http://publications.mfo.de/handle/mfo/4367 Oberwolfach Reports Volume 23 (2026) Thu, 15 Oct 2026 06:15:28 GMT 2026-10-15T06:15:28Z http://publications.mfo.de/handle/mfo/4482 Topologie The workshop covered the areas of algebraic and geometric topology, homotopy theory, geometric group theory, and manifold topology, as well as applications of topological methods to neighbouring areas of mathematics. The goal of the workshop was to survey recent trends in the field of Topology as a whole, to explain breakthrough results in an accessible way, and to strengthen ties between topologists who might otherwise not have an opportunity to interact. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4482 2026-01-01T00:00:00Z The workshop covered the areas of algebraic and geometric topology, homotopy theory, geometric group theory, and manifold topology, as well as applications of topological methods to neighbouring areas of mathematics. The goal of the workshop was to survey recent trends in the field of Topology as a whole, to explain breakthrough results in an accessible way, and to strengthen ties between topologists who might otherwise not have an opportunity to interact. http://publications.mfo.de/handle/mfo/4481 Mathematical Foundation of Digital Twins A digital twin (DT) is a computational model that evolves over time to persistently represent the structure, behavior, and context of a unique physical system or process. DTs are characterized by a dynamic and continuous two-way flow of information between the computational model and the physical system. Data streams from the physical system are assimilated into the computational model to reduce uncertainties and improve predictions of the model, which in turn is used as a basis for controlling the physical system, optimizing data acquisition, and providing decision support. The DT paradigm brings along challenging mathematical and statistical questions which were at the center of the workshop. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4481 2026-01-01T00:00:00Z A digital twin (DT) is a computational model that evolves over time to persistently represent the structure, behavior, and context of a unique physical system or process. DTs are characterized by a dynamic and continuous two-way flow of information between the computational model and the physical system. Data streams from the physical system are assimilated into the computational model to reduce uncertainties and improve predictions of the model, which in turn is used as a basis for controlling the physical system, optimizing data acquisition, and providing decision support. The DT paradigm brings along challenging mathematical and statistical questions which were at the center of the workshop. http://publications.mfo.de/handle/mfo/4480 Geometrie The workshop Geometrie was well attended with 47 in-person and two virtual participants whose interests covered a considerable part of current research in Differential Geometry. We had talks in such diverse subjects as geometric evolution equations, Kähler geometry, minimal surfaces, RCD spaces and positive scalar curvature metrics. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4480 2026-01-01T00:00:00Z The workshop Geometrie was well attended with 47 in-person and two virtual participants whose interests covered a considerable part of current research in Differential Geometry. We had talks in such diverse subjects as geometric evolution equations, Kähler geometry, minimal surfaces, RCD spaces and positive scalar curvature metrics. http://publications.mfo.de/handle/mfo/4479 Random Matrices Random matrix theory (RMT) has emerged as a central object of study across physics, computer science, and mathematics. Originally introduced by Wigner to describe energy levels in heavy nuclei, RMT reveals universal statistical properties of eigenvalues, dependent solely on the matrix symmetry class. Over the past three decades, it has found applications in diverse fields, from growth models and number theory to multivariate statistics and data science. Research in RMT follows two interconnected paths: integrability, which seeks explicit distributions for specific models, and universality, which extends these properties to broader classes of probabilistic systems. Recent breakthroughs include the proof of the Anderson transition for band matrices in any dimension, universality for non-Hermitian matrices and $d$-regular graphs, and the resolution of long-standing conjectures in number theory. Key open challenges persist, particularly in random Schrödinger operators (e.g., proving delocalization in physical dimension 3), Coulomb gases (crystallization phenomena), large deviations, and random tensors. The interplay between universality and integrability continues to drive innovation, with new connections to machine learning, extreme value theory, and quantum field theory. This workshop brought together outstanding researchers from a variety of mathematical backgrounds whose areas of research are linked to random matrices. While there are strong links between their motivations, the techniques used by these researchers span a large swath of mathematics, ranging from purely algebraic techniques to stochastic analysis, classical probability theory, operator algebra, supersymmetry, orthogonal polynomials, etc. