http://publications.mfo.de/handle/mfo/4107 Oberwolfach Reports Volume 21 (2024) Mon, 06 Apr 2026 23:20:16 GMT 2026-04-06T23:20:16Z http://publications.mfo.de/handle/mfo/4274 Convex Geometry and its Applications The geometry of convex domains in Euclidean space plays a central role in several branches of mathematics: functional and harmonic analysis, the theory of PDEs, linear programming and, increasingly, in the study of algorithms in computer science. Convex Geometry has experienced a series of striking developments in the past few years: for example, the new tools from stochastic localization, the huge progress around the slicing problem, the measure transportation perspective on old problems, progress on conjectured geometric and functional inequalities and new applications of methods and results to a wide range of fields, including random matrices and statistical learning. The purpose of this meeting is to bring together researchers from the analytic, geometric and probabilistic groups who have contributed to the latest exciting results, to exchange ideas and pave the path for future developments. The meeting will continue a tradition of more than 50 years of Oberwolfach meetings with Convex Geometry in the title, at the same time emphasizing the new directions and developments, and new connections to other mathematical fields. Mon, 01 Jan 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4274 2024-01-01T00:00:00Z The geometry of convex domains in Euclidean space plays a central role in several branches of mathematics: functional and harmonic analysis, the theory of PDEs, linear programming and, increasingly, in the study of algorithms in computer science. Convex Geometry has experienced a series of striking developments in the past few years: for example, the new tools from stochastic localization, the huge progress around the slicing problem, the measure transportation perspective on old problems, progress on conjectured geometric and functional inequalities and new applications of methods and results to a wide range of fields, including random matrices and statistical learning. The purpose of this meeting is to bring together researchers from the analytic, geometric and probabilistic groups who have contributed to the latest exciting results, to exchange ideas and pave the path for future developments. The meeting will continue a tradition of more than 50 years of Oberwolfach meetings with Convex Geometry in the title, at the same time emphasizing the new directions and developments, and new connections to other mathematical fields. http://publications.mfo.de/handle/mfo/4273 Mini-Workshop: Data-driven Modeling, Analysis, and Control of Dynamical Systems With the rapid increase in data resources and computational power as well as the accompanying current trend to incorporate machine learning into existing methods, data-driven approaches for modelling, analysis, and control of dynamical systems have attracted new interest and opened doors to novel applications. However, there is always a discrepancy between mathematical models and reality such that rigorously-shown error bounds and uncertainty quantification are indispensable for a reliable use of data-driven techniques, e.g., using surrogate models in optimisation-based control. Similar comments apply to data-enhanced models. Consequently, uncertainty about parameters, the model itself and numerous other aspects need to be taken into account, e.g., in data-driven control of (stochastic) dynamical systems. Hence, the respective paradigm changes have led to a variety of novel concepts which, however, still suffer from limitations: many concentrate only on a single aspect, are only applicable to systems of limited complexity, or lack a sound mathematical foundation including guarantees on feasibility, robustness, or the overall performance. Pushing these limits, we face a wide spectrum of theoretic and algorithmic challenges in modeling, analysis, and control under uncertainty using data-driven methods. Mon, 01 Jan 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4273 2024-01-01T00:00:00Z With the rapid increase in data resources and computational power as well as the accompanying current trend to incorporate machine learning into existing methods, data-driven approaches for modelling, analysis, and control of dynamical systems have attracted new interest and opened doors to novel applications. However, there is always a discrepancy between mathematical models and reality such that rigorously-shown error bounds and uncertainty quantification are indispensable for a reliable use of data-driven techniques, e.g., using surrogate models in optimisation-based control. Similar comments apply to data-enhanced models. Consequently, uncertainty about parameters, the model itself and numerous other aspects need to be taken into account, e.g., in data-driven control of (stochastic) dynamical systems. Hence, the respective paradigm changes have led to a variety of novel concepts which, however, still suffer from limitations: many concentrate only on a single aspect, are only applicable to systems of limited complexity, or lack a sound mathematical foundation including guarantees on feasibility, robustness, or the overall performance. Pushing these limits, we face a wide spectrum of theoretic and algorithmic challenges in modeling, analysis, and control under uncertainty using data-driven methods. http://publications.mfo.de/handle/mfo/4272 Mini-Workshop: High-Dimensional Control Problems and Mean-Field Equations with Applications in Machine Learning High-dimensional control problems and mean field equations have been of increased interest in recent years and novel numerical tools tackling the curse of dimensionality have been developed. These optimization tasks are strongly related to learning problems such as data-driven optimal control and learning of deep neural networks. As a consequence, there is a huge potential to employ control theoretical techniques in Machine Learning. The Mini-Workshop