This workshop on cohomology of finite groups has traditionally a special emphasis on interactions with neighboring fields, such as group theory, algebraic topology, commutative algebra, and representation theory. This subject has been the topic of five workshops held at Oberwolfach in the last twenty-five years. The workshop aims to highlight recent progress and explore new connections, fostering the continued development of the interactions that have been so productive in recent years. Special emphasis was given to topics which have recently seen important breakthroughs, such as higher representation theory, homotopy theory of permutation modules, and the solution of a conjecture of Quillen.
Flows on measure spaces have long been examined in stochastic analysis and have recently attracted significant interest in machine learning, leading to intriguing research questions that often fall outside the scope of existing theory. Normalizing flows, score-based diffusion, and flow matching models are among the most powerful generative neural methods and rely on the geometry of measure spaces. In particular, the Wasserstein metric and optimal transport techniques have advanced the field in recent years. However, involving different Riemannian-like metrics on measure spaces, e.g., by the framework of right-invariant metrics on the group of diffeomorphisms and their action on objects, e.g., densities, and designing transport inference functionals with advanced properties like equivariance led to new neural models. Generative models can be conditioned on (degraded) data, which leads to new developments in the solution of Bayesian inverse problems. Viewing transformers as interacting particle systems introduced a new mathematical perspective on these complex systems and shed light on their clustering behavior. Finally, learning neural models comes with new challenges in (stochastic) optimization, such as accelerated optimization, operator splitting, and mirror descent on measure spaces, ensemble filtering methods, the treatment of high dimensions via slicing or Fourier random features, as well as scalability questions and related lifting to infinite-dimensional spaces. The workshop will bring together scientists interested in different aspects of flows on measure spaces to further understand and develop their analysis, in particular to address questions in deep generative learning and to develop improved optimization methods for measure spaces.
The core topic of the workshop was the representation theory of quivers and finite-dimensional (associative) algebras with a focus on internal structures of module categories, preprojective algebras, geometric aspects of quiver representations, Cohen–Macaulay representations, and incidence algebras, posets and combinatorics. The workshop also covered several links to other areas of mathematics, including homological algebra, commutative algebra, algebraic geometry, Fukaya categories of surfaces, and Lie theory.
Quantum information theory deals with the way information can be exchanged using the laws of quantum mechanics. The field is relatively young, but has seen an extaordinary rapid development over the past decade. The particular focus of this workshop is the analysis of quantum information theory and its connections to infinite dimensional systems and operator algebras. On the other hand the theory of noncommutative harmonic analysis has established itself further in a parallel development, with successful attempts at generalizing the theory of singular integrals and Fourier multipliers in the noncommutative realm. While there are close ties between the mathematics underpinning noncommutative harmonic analysis and quantum information theory, the two fields have so far remained largely disjoint, despite ample evidence that there could be a fruitful cross pollination between the two. In this workshop we have brought together some of the leading experts in both fields, as well as talented young mathematicians.
Combinatorics is a thriving branch of Mathematics with strong links to other areas of Mathematics and the sciences. Many of the modern techniques employed in Combinatorics draw on diverse ideas from Probability, Analysis, Geometry, Topology, and conversely, combinatorial approaches often lead to new insights in these other areas, and in the broader scientific arena (particularly Theoretical Computer Science). The most recent years in Combinatorics have been particularly exciting with spectacular solutions of many longstanding open problems, via an array of new ideas that promise to have many future applications. Leading experts and new talents came together to share and further develop these exciting ideas. Talks emphasized recent breakthroughs in discrete geometry, graph theory, combinatorial applications in Fourier analysis, group theory, combinatorial probability, Ramsey theory, nonabelian additive combinatorics, and design theory.
Modern machine learning systems exhibit a range of phenomena that are not adequately explained by classical theories. The workshop examined recent progress toward a mathematical understanding of these phenomena, bringing together researchers with expertise in probability, analysis, geometry, optimization, statistics, and theoretical computer science. Discussions addressed complexity and parametrization, optimization dynamics, representation learning, robustness, and the mathematical structure of modern architectures. Particular emphasis was placed on identifying intrinsic geometric, probabilistic, and computational principles underlying learning systems. The workshop fostered interactions across disciplines and highlighted a number of open problems and future directions for the mathematical foundations of machine learning.