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4479 2026-01-01T00:00:00Z Random matrix theory (RMT) has emerged as a central object of study across physics, computer science, and mathematics. Originally introduced by Wigner to describe energy levels in heavy nuclei, RMT reveals universal statistical properties of eigenvalues, dependent solely on the matrix symmetry class. Over the past three decades, it has found applications in diverse fields, from growth models and number theory to multivariate statistics and data science. Research in RMT follows two interconnected paths: integrability, which seeks explicit distributions for specific models, and universality, which extends these properties to broader classes of probabilistic systems. Recent breakthroughs include the proof of the Anderson transition for band matrices in any dimension, universality for non-Hermitian matrices and $d$-regular graphs, and the resolution of long-standing conjectures in number theory. Key open challenges persist, particularly in random Schrödinger operators (e.g., proving delocalization in physical dimension 3), Coulomb gases (crystallization phenomena), large deviations, and random tensors. The interplay between universality and integrability continues to drive innovation, with new connections to machine learning, extreme value theory, and quantum field theory. This workshop brought together outstanding researchers from a variety of mathematical backgrounds whose areas of research are linked to random matrices. While there are strong links between their motivations, the techniques used by these researchers span a large swath of mathematics, ranging from purely algebraic techniques to stochastic analysis, classical probability theory, operator algebra, supersymmetry, orthogonal polynomials, etc. http://publications.mfo.de/handle/mfo/4478 Group Actions and Harmonic Analysis in Number Theory Rooted in number theory, this workshop combines recent methodological advances in a variety of adjacent fields including automorphic representation theory, ergodic theory, Fourier analysis and algebraic geometry. It focuses on a selected set of problems that could recently be solved quite surprisingly in a number of cases, but are awaiting a more general framework. This includes in particular equidistribution problems in homogeneous dynamics, automorphic forms in higher rank and exponential sums and arithmetic over function fields. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4478 2026-01-01T00:00:00Z Rooted in number theory, this workshop combines recent methodological advances in a variety of adjacent fields including automorphic representation theory, ergodic theory, Fourier analysis and algebraic geometry. It focuses on a selected set of problems that could recently be solved quite surprisingly in a number of cases, but are awaiting a more general framework. This includes in particular equidistribution problems in homogeneous dynamics, automorphic forms in higher rank and exponential sums and arithmetic over function fields. http://publications.mfo.de/handle/mfo/4477 Conic Linear Optimization for Computer-assisted Proofs Conic (linear) optimization is the problem of minimizing a linear functional over the intersection of a convex cone with an affine subspace of a topological vector space. Examples of conic linear optimization problems include semidefinite programming in the finite-dimensional case, and the generalised problem of moments in the infinite-dimensional case. Here, the underlying cones are the cone of positive semidefinite matrices, and the cone of positive Borel measures (with given support), respectively. Since the 1980s, efficient algorithms have been developed for (certain) conic optimization problems. The use of computers makes it possible to prove inequalities with (near-) optimal constants or (near-) optimal weights which otherwise would be completely out of reach with classical pen and paper reasoning. Also the field of conic optimization has a powerful duality theory. This duality theory makes it relatively easy to verify rigorously the obtained numerical computer-assisted proofs without for instance using extensive case by case checking. Thus there has been a flood of new proofs using conic optimization in discrete geometry, (extremal) graph theory, numerical analysis, and other fields, the most famous example perhaps being the proof of the Kepler Conjecture. The aim of this workshop was to bring researchers from these diverse fields together to work towards expanding the current scope of conic optimization as a method of generating proofs, and to identify problems and challenges to work on together. This workshop was a follow-up on the (half) workshop in 2022 of the same name (Oberwolfach Workshop 2215b). Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4477 2026-01-01T00:00:00Z Conic (linear) optimization is the problem of minimizing a linear functional over the intersection of a convex cone with an affine subspace of a topological vector space. Examples of conic linear optimization problems include semidefinite programming in the finite-dimensional case, and the generalised problem of moments in the infinite-dimensional case. Here, the underlying cones are the cone of positive semidefinite matrices, and the cone of positive Borel measures (with given support), respectively. Since the 1980s, efficient algorithms have been developed for (certain) conic optimization problems. The use of computers makes it possible to prove inequalities with (near-) optimal constants or (near-) optimal weights which otherwise would be completely out of reach with classical pen and paper reasoning. Also the field of conic optimization has a powerful duality theory. This duality theory makes it relatively easy to verify rigorously the obtained numerical computer-assisted proofs without for instance using extensive case by case checking. Thus there has been a flood of new proofs using conic optimization in discrete geometry, (extremal) graph theory, numerical analysis, and other fields, the most famous example perhaps being the proof of the Kepler Conjecture. The aim of this workshop was to bring researchers from these diverse fields together to work towards expanding the current scope of conic optimization as a method of generating proofs, and to identify problems and challenges to work on together. This workshop was a follow-up on the (half) workshop in 2022 of the same name (Oberwolfach Workshop 2215b). http://publications.mfo.de/handle/mfo/4476 S^7 : Statistics and Stochastics for Sheaves, Shapes, Stratified and Sub-Riemannian Structures The mathematical description of shapes as elements of non-linear geometric spaces has been central to the field of geometric statistics and has spawned a variety of new links to other fields of mathematics: Geometric data requires models for stochastic processes in geometric spaces, for example, for data in structural and evolutionary biology, in biomedical imaging, and in morphology such as plant sciences, and non-holonomic constraints lead to sub-Riemannian geometric structures. The prevalence of shape data has created a need for statistical inference tools to draw conclusions from observed shapes in complex, non-linear settings, and new numerical techniques now allow for studying geometries experimentally in novel ways, for example, using neural networks to allow simulation of diffusion bridges in sub-Riemannian spaces. The ability to model local information with sheaves gives the ability to analyze shapes with neural networks. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4476 2026-01-01T00:00:00Z The mathematical description of shapes as elements of non-linear geometric spaces has been central to the field of geometric statistics and has spawned a variety of new links to other fields of mathematics: Geometric data requires models for stochastic processes in geometric spaces, for example, for data in structural and evolutionary biology, in biomedical imaging, and in morphology such as plant sciences, and non-holonomic constraints lead to sub-Riemannian geometric structures. The prevalence of shape data has created a need for statistical inference tools to draw conclusions from observed shapes in complex, non-linear settings, and new numerical techniques now allow for studying geometries experimentally in novel ways, for example, using neural networks to allow simulation of diffusion bridges in sub-Riemannian spaces. The ability to model local information with sheaves gives the ability to analyze shapes with neural networks. http://publications.mfo.de/handle/mfo/4475 Complex Geometry and Dynamical Systems The workshop focused on recent developments in complex geometry, several complex variables, and holomorphic dynamics. The topics of the talks included complex Monge-Ampère and Hessian equations, pluripotential theory and the theory of currents, Bergman and Szegő kernels together with their asymptotic and spectral aspects, as well as complex dynamics in higher dimensions. Further themes were random and asymptotic questions in complex geometry, hyperbolicity, non-Kähler geometry, period maps, moduli problems, and interactions with birational geometry, arithmetic geometry, and mathematical physics. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4475 2026-01-01T00:00:00Z The workshop focused on recent developments in complex geometry, several complex variables, and holomorphic dynamics. The topics of the talks included complex Monge-Ampère and Hessian equations, pluripotential theory and the theory of currents, Bergman and Szegő kernels together with their asymptotic and spectral aspects, as well as complex dynamics in higher dimensions. Further themes were random and asymptotic questions in complex geometry, hyperbolicity, non-Kähler geometry, period maps, moduli problems, and interactions with birational geometry, arithmetic geometry, and mathematical physics. http://publications.mfo.de/handle/mfo/4474 Geometry and Combinatorics: The Model Theoretic Perspective Our meeting focused mainly on geometric and combinatorial aspects in model theory and their interactions with geometric group theory, algebraic geometry, valued fields, and combinatorics. The model-theoretic perspective provided a unifying point of view for applications in seemingly different mathematical areas. At the same time, phenomena and problems encountered in applications provided fresh impetus for further develoments in abstract model theory. It also provided a welcome meeting point for reasearchers from different areas of mathematics. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4474 2026-01-01T00:00:00Z Our meeting focused mainly on geometric and combinatorial aspects in model theory and their interactions with geometric group theory, algebraic geometry, valued fields, and combinatorics. The model-theoretic perspective provided a unifying point of view for applications in seemingly different mathematical areas. At the same time, phenomena and problems encountered in applications provided fresh impetus for further develoments in abstract model theory. It also provided a welcome meeting point for reasearchers from different areas of mathematics. http://publications.mfo.de/handle/mfo/4473 Low-dimensional Topology and Number Theory The workshop brought together researchers in low-dimensional topology and number theory to explore the rich and increasingly deep interactions between these fields. Central themes included quantum invariants of knots and 3-manifolds, their asymptotic behavior, and the framework of Arithmetic Topology, all of which reveal striking parallels and connections between geometric, quantum, and arithmetic phenomena. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4473 2026-01-01T00:00:00Z The workshop brought together researchers in low-dimensional topology and number theory to explore the rich and increasingly deep interactions between these fields. Central themes included quantum invariants of knots and 3-manifolds, their asymptotic behavior, and the framework of Arithmetic Topology, all of which reveal striking parallels and connections between geometric, quantum, and arithmetic phenomena.