October 2023. arXiv:2310.12770.

Benjamin Antieau, Akhil Mathew, Matthew Morrow, and Thomas Nikolaus. On the Beilinson fiber square. Duke Math. J., 171(18):3707–3806, December 2022. doi:10.1215/00127094-2022-0037.

Thomas Nikolaus and Peter Scholze. Correction to “On topological cyclic homology”. Acta Math., 222(1):215–218, March 2019. doi:10.4310/ACTA.2019.v222.n1.a2.

Benjamin Antieau and Thomas Nikolaus. Cartier modules and cyclotomic spectra. J. Amer. Math. Soc., 34(1):1–78, January 2021. doi:10.1090/jams/951.

Shimura varieties and moduli spaces of G-shtukas

A major result achieved by Viehmann is the theorem determining the topology on ${\rm Bun}_G$ and the non-emptiness result for intersections of Newton strata and the weakly admissible locus [1], as well as joint work with Nguyen on the weak Harder–Narasimhan stratification [2].

Viehmann was awarded an ERC Consolidator Grant in 2018. For her influential work on arithmetic algebraic geometry as part of the Langlands programme, she received the Leibniz Prize in 2024.

In 2024, Trentin graduated in Viehmann’s group with a remarkable thesis describing affine Deligne–Lusztig varieties in a ramified case that goes beyond the fully Hodge–Newton decomposable situation [3]. This work is a starting point for new approaches in Topic T4.

Eva Viehmann. On Newton strata in the $B_dR^+$-Grassmannian. Duke Mathematical Journal, 173(1):177–225, January 2024. doi:10.1215/00127094-2024-0005.

Kieu Hieu Nguyen and Eva Viehmann. A Harder-Narasimhan stratification of the $B^+_dR$-Grassmannian. Compositio Mathematica, 159(4):711–745, March 2023. doi:10.1112/S0010437X23007066.

Stefania Trentin. On the Rapoport-Zink space for $\mathrm{GU}(2, 4)$ over a ramified prime. arXiv e-prints, September 2023. arXiv:2309.11290.

In set theory, Schindler answered a long-standing open question about the relationship of Martin’s Maximum++ and Woodin’s P_max axiom (*) by showing that Martin’s Maximum++ outright implies (*) [1]. This fundamental result has been awarded the Hausdorff Medal of the European Set Theory Society. Together with the doctoral researchers Sun and Yasuda, Schindler proved generalisations and applications of this result [2], and Lietz exploited the proof method to answer another long-standing open problem about being able to force the non-stationary ideal on $\omega_1$ to be dense [3]. Lietz graduated in Schindler's group in 2023 and won the Sacks Prize for the most outstanding doctoral thesis in mathematical logic.

David Asperó and Ralf Schindler. Martin's Maximum++ implies Woodin's axiom ($\ast$). Ann. Math., 193(3):793–835, May 2021. doi:10.4007/annals.2021.193.3.3.

Ralf Schindler and Taichi Yasuda. Martin's maximum${}^{\ast, ++}_{\mathfrak{c}}$ in $\mathbb{P}_{\max}$ extensions of strong models of determinacy. arXiv e-prints, April 2024. arXiv:2404.12836.

Andreas Lietz. Forcing "$\mathrm{NS}_{ω_1}$ is $ω_1$-dense" from large cardinals. arXiv e-prints, March 2024. arXiv:2403.09020.

Constructions for sharply 2-transitive groups & model-theoretic approach to perfectoid fields

On the side of group constructions, Tent with co-authors, including Amelio (doctoral researcher in Tent's group) , and André (now associate professor at Sorbonne Universtiy), succeeded in providing various constructions for sharply 2-transitive groups in characteristic 0 and in large odd characteristic [1, 2, 3]. This also required a better understanding of the free Burnside groups for odd exponent, for which they provided the best currently known lower bound in [4]. The model theoretic aspects of sharply 2-transitive groups were investigated in [5].

Applying model-theoretic methods towards arithmetic geometry, Jahnke with co-authors developed a model-theoretic approach to perfectoid fields, gaining an unexpected understanding of which first-order properties (un)tilt [6]. Moreover, they generalised the classical Ax–Kochen/Ershov results for unramified fields to both the imperfect residue field as well as the finitely ramified settings and obtained a characterisation of henselian NIP fields [7].

Marco Amelio, Simon André, and Katrin Tent. Non-split sharply 2-transitive groups of odd positive characteristic. arXiv e-prints, December 2023. arXiv:2312.16992.

Simon André and Katrin Tent. Simple sharply 2-transitive groups. Transactions of the American Mathematical Society, 376(06):3965–3993, June 2023. doi:10.1090/tran/8846.