Algebraic K -theory is a generalization of linear algebra to rings and to geometric objects, leading to numerous applications across various mathematical fields. It plays a crucial role in number theory, algebraic and geometric topology, algebraic geometry, and analysis. Moreover, algebraic K -theory is intimately connected with motivic cohomology and motivic homotopy theory, which provide a deeper structural understanding of algebraic K -groups.Recent advancements in the field have been significantly driven by the algebraic calculus of \infty -categories. This abstract framework has yielded substantial benefits, including applications to p -adic Hodge theory through the computation of p -adic K -theory and the utilization of trace methods, which effectively linearize algebraic K -theory. The perspective of stable \infty -categories has also enhanced our understanding of algebraic K -theory with respect to various localization constructions.Furthermore, algebraic K -theory has made remarkable contributions to stable homotopy theory, notably in relation to the telescope conjecture. These advances and their implications will be thoroughly explored at the upcoming workshop. The event will feature several presentations on the applications of algebraic K -theory to neighboring areas of mathematics, showcasing the field’s broad impact and ongoing developments.
Coding theory is concerned with the design and analysis of error-correcting codes, a method for protecting data from noise or corruption. In addition to their wide practical applicability, error-correcting codes are also supported by a rich theory, with connections to diverse disciplines in mathematics, science, and engineering. This workshop focused on exciting mathematical methods in the design of error-correcting codes, including high-dimensional expanders, convex optimization, and structured random ensembles. These methods have led to recent breakthroughs in coding theory. The goal of this workshop was to bring together researchers from multiple communities to exchange ideas around these and other mathematical techniques, hopefully leading to further advances.
This workshop in real and harmonic analysis surveyed recent investigations in areas including geometric measure theory and restriction theory, multiple ergodic averages, local smoothing estimates, Schrödinger and Hörmander-type oscillatory integral operators, multilinear estimates and analysis on the Hamming cube. In particular, the workshop emphasized the recent solutions of two longstanding problems: the Kakeya phenomenon in dimension three, and almost everywhere convergence of long time averages associated to multiple commuting measure-preserving transformations. The methods presented during the workshop have yielded applications in ergodic theory, number theory, and computer science.
This workshop brought together researchers working in various areas of differential geometry, with an emphasis on problems that connect local geometric analysis with global structure and topology. Talks covered a broad range of topics, including geometric flows, scalar curvature, minimal surfaces, singular spaces, and large-scale geometric phenomena. While the methods and settings varied widely, a recurring theme was the use of analytic tools to understand spaces with curvature or topological constraints, especially in the presence of singularities. The format encouraged informal discussion and exchange of ideas across different subfields.
The theory of subfactors plays an important role in the discovery and analysis of quantum symmetries that seem to be ubiquitous in mathematics and physics. Subfactors profoundly interact with a wide range of different areas such as quantum topology, vertex operator algebras, quantum groups, free probability theory, quantum computing and quantum information theory, quantum field theory, conformal field theory, tensor categories, condensed matter physics, and, of course, operator algebras.The aim of this workshop was to bring together an international group of researchers from these fields to disseminate recent results and to stimulate new collaborations. The focus was on operator algebraic, vertex operator algebraic and categorical aspects of quantum symmetries and their applications to open questions in mathematical physics. A substantial group of young mathematicians attended the workshop and were given the opportunity to present their work.
Stein’s method, a powerful tool rooted in probability and stochastic analysis, has recently showcased its efficacy in addressing diverse challenges encountered in deep learning, optimisation, sampling, and causal inference. The primary focus of the workshop is to strengthen the probabilistic and analytic foundations of Stein’s method, while simultaneously exploring novel avenues for its application. Bringing together researchers from the analysis, probability, statistics, and machine learning communities, who share a common interest in Stein’s method, the workshop aims to facilitate idea exchange, tackle open problems, and foster collaborations to advance the forefront of knowledge in these fields. Of particular importance is the emphasis placed on the intersection of these disciplines, where Stein’s method plays a pivotal role.
This workshop was part of the long-standing biannual series on Dynamical Systems at Oberwolfach, which began in 1981 with a meeting organized by Moser and Zehnder. It focused on recent advances and developments in dynamical systems, with particular attention to Hamiltonian systems and symplectic geometry. This year, special emphasis was placed on rigidity problems, periodic and connecting orbits.
