Oberwolfach Reports (OWR)

Permanent URI for this collectionhttps://oa.tib.eu/renate/handle/123456789/14627

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Homogeneous Structures: Model Theory meets Universal Algebra
(Zürich : EMS Publ. House, 2025) Barto, Libor; Bodirsky, Manuel; Kwiatkowska, Aleksandra; Pinsker, Michael
Many fundamental mathematical structures, such as the rationals or the random graph, are homogeneous, meaning that local isomorphisms extend to global automorphisms. Such structures arise as limits of classes of finite structures and encode these classes in a single object. This viewpoint has proved fruitful in model theory, universal algebra, and computer science, with applications to constraint satisfaction, automata theory, and verification. Homogeneous structures have rich automorphism groups, which makes them interesting for topological dynamics. For many applications, however, automorphism groups do not store enough information about the homogeneous structure, and one must instead consider polymorphism clones. Universal algebra has recently achieved major results for polymorphism clones on finite structures, culminating in the 2017 resolution of the Feder–Vardi dichotomy conjecture. An analogous conjecture for homogeneous structures remains open despite growing structural insights.
Mini-Workshop: Algebraic Foliations: Analytic and Birational Viewpoint
(Zürich : EMS Publ. House, 2025) Di Nezza, Eleonora; Floris, Enrica; Zimmermann, Susanna
The main goal of the mini-workshop was starting strong collaborations between outstanding women in geometry with a broad spectrum of expertise. The focus was on the interplay of three topics in Geometry: analytic methods in Kähler geometry, Foliations and Cremona groups. More precisely, Foliation theory was a unifying theme.
Mini-Workshop: Approximation of Manifold-Valued Functions
(Zürich : EMS Publ. House, 2025) Sharon, Nir; Wendland, Holger; Zimmermann, Ralf
The approximation of unknown functions from scattered, possibly high-dimensional data is central to many scientific applications. Advances in data acquisition have driven the need for flexible nonlinear models, including manifold-valued functions. Approximating and learning such functions differs fundamentally from classical linear methods and requires tools from numerical analysis, linear algebra, and differential geometry. This interdisciplinary framework has applications ranging from data science and machine learning to numerical PDEs and quantum chemistry. This mini-workshop brings together researchers developing constructive approximation methods for manifold-valued functions, their theory, and applications.
Mini-Workshop: Hyperbolic meets Stochastic Geometry
(Zürich : EMS Publ. House, 2025) Kellerhals, Ruth; Thäle, Christoph
The mini-workshop brought together researchers from hyperbolic geometry and stochastic geometry with the aim of advancing the emerging field of hyperbolic stochastic geometry. It focused on understanding how negative curvature fundamentally influences the behaviour of random geometric models. Particular emphasis was placed on limit theorems, phase transitions, and scaling phenomena that differ substantially from those observed in Euclidean settings. The program combined survey lectures, research presentations, and discussion sessions to link geometric methods with probabilistic techniques tailored to hyperbolic spaces. As a result, the workshop clarified central challenges in the field, identified key open problems, and initiated new collaborations spanning geometry, probability, and related areas.
Functional Inequalities: Geometric Calculus meets Stochastic Analysis
(Zürich : EMS Publ. House, 2025) Gordina, Masha; Lin, Jessica; Milman, Emanuel; Sturm, Karl-Theodor
Functional inequalities form a unifying theme across a wide spectrum of modern analysis, geometry, and probability. They encode deep geometric and analytic information – for instance through Poincaré, log-Sobolev, transportation, isoperimetric and curvature-dimension inequalities – and serve as crucial tools in the study of Markov semigroups, diffusion processes, metric measure spaces, and geometric flows. The workshop brought together researchers working in geometric analysis, stochastic analysis, and optimal transport in order to promote exchange of ideas and further strengthen the interaction between these rapidly developing fields. Substantial emphasis was placed on non-smooth or singular geometric structures, stochastic dynamics with degeneracies, and new bridges between discrete, fractal, and continuum settings.
Arithmetic Statistics for Algebraic Objects
(Zürich : EMS Publ. House, 2025) Bary-Soroker, Lior; Ostafe, Alina; Sarnak, Peter
The workshop focused on various directions of arithmetic statistics in algebra and number theory. These include statistical problems for random polynomials and varieties, probabilistic Galois theory, and counting and distribution problems for algebraic functions, algebraic number fields, elliptic curves, L-functions, as well as arithmetic problems in non-abelian settings (eg, arithmetic statistics for algebraic groups).
Recent Developments in SPDEs and BSDEs meet Harmonic and Functional Analysis
(Zürich : EMS Publ. House, 2025) Geiß, Stefan; Gess, Benjamin; Petermichl, Stefanie; Veraar, Mark
The purpose of this workshop is the strengthening of the interaction between the fields of Stochastic Partial Differential Equations (SPDEs), Backward Stochastic Differential Equations (BSDEs), and Harmonic and Functional Analysis. We focus on the essential role of analytic techniques (including function spaces, weighted inequalities, and Ap-weights) in solving problems in critical stochastic settings. Special topics include SPDEs in critical spaces, regularization by noise, singular SPDEs, quadratic and nonlocal BSDEs.
Mini-Workshop: Probabilistic Perspectives in Neural Network-Based Machine Learning
(Zürich : EMS Publ. House, 2025) Dereich, Steffen; Dieuleveut, Aymeric; Kassing, Sebastian; Langer, Sophie