Eliyahu Rips and Katrin Tent. Sharply 2-transitive groups of characteristic 0. J. Reine Angew. Math., 2019(750):227–238, May 2019. doi:10.1515/crelle-2016-0054.

Agatha Atkarskaya, Eliyahu Rips, and Katrin Tent. The Burnside problem for odd exponents. arXiv e-prints, April 2023. arXiv:2303.15997.

Tim Clausen and Katrin Tent. Mock hyperbolic reflection spaces and Frobenius groups of finite Morley rank. arXiv e-prints, April 2021. arXiv:2104.10096.

Franziska Jahnke and Konstantinos Kartas. Beyond the Fontaine-Wintenberger theorem. arXiv e-prints, April 2023. arXiv:2304.05881.

Sylvy Anscombe and Franziska Jahnke. Characterizing NIP henselian fields. Journal of the London Mathematical Society, March 2024. doi:10.1112/jlms.12868.

Non-linear stability in general relativity

In the first funding period, we significantly strengthened the interplay between differential geometry, mathematical physics and PDEs through the newly installed Bridging the Gaps – Alexander von Humboldt professorship of Holzegel in general relativity. Concretely, [1] connects to the so-called ‘AdS-CFT correspondence’ in theoretical physics, and Holzegel’s work with co-authors on stability [2] exploits crucial insights of the physics literature of the 1970s. Together with co-authors, Holzegel completed the proof of the non-linear stability of the Schwarzschild spacetime in general relativity [3]. This work will serve as a blueprint to attack a famous conjecture in general relativity and PDEs, namely to establish the non-linear stability of the Kerr family of solutions, cf. Topic T6. Holzegel was awarded an ERC Consolidator Grant in 2018.

Gustav Holzegel and Arick Shao. The bulk-boundary correspondence for the Einstein equations in asymptotically anti-de Sitter spacetimes. Archive for Rational Mechanics and Analysis, 247(3):56, May 2023. doi:10.1007/s00205-023-01890-9.

Mihalis Dafermos, Gustav Holzegel, Igor Rodnianski, and Martin Taylor. Quasilinear wave equations on asymptotically flat spacetimes with applications to Kerr black holes. arXiv e-prints, December 2022. arXiv:2212.14093.

Mihalis Dafermos, Gustav Holzegel, Igor Rodnianski, and Martin Taylor. The non-linear stability of the Schwarzschild family of black holes. arXiv e-prints, April 2021. arXiv:2104.08222.

Proof of the Hopf conjecture under symmetry assumptions

Kennard, Wiemeler, and Wilking proved the Hopf conjecture of 1931 under very mild symmetry assumptions [1, 2]: the Euler characteristic of a Riemannian manifold with positive sectional curvature must be positive provided that it admits an effective and isometric action of $T^5=(S^1)^5$. Nienhaus was able to lower the symmetry assumption in his thesis to $T^4$-actions. This builds on an intense interplay between differential geometry, algebraic topology, representation theory of tori and matroids. Longterm goals are the proof of the Hopf conjecture assuming only an effective, isometric $S^1$-action and of an old conjecture of Yau claiming that a positively curved Riemannian manifold indeed admits an $S^1$-action. Wilking holds a Leibniz Prize and was awarded the Staudt Prize in 2022.

Lee Kennard, Michael Wiemeler, and Burkhard Wilking. Splitting of torus representations and applications in the Grove symmetry program. arXiv e-prints, June 2021. arXiv:2106.14723.

Lee Kennard, Michael Wiemeler, and Burkhard Wilking. Positive curvature, torus symmetry, and matroids. arXiv e-prints, December 2022. arXiv:2212.08152.

Proof of cases of the Farrell-Jones conjecture & disproof of the unit conjecture for group rings

As major contributions to long-standing open questions in topology, Bartels–Bestvina proved the Farrell–Jones conjecture for mapping class groups [1]. Moreover, together with Lück, Bartels has extended results for the Farrell–Jones conjecture from group rings of discrete groups to Hecke algebras of totally disconnected groups [2], proving a conjecture of Dat on the K-theory of rational Hecke algebras of reductive p-adic groups and thereby connecting methods developed in topology to smooth representation theory and the Langlands programme.

Gardam, before he received a professorship in Bonn, provided as postdoctoral researcher a concrete counterexample to the long-standing unit conjecture for group rings [3]. This conjecture was first formulated by Higman in 1940 and popularised by Kaplansky in the 50s. Today he is a professor in Bonn.

Arthur Bartels and Mladen Bestvina. The Farrell-Jones conjecture for mapping class groups. Invent. Math., 215(2):651–712, January 2019. doi:10.1007/s00222-018-0834-9.