was devoted to discuss possible synergies among the various tools developed in these fields. Mon, 01 Jan 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4272 2024-01-01T00:00:00Z High-dimensional control problems and mean field equations have been of increased interest in recent years and novel numerical tools tackling the curse of dimensionality have been developed. These optimization tasks are strongly related to learning problems such as data-driven optimal control and learning of deep neural networks. As a consequence, there is a huge potential to employ control theoretical techniques in Machine Learning. The Mini-Workshop was devoted to discuss possible synergies among the various tools developed in these fields. http://publications.mfo.de/handle/mfo/4271 Mini-Workshop: Geometry of Random Fields and Random Walk Clusters: New Horizons Critical phenomena represent a central theme in probability. Research has been going on for many decades and remains very active to date. Recently, models involving natural probabilistic objects such as random walks, loop soups, random interlacements and the Gaussian free field have witnessed exciting developments, both in two- and higher-dimensional setups. The purpose of the workshop was to provide an overview of the state of the art in this rapidly evolving research area. The workshop enabled participants to communicate about the most recent advances in the field, and to discuss propitious avenues for future research. Mon, 01 Jan 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4271 2024-01-01T00:00:00Z Critical phenomena represent a central theme in probability. Research has been going on for many decades and remains very active to date. Recently, models involving natural probabilistic objects such as random walks, loop soups, random interlacements and the Gaussian free field have witnessed exciting developments, both in two- and higher-dimensional setups. The purpose of the workshop was to provide an overview of the state of the art in this rapidly evolving research area. The workshop enabled participants to communicate about the most recent advances in the field, and to discuss propitious avenues for future research. http://publications.mfo.de/handle/mfo/4270 Representations of p-adic Groups Representation theory of $p$-adic groups is a topic at a crossroads. It links among others to harmonic analysis, algebraic geometry, number theory, Lie theory, and homological algebra. The atomic objects in the theory are supercuspidal representations. Most of their aspects have a strong arithmetic flavour, related to Galois groups of local fields. All other representations are built from these atoms by parabolic induction, whose study involves Hecke algebras and complex algebraic geometry. In the local Langlands program, connections between various aspects of representations of $p$-adic groups have been conjectured and avidly studied. This workshop brought together mathematicians from various backgrounds, who hold the promise to contribute to the solution of open problems in the representation theory of $p$-adic groups. Topics included explicit local Langlands correspondences, Hecke algebras for Bernstein components, harmonic analysis, covering groups and $\ell$-modular representations of reductive $p$-adic groups. Mon, 01 Jan 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4270 2024-01-01T00:00:00Z Representation theory of $p$-adic groups is a topic at a crossroads. It links among others to harmonic analysis, algebraic geometry, number theory, Lie theory, and homological algebra. The atomic objects in the theory are supercuspidal representations. Most of their aspects have a strong arithmetic flavour, related to Galois groups of local fields. All other representations are built from these atoms by parabolic induction, whose study involves Hecke algebras and complex algebraic geometry. In the local Langlands program, connections between various aspects of representations of $p$-adic groups have been conjectured and avidly studied. This workshop brought together mathematicians from various backgrounds, who hold the promise to contribute to the solution of open problems in the representation theory of $p$-adic groups. Topics included explicit local Langlands correspondences, Hecke algebras for Bernstein components, harmonic analysis, covering groups and $\ell$-modular representations of reductive $p$-adic groups. http://publications.mfo.de/handle/mfo/4269 Mini-Workshop: Infinite-dimensional Kac-Moody Lie Algebras in Supergravity and M Theory This mini-workshop explores the role played by infinite-dimensional Lie algebra of Kac-Moody type in supergravity and M theory. Deep conjectures about the role of affine, hyperbolic and other indefinite Kac-Moody algebras could both provide new methods and models for low-dimensional gravity systems and shed new light on the fundamental structure of gravity and unified theories. The elusive precise mathematical structure of some of the algebras can in turn be informed by physical ideas. Mon, 01 Jan 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4269 2024-01-01T00:00:00Z This mini-workshop explores the role played by infinite-dimensional Lie algebra of Kac-Moody type in supergravity and M theory. Deep conjectures about the role of affine, hyperbolic and other indefinite Kac-Moody algebras could both provide new methods and models for low-dimensional gravity systems and shed new light on the fundamental structure of gravity and unified theories. The elusive precise mathematical structure of some of the algebras can in turn be informed by physical ideas. http://publications.mfo.de/handle/mfo/4268 Mini-Workshop: Critical Phenomena of the XY Model This mini-workshop focused on recent advances in probability theory concerned with the critical phenomena of the XY model and other related spin systems. There were 18 participants, all working at the forefront of this dynamic field, and from various career stages and a diverse range of institutions as well as backgrounds