The main theme of the workshop was the use of probabilistic methods in combinatorics and related fields. This area is evolving extremely quickly, with the introduction of powerful new methods and the development of increasingly sophisticated techniques, and there have been a number of very significant breakthroughs in the area in recent years. The workshop emphasized several of these recent breakthroughs, which include foundational results in the theory of random graphs and processes, and also applications of probabilistic techniques in Ramsey theory, design theory, and group theory, and of combinatorial techniques to problems in number theory, functional analysis, and high-dimensional geometry.
Cohomology theories have long proven to be powerful, unifying tools in numerous areas of mathematics, specifically in algebra, geometry, and number theory. Historically, the invention of the “right” cohomology theory often proved to be the key to uniform, conceptual proofs of conjectures and explanations of heuristic phenomena, bringing together seemingly unrelated mathematical areas. This workshop concentrated on the areas of automorphic forms and enumerative algebra, two fields in which cohomology theories already had a particular impact. It brought together mathematicians from various subdirections of these areas, both junior and seniorr
Computations with functions depending on a large numbers of variables are at the core of many problems in science and engineering. They arise naturally in physical models described by partial differential equations (PDEs) depending on many parameters, in purely data-driven tasks such as optimization and machine learning, and in hybrid contexts combining physical models with data.Traditionally, dealing with such high dimensionality was avoided by the use of simplified models. With the availability of more computational power and the development of sophisticated approximation schemes and algorithms, however, such tasks in high dimensions are increasingly treated directly on the basis of general mathematical principles.The naive use of classical approximation methods for such problems typically leads to computational costs that scale exponentially with respect to the dimension, an effect known as the curse of dimensionality . To make computations tractable, nonlinear strategies that leverage in more subtle ways inherent properties of the problem are inevitably required. In recent years, many new and diverse approaches have emerged from different fields. Shaping up the theoretical foundations for the analysis and development of these approaches requires new interactions between approximation theory, numerical analysis, probability theory, mathematical and statistical learning theory, and optimization. This workshop aimed to deepen the mathematical foundations of the underlying numerical concepts that drive this new evolution of computational methods, and to promote the exchange of ideas arising from various disciplines about how to treat high-dimensional problems.
The subject of operator algebras is a very active area of mathematics which, since its inception in the 1940s, has always been driven by interactions with other fields of mathematics and physics. The scope of these interactions is very wide, ranging over dynamical systems, (non-commutative) geometry, functional analysis, (geometric) group theory, topology, random matrices, harmonic analysis and quantum information theory.The goals of this workshop were to stimulate new collaborations across these fields of mathematics, to disseminate recent progress by giving participants a global view on the subject and to specially focus on several important developments, including progress on Connes’ rigidity conjecture for property (T) groups, a deeper understanding of the analogs of group boundaries in both C ^{*} and von Neumann algebra theory, C ^{*} -simplicity and selflessness of groups, and progress in (equivariant) classification of C ^{*} -algebras.
The goal of this workshop was to explore the recent advances in the mathematical understanding of the macroscopic properties which emerge on large space-time scales from interacting microscopic particle systems. The talks addressed the following topics: stochastic homogeneization, hydrodynamic limits, Markov chain mixing times, superdiffusivity in out-of-equilibrium 2-dimensional systems, random walks in random environments.
This workshop explored how modern machine learning can both accelerate mathematical discovery and preserve rigorous standards. It focused on three angles: using AI techniques to help mathematicians make advances on challenging problems; using mathematics to understand AI predictions; and using deep-learning models for automated theorem proving. Key discussions included using machine learning as a tool for constructing interesting mathematical constructions and navigating in mathematical search spaces, to uncover conjectures and high-quality examples (e.g., sphere packings via DiffuseBoost, combinatorial objects via AlphaEvolve); Integrating Large Language Models (LLMs) with formal systems (e.g., Lean/mathlib) to create scalable, certifiable AI-based automated theorem prover; Collaborative formalization (e.g., the Carleson theorem project), autoformalization for high-quality supervised data, and reinforcement learning/search methods for proof generation and algorithmic reasoning.
This workshop focused on nonlinear elliptic and parabolic partial differential equations, touching topics such as geometric flows, geometric variational problems and minimal surfaces, free boundaries, and geometric measure theory.