Arthur Bartels and Wolfgang Lueck. Algebraic K-theory of reductive p-adic groups. arXiv e-prints, June 2023. arXiv:2306.03452.

Giles Gardam. A counterexample to the unit conjecture for group rings. Ann. Math., 194(3):967–979, November 2021. doi:10.4007/annals.2021.194.3.9.

Structure and classification of C*-algebras

Major advances were achieved on the structure and classification theory of nuclear C*-algebras and their underlying dynamics. A particular focus lay on topological dynamical systems and the classifiability of their associated C*-algebras, both in the stably finite setting, by Kerr and Naryshkin [1, 2, 3], and purely infinite setting, by Geffen, Gardella, Kranz, and Naryshkin [4].

A breakthrough of Winter with co-authors was the proof that the nuclear dimension of separable, simple, unital, nuclear, Jiang–Su-stable C*-algebras is finite [5]. This makes classification accessible from Jiang–Su-stability and in particular brings large classes of C*-algebras associated to free and minimal actions of amenable groups on finitedimensional spaces within the scope of the Elliott classification programme.

A notion of nuclear dimension for diagonal pairs of C*-algebras was established in [6] to express dynamical dimension-type properties entirely in terms of the associated C*-algebra pair. For free actions of amenable groups on Cantor spaces, the diagonal dimension agrees with Kerr’s tower dimension.

Winter was awarded an ERC Advanced Grant in 2019.

David Kerr and Petr Naryshkin. Elementary amenability and almost finiteness. arXiv e-prints, July 2021. arXiv:2107.05273.

Petr Naryshkin. Polynomial growth, comparison, and the small boundary property. Adv. Math., 406:108519, September 2022. doi:10.1016/j.aim.2022.108519.

Petr Naryshkin. Group extensions preserve almost finiteness. Journal of Functional Analysis, 286(7):110348, April 2024. doi:10.1016/j.jfa.2024.110348.

Eusebio Gardella, Shirly Geffen, Julian Kranz, and Petr Naryshkin. Classifiability of crossed products by nonamenable groups. J. Reine Angew. Math., 797(0):285–312, April 2023. doi:10.1515/crelle-2023-0012.

Jorge Castillejos, Samuel Evington, Aaron Tikuisis, Stuart White, and Wilhelm Winter. Nuclear dimension of simple C∗-algebras. Inventiones mathematicae, 224(1):245–290, April 2021. doi:10.1007/s00222-020-01013-1.

Kang Li, Hung-Chang Liao, and Wilhelm Winter. The diagonal dimension of sub-C∗-algebras. arXiv e-prints, March 2023. arXiv:2303.16762.

Research highlights in quantum field theories & stochastic analysis

The construction of non-linear quantum field theories (QFTs) in critical dimension is an enormous challenge. In the first funding period, Grosse, Hock and Wulkenhaar found the exact solution of the planar sector of a QFT on a non-commutative geometry of critical dimension, proved that the non-linearity reduces the effective dimension into the subcritical regime [1] and discovered a recursive procedure for an approximation by finite matrices [2,3] that provides the exact solution of any topological sector.

QFTs can be analysed through the long-term behaviour of specific non-linear stochastic PDEs. Hairer and Gubinelli’s groundbreaking work developed a renormalisation procedure and a local-in-time well-posedness theory. Jointly with co-authors, Weber improved these local-in-time results and studied the large-scale behaviour of solutions through suitable a-priori-estimates [4, 5, 6], thereby achieving the first construction of the Euclidean ϕ4-theory in three dimensions using PDE methods.

In Topic T7 we aim to combine these groundbreaking recent results to achieve the very first construction of a non-linear critical QFT. Weber was awarded an ERC Consolidator Grant in 2022.

Harald Grosse, Alexander Hock, and Raimar Wulkenhaar. Solution of the self-dual $\Phi^4$ QFT-model on four-dimensional Moyal space. J. High Energy Phys., 01:081, January 2020. doi:10.1007/jhep01(2020)081.

Johannes Branahl, Alexander Hock, and Raimar Wulkenhaar. Blobbed topological recursion of the quartic Kontsevich model I: Loop equations and conjectures. Commun. Math. Phys., 393(3):1529–1582, August 2022. doi:10.1007/s00220-022-04392-z.

Jörg Schürmann and Raimar Wulkenhaar. An algebraic approach to a quartic analogue of the Kontsevich model. Mathematical Proceedings of the Cambridge Philosophical Society, 174(3):471–495, May 2023. doi:10.1017/S0305004122000366.