and gender. The mini-workshop featured talks by almost all participants except organizers as well as an open problem session. The talks consisted of: a) three mini-courses of four lectures each and b) hour long seminars from the remaining participants. The mini-courses covered random currents for the XY model, random walk representations and triviality in the XY and Ising models, and connections between the spherical model and the Gaussian Free Field. The remaining seminars spanned a diverse range of topics, such as novel probabilistic approaches to the BKT transition in 2d, recent progress on high-dimensional spin systems and finite-size effects, and novel geometric representations for correlations in spin systems. Mon, 01 Jan 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4268 2024-01-01T00:00:00Z This mini-workshop focused on recent advances in probability theory concerned with the critical phenomena of the XY model and other related spin systems. There were 18 participants, all working at the forefront of this dynamic field, and from various career stages and a diverse range of institutions as well as backgrounds and gender. The mini-workshop featured talks by almost all participants except organizers as well as an open problem session. The talks consisted of: a) three mini-courses of four lectures each and b) hour long seminars from the remaining participants. The mini-courses covered random currents for the XY model, random walk representations and triviality in the XY and Ising models, and connections between the spherical model and the Gaussian Free Field. The remaining seminars spanned a diverse range of topics, such as novel probabilistic approaches to the BKT transition in 2d, recent progress on high-dimensional spin systems and finite-size effects, and novel geometric representations for correlations in spin systems. http://publications.mfo.de/handle/mfo/4267 Mini-Workshop: Mixing Times in the Kardar-Parisi-Zhang Universality Class The workshop covered new developments in the Kardar-Parisi-Zhang (KPZ) class of growth models and markov chain mixing, broadened by talks on random dimers, matrices and parking functions. Mixing is often related to large time fluctuations, which are governed by universal limits such as the KPZ fixed point. The workshop thus focused on the asymptotic behaviour of KPZ models, the characterization of limiting objects, and cutoff. Mon, 01 Jan 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4267 2024-01-01T00:00:00Z The workshop covered new developments in the Kardar-Parisi-Zhang (KPZ) class of growth models and markov chain mixing, broadened by talks on random dimers, matrices and parking functions. Mixing is often related to large time fluctuations, which are governed by universal limits such as the KPZ fixed point. The workshop thus focused on the asymptotic behaviour of KPZ models, the characterization of limiting objects, and cutoff. http://publications.mfo.de/handle/mfo/4266 Combinatorial Optimization Combinatorial optimization deals with optimization problems defined on polyhedral constraints or discrete structures such as graphs and networks. In the past thirty years the topic has developed into a rich mathematical discipline with many connections to other fields of mathematics such as combinatorics, group theory, geometry of numbers, convex analysis or real algebraic geometry. It also has strong ties to theoretical computer science and other more applied sciences (such as game theory and operations research). Mon, 01 Jan 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4266 2024-01-01T00:00:00Z Combinatorial optimization deals with optimization problems defined on polyhedral constraints or discrete structures such as graphs and networks. In the past thirty years the topic has developed into a rich mathematical discipline with many connections to other fields of mathematics such as combinatorics, group theory, geometry of numbers, convex analysis or real algebraic geometry. It also has strong ties to theoretical computer science and other more applied sciences (such as game theory and operations research). http://publications.mfo.de/handle/mfo/4265 Directions in Rough Analysis Rough path theory emerged in the 1990s and was developed in the 2000s as an improved approach to understanding the interaction of complex random systems. As a broader alternative to stochastic calculus, it simultaneously settled significant questions and substantially expanded the scope of classical methods in stochastic analysis. Subsequent related developments have had an impact at the highest level, Martin Hairer's work on regularity structures being among the most prominent. In 2020, rough analysis gained its own AMS classification code, 60L, and this workshop focused on the currently most active areas of the subject among two central strands: (1) the mathematics of the signature transform, including its applications to data science and finance, and (2) rough path theory applied to novel areas in stochastic analysis, such as homogenization, SLE and rough PDEs. Mon, 01 Jan 2024 00:00:00 GMT http://publications.mfo.de/handle/mfo/4265 2024-01-01T00:00:00Z Rough path theory emerged in the 1990s and was developed in the 2000s as an improved approach to understanding the interaction of complex random systems. As a broader alternative to stochastic calculus, it simultaneously settled significant questions and substantially expanded the scope of classical methods in stochastic analysis. Subsequent related developments have had an impact at the highest level, Martin Hairer's work on regularity structures being among the most prominent. In 2020, rough analysis gained its own AMS classification code, 60L, and this workshop focused on the currently most active areas of the subject among two central strands: (1) the mathematics of the signature transform, including its applications to data science and finance, and (2) rough path theory applied to novel areas in stochastic analysis, such as homogenization, SLE and rough PDEs.