http://publications.mfo.de/handle/mfo/4226 Oberwolfach Reports Volume 22 (2025) Tue, 07 Apr 2026 16:31:14 GMT 2026-04-07T16:31:14Z http://publications.mfo.de/handle/mfo/4407 Homogeneous Structures: Model Theory meets Universal Algebra 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. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4407 2025-01-01T00:00:00Z 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. http://publications.mfo.de/handle/mfo/4406 Mini-Workshop: Hyperbolic meets Stochastic Geometry 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. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4406 2025-01-01T00:00:00Z 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. http://publications.mfo.de/handle/mfo/4405 Mini-Workshop: Approximation of Manifold-Valued Functions 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. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4405 2025-01-01T00:00:00Z 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. http://publications.mfo.de/handle/mfo/4404 Mini-Workshop: Algebraic Foliations: Analytic and Birational Viewpoint 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. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4404 2025-01-01T00:00:00Z 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. http://publications.mfo.de/handle/mfo/4403 Functional Inequalities: Geometric Calculus meets Stochastic Analysis 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. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4403 2025-01-01T00:00:00Z 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. http://publications.mfo.de/handle/mfo/4402 Recent Developments in SPDEs and BSDEs meet Harmonic and Functional Analysis 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 $A_p$-weights) in solving problems in critical stochastic settings. Special topics include SPDEs in critical spaces, regularization by noise, singular SPDEs, quadratic and nonlocal BSDEs. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4402 2025-01-01T00:00:00Z 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 $A_p$-weights) in solving problems in critical stochastic settings. Special topics include SPDEs in critical spaces, regularization by noise, singular SPDEs, quadratic and nonlocal BSDEs. http://publications.mfo.de/handle/mfo/4401 Arithmetic Statistics for Algebraic Objects 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). Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4401 2025-01-01T00:00:00Z 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). http://publications.mfo.de/handle/mfo/4400 Analytic Number Theory 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. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4400 2025-01-01T00:00:00Z 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. http://publications.mfo.de/handle/mfo/4399 Mini-Workshop: Probabilistic Perspectives in Neural Network-Based Machine Learning 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. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4399 2025-01-01T00:00:00Z 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. http://publications.mfo.de/handle/mfo/4398 Mini-Workshop: The Yang-Baxter Equation and Representations of Braid Groups 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. Wed, 01 Jan 2025 00:00:00 GMT http://publications.mfo.de/handle/mfo/4398 2025-01-01T00:00:00Z 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.