Ajay Chandra, Augustin Moinat, and Hendrik Weber. A priori bounds for the $\Phi^4$ equation in the full sub-critical regime. Archive for Rational Mechanics and Analysis, 247(3):48, May 2023. doi:10.1007/s00205-023-01876-7.

Jean-Christophe Mourrat and Hendrik Weber. The dynamic $\Phi^4_3$ model comes down from infinity. Comm. Math. Phys., 356(3):673–753, October 2017. doi: 10.1007/s00220-017-2997-4.

A. Moinat and Hendrik Weber. Space-Time Localisation for the Dynamic $\Phi^4_3$ Model. Comm. Pure Appl. Math., 73(12):2519–2555, July 2020. doi: 10.1002/cpa.21925.

New formulas in stochastic geometry

In stochastic geometry, Kabluchko derived explicit formulas for the expected angle sums of the $d$-dimensional simplex whose vertices are $d+1$ points placed uniformly at random in the unit ball in $\mathbb R^d$ or on the unit sphere in $\mathbb R^d$. On this basis, he obtained explicit formulas for the expected number of faces of a random polytope generated by $n$ points placed uniformly at random in the ball or on the sphere [2]. He also determined explicitly the expected face numbers of the typical cell in the Poisson-Voronoi tessellation [2] and of the zero cell in the Poisson hyperplane tessellation [1], which were previously known only in low dimensions and some simple special cases.

Zakhar Kabluchko. Angles of random simplices and face numbers of random polytopes. Adv. Math., March 2021. doi:10.1016/j.aim.2021.107612.

Zakhar Kabluchko. Expected f-vector of the Poisson zero polytope and random convex hulls in the half-sphere. Mathematika, 66(4):1028–1053, August 2020. doi:10.1112/mtk.12056.

Numerics in high-dimensional non-linear geometric spaces

Numerics in high-dimensional non-linear geometric spaces, whose importance for data analysis and exploration continually increases, are notoriously challenging. Wirth and coauthors developed and analysed fundamental corresponding tools, e.g. spline interpolation or curvature exploration [1, 2].

Data and geometry processing require input from learning theory, geometry, variational analysis, shape space and PDE methods. Wirth and co-authors, combined these different viewpoints to develop and analyse numerical methods for exploring and simplifying data and shape manifolds [3] (cf. Topic T10). Wirth is a recipient of the Alfried Krupp Award.

Hanne Hardering and Benedikt Wirth. Quartic $L^p$-convergence of cubic Riemannian splines. IMA J. Numer. Anal., 42(4):3360–3385, October 2022. doi:10.1093/imanum/drab077.

Alexander Effland, Behrend Heeren, Martin Rumpf, and Benedikt Wirth. Consistent curvature approximation on Riemannian shape spaces. IMA J. Numer. Anal., 42(1):78–106, January 2022. doi:10.1093/imanum/draa092.

Juliane Braunsmann, Marko Rajković, Martin Rumpf, and Benedikt Wirth. Convergent autoencoder approximation of low bending and low distortion manifold embeddings. ESAIM: Mathematical Modelling and Numerical Analysis, 58(1):335–361, January 2024. doi:10.1051/m2an/2023088.

Deep Learning methods for high-dimensional PDEs

Jentzen and co-authors made pioneering contributions regarding the design and analysis of deep learning (DL) approximation methods for solving mathematical equations such as PDEs. They were the first to propose and design DL methods for solving high-dimensional PDEs [1]. The article generated a major impact in the scientific community with many follow-up works, and it had been recognized by a Frontiers of Science Award in Mathematics at the 2024 International Congress of Basic Science. They also made fundamental contributions regarding the rigorous mathematical analysis of such DL methods for PDEs.
In particular, they revealed that artificial neural networks (ANNs) can provably overcome the curse of dimensionality (COD) in the numerical approximation of PDEs in the sense that the number of parameters of the approximating ANN grows at most polynomially in, both, the reciprocal of the prescribed approximation accuracy and the PDE dimension [2, 3]. These mathematical contributions were presented within Jentzen’s prize lecture for the Felix–Klein–Prize at ECM 2021. In addition, Jentzen was awarded an ERC Consolidator Grant in 2022.

Jiequn Han, Arnulf Jentzen, and Weinan E. Solving highdimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci. USA, 115(34):8505–8510, August 2018. DOI: 10.1073/pnas.1718942115.

Philipp Grohs, Fabian Hornung, Arnulf Jentzen, and Philippe von Wurstemberger. A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black–Scholes partial differential equations. Memoirs of the American Mathematical Society, April 2023. doi:10.1090/memo/1410.

Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse, Tuan Anh Nguyen, and Philippe von Wurstemberger. Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations. Proc. A., 476(2244):20190630, 25, December 2020. doi:10.1098/rspa.2019.0630.