http://publications.mfo.de/handle/mfo/4014 Oberwolfach Reports Volume 20 (2023) Wed, 08 Apr 2026 23:46:25 GMT 2026-04-08T23:46:25Z http://publications.mfo.de/handle/mfo/4147 Geometric, Algebraic, and Topological Combinatorics The 2023 Oberwolfach meeting "Geometric, Algebraic, and Topological Combinatorics'' was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference were (1) Federico Ardila and Tom Braden discussed recent exciting developments in the intersection theory of matroids; (2) Stavros Papadakis and Vasiliki Petrotou presented their proof of the Lefschetz property for spheres, and, more generally, for pseudomanifolds and cycles (this second part is joint with Karim Adiprasito); (3) Gaku Liu reported on his joint work with Spencer Backman that establishes the existence of a regular unimodular triangulation of an arbitrary matroid base polytope. Sun, 01 Jan 2023 00:00:00 GMT http://publications.mfo.de/handle/mfo/4147 2023-01-01T00:00:00Z The 2023 Oberwolfach meeting "Geometric, Algebraic, and