Artificial neural networks (ANNs) have emerged as a powerful tool in modern machine learning, yet their mathematical foundations remain only partially understood. A key challenge is the inherently stochastic nature of ANN training: optimization occurs in high-dimensional parameter spaces with complex loss landscapes, influenced by stochastic initialization and noisy gradient updates. Understanding these dynamics requires probabilistic methods and asymptotic frameworks. This workshop explored recent advances in stochastic training dynamics, emphasizing probabilistic techniques and limit theorems. By bringing together researchers from probability, optimization, and deep learning theory, this workshop laid the groundwork for new directions in understanding neural network training from a stochastic perspective.
Analytic Number Theory
(Zürich : EMS Publ. House, 2025) Matomäki, Kaisa; Maynard, James; Soundararajan, Kannan; Wooley, Trevor D.
Analytic number theory is a subject which continues to flourish and grow with several significant developments over the past few years making progress on some of the most famous open problems in mathematics. This workshop brought together world experts and young talent to discuss the various branches and recent developments in the subject.
Mini-Workshop: The Yang–Baxter Equation and Representations of Braid Groups
(Zürich : EMS Publ. House, 2025) Colazzo, Ilaria; Plavnik, Julia; Rowell, Eric; Vendramin, Leandro
The Yang-Baxter equation is a famous equation in mathematics and mathematical physics. It plays a central role in several areas of mathematics, including algebra, topology, and quantum field theory. The aim of the workshop is to review recent developments in areas where the Yang-Baxter equation is crucial to discuss new research directions and ideas for addressing open problems.
Arbeitsgemeinschaft: Combinatorial Hodge Theory
(Zürich : EMS Publ. House, 2025) Eur, Chris; Huh, June; Larson, Matt
Combinatorial Hodge theory has undergone rapid development in recent years, revealing deep connections between matroid theory, tropical geometry, toric geometry, and convex geometry. Building on the Kähler package for matroids and the emergence of Lorentzian structures across combinatorics, the field now encompasses a broad family of Hodge-theoretic phenomena arising from purely combinatorial objects. The goal of this Arbeitsgemeinschaft was to provide participants with a structured and accessible overview of these developments, emphasizing both foundational material and current research directions. The program was organized around four major themes–matroids, Hodge theory, toric methods, and Lorentzian polynomials–with lectures highlighting topics such as Baker–Bowler framework for matroids with coefficients, Chow rings of matroids and wonderful compactifications of hyperplane arrangements, Lorentzian polynomials and volume polynomials, and matroids over triangular hyperfields. Together, these lectures aimed to articulate the unifying principles underlying the subject and to prepare participants for further research in this evolving area.
Mini-Workshop: Resurgence, Difference Equations and Quantum Modularity
(Zürich : EMS Publ. House, 2025) Alim, Murad; Fantini, Veronica; Hollands, Lotte; Rella, Claudia
The resurgent analysis of asymptotic series has found diverse applications and led to profound insights into our understanding of enumerative geometry, difference equations, and quantum modular forms. Consequently, interactions among these three independent yet intertwined avenues of contemporary research have recently soared. This mini-workshop brought together both senior and junior researchers from each area to clarify the state of the art and establish common ground for collectively addressing the most relevant open questions in the field.
Arbeitsgemeinschaft: Analysis of Many-body Quantum Systems
(Zürich : EMS Publ. House, 2025) Hainzl, Christian; Schlein, Benjamin; Seiringer, Robert; Solovej, Jan Philip
This Oberwolfach Arbeitsgemeinschaft focuses on the mathematical analysis of quantum many-body systems. Of particular interest is the ground state energy of dilute quantum gases, either fermions or bosons, and its low-density expansion in the thermodynamic limit.
Mini-Workshop: Renormalisation and Randomness
(Zürich : EMS Publ. House, 2025) Aschieri, Paolo; Gurau, Razvan-Gheorghe; Paycha, Sylvie; Rejzner, Kasia
In recent decades, renormalisation, which has long been a workman’s tool for theoretical physicists, has also become an essential mathematical tool that appears in many guises. Within mathematics, renormalisation bridges across topics such as combinatorics and stochastic analysis. Yet, in part due to a lack of a common language, the advances on the mathematical side do not seem to fully reach out to the theoretical physicist. Conversely, the mathematician rarely benefits from the physicist’s expertise in renormalisation. The goal of this workshop is to bridge this gap and provide a platform for communication and exchange of ideas, a first step in the direction of increased interaction and cross fertilisation between the two communities.
Singularities
(Zürich : EMS Publ. House, 2025) de Bobadilla, Javier Fernández; Loeser, François; Némethi, András; van Straten, Duco
Singularity theory concerns local and global structure of singularities of algebraic varieties and maps, often focusing on the interplay between algebraic geometry, symplectic topology, algebra, combinatorics, etc.
MATRIX-MFO Tandem Workshop: Machine Learning and AI for Mathematics
(Zürich : EMS Publ. House, 2025) Charton, François; de Gier, Jan; Hayat, Amaury; Kempe, Julia; Williamson, Geordie
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.
New Mathematical Directions in Coding Theory
(Zürich : EMS Publ. House, 2025) Ron-Zewi, Noga; Wootters, Mary
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.
Combinatorics, Probability and Computing
(Zürich : EMS Publ. House, 2025) Morris, Robert; Riordan, Oliver; Sauermann, Lisa
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.
Large Scale Stochastic Dynamics
(Zürich : EMS Publ. House, 2025) Caputo, Pietro; Toninelli, Fabio; Tóth, Bálint
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.
Stein’s Method in Stochastic Geometry, Statistical Learning, and Optimisation
(Zürich : EMS Publ. House, 2025) Balasubramanian, Krishnakumar; Erdogdu, Murat A.; Goldstein, Larry; Reinert, Gesine
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.

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