http://publications.mfo.de/handle/mfo/2804 starting 2004 Thu, 23 Jul 2026 00:19:40 GMT 2026-07-23T00:19:40Z http://publications.mfo.de:80/bitstream/id/22cb835a-bf96-4db8-8054-fb7a8ad70db1/ http://publications.mfo.de/handle/mfo/2804 http://publications.mfo.de/handle/mfo/4447 Flows on Measure Spaces and Applications in Machine Learning 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. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4447 2026-01-01T00:00:00Z 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. http://publications.mfo.de/handle/mfo/4446 Higher Structures from Symmetries in Quantum Field Theory Symmetries in the form of actions of groups and Lie algebras have been an important tool for the analysis of physical systems for a very long time. It has been realized only relatively recently that quantum field theories admit symmetries that are described by more general mathematical structures: fusion categories and their higher-categorical generalizations. The understanding of such higher categories and the construction of concrete examples raise many mathematical challenges which are at present addressed with a wide variety of mathematical tools, including homotopy theory, the theory of $C^*$-algebras, factorization algebras and operads. An improved mathematical understanding of higher categories has an immediate impact on the classification and characterization of quantum systems which can realize these categorical structures as symmetries. Particularly note-worthy applications of these symmetries include integrable systems with fusion category symmetries, defects in topological and conformal field theories, the characterization of three-dimensional topological order and systems in quantum information theory realizing quantum codes. The workshop "Higher Structures from Symmetries in Quantum Field Theory", brought a wide range of mathematical researchers together, working on a broad range of topics, from category theory to condensed matter physics, and provided an extremely fruitful interaction across disciplines. For some leading researchers from neighbouring fields, this was their first visit to Oberwolfach. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4446 2026-01-01T00:00:00Z Symmetries in the form of actions of groups and Lie algebras have been an important tool for the analysis of physical systems for a very long time. It has been realized only relatively recently that quantum field theories admit symmetries that are described by more general mathematical structures: fusion categories and their higher-categorical generalizations. The understanding of such higher categories and the construction of concrete examples raise many mathematical challenges which are at present addressed with a wide variety of mathematical tools, including homotopy theory, the theory of $C^*$-algebras, factorization algebras and operads. An improved mathematical understanding of higher categories has an immediate impact on the classification and characterization of quantum systems which can realize these categorical structures as symmetries. Particularly note-worthy applications of these symmetries include integrable systems with fusion category symmetries, defects in topological and conformal field theories, the characterization of three-dimensional topological order and systems in quantum information theory realizing quantum codes. The workshop "Higher Structures from Symmetries in Quantum Field Theory", brought a wide range of mathematical researchers together, working on a broad range of topics, from category theory to condensed matter physics, and provided an extremely fruitful interaction across disciplines. For some leading researchers from neighbouring fields, this was their first visit to Oberwolfach. http://publications.mfo.de/handle/mfo/4445 Modern and Emerging Phenomena in Machine Learning 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. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4445 2026-01-01T00:00:00Z 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. http://publications.mfo.de/handle/mfo/4444 Nonlinear Waves and Dispersive Equations Nonlinear dispersive equations describe nonlinear wave phenomena that arise in many physical systems, for instance, in the context of general relativity, quantum mechanics, or water waves. Linear dispersive equations have solutions that spread out and decay while keeping a constant $L^2$-norm. The interaction with nonlinear effects leads to a rich variety of behaviors, including finite-time blow-up, soliton formation, and scattering. These equations are connected to many branches of mathematics, such as integrable systems, harmonic analysis, geometry, and probability. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4444 2026-01-01T00:00:00Z Nonlinear dispersive equations describe nonlinear wave phenomena that arise in many physical systems, for instance, in the context of general relativity, quantum mechanics, or water waves. Linear dispersive equations have solutions that spread out and decay while keeping a constant $L^2$-norm. The interaction with nonlinear effects leads to a rich variety of behaviors, including finite-time blow-up, soliton formation, and scattering. These equations are connected to many branches of mathematics, such as integrable systems, harmonic analysis, geometry, and probability. http://publications.mfo.de/handle/mfo/4443 Cohomology of Finite Groups: Interactions and Applications 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. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4443 2026-01-01T00:00:00Z 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. http://publications.mfo.de/handle/mfo/4442 Median Geometry and Applications This was the first-ever workshop on median geometry and its applications, bringing together the leading experts from areas of pure and applied expertise in fields currently seen as distinct. Invited experts from mathematical biology, and computer science interacted with specialists from geometry, combinatorics and group theory, all fields in which median geometry plays a key role. The participants worked to understand the similarities and differences in the way in which they modelled and exploited median geometry in their work, developing a dictionary of terms that facilitated communication and fostering collaboration. As well as exploring recent activity in their fields, the participants developed a problem list to guide future development and promote cross disciplinary research. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4442 2026-01-01T00:00:00Z This was the first-ever workshop on median geometry and its applications, bringing together the leading experts from areas of pure and applied expertise in fields currently seen as distinct. Invited experts from mathematical biology, and computer science interacted with specialists from geometry, combinatorics and group theory, all fields in which median geometry plays a key role. The participants worked to understand the similarities and differences in the way in which they modelled and exploited median geometry in their work, developing a dictionary of terms that facilitated communication and fostering collaboration. As well as exploring recent activity in their fields, the participants developed a problem list to guide future development and promote cross disciplinary research. http://publications.mfo.de/handle/mfo/4441 Multiple Zeta Values and Modular Forms The workshop explored recent developments and clarified relationships between multiple zeta values (MZVs) and modular forms. Central themes included algebraic and linear relations among MZVs and their $q$-analogues, the role of cusp forms, and links with partitions, representation theory, and mathematical physics. Several talks investigated variations of modular forms, including multiple Eisenstein series, vector-valued modular forms, and harmonic Maass forms, as well as variants of zeta functions and their functional equations. The workshop brought together junior and senior mathematicians from various subdirections of these areas. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4441 2026-01-01T00:00:00Z The workshop explored recent developments and clarified relationships between multiple zeta values (MZVs) and modular forms. Central themes included algebraic and linear relations among MZVs and their $q$-analogues, the role of cusp forms, and links with partitions, representation theory, and mathematical physics. Several talks investigated variations of modular forms, including multiple Eisenstein series, vector-valued modular forms, and harmonic Maass forms, as well as variants of zeta functions and their functional equations. The workshop brought together junior and senior mathematicians from various subdirections of these areas. http://publications.mfo.de/handle/mfo/4440 Representation Theory of Quivers and Finite-Dimensional Algebras 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. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4440 2026-01-01T00:00:00Z 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. http://publications.mfo.de/handle/mfo/4439 Numerical Analysis for Geometric and Nonlinear PDEs A growing and impactful area of computational mathematics and numerical analysis is the solution of PDEs modeled from underlying geometric principles. This field covers a broad range of PDE problems, including geometric evolution equations, PDEs on surfaces, nonlinear bending models, and fully nonlinear Monge-Ampère in optimal transport. These mathematical formulations are found in numerous applications in machine learning, meteorology, medical imaging, cell biology, geophysics, and computer graphics. This workshop provided an opportunity for interactions between senior and early-career researchers working on numerical methods for geometric and nonlinear PDEs. The expertise of the participants spanned the numerical analysis, computational implementation, and practical applications of these problems. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4439 2026-01-01T00:00:00Z A growing and impactful area of computational mathematics and numerical analysis is the solution of PDEs modeled from underlying geometric principles. This field covers a broad range of PDE problems, including geometric evolution equations, PDEs on surfaces, nonlinear bending models, and fully nonlinear Monge-Ampère in optimal transport. These mathematical formulations are found in numerous applications in machine learning, meteorology, medical imaging, cell biology, geophysics, and computer graphics. This workshop provided an opportunity for interactions between senior and early-career researchers working on numerical methods for geometric and nonlinear PDEs. The expertise of the participants spanned the numerical analysis, computational implementation, and practical applications of these problems. http://publications.mfo.de/handle/mfo/4438 Noncommutative Harmonic Analysis and Quantum Information 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. Thu, 01 Jan 2026 00:00:00 GMT http://publications.mfo.de/handle/mfo/4438 2026-01-01T00:00:00Z 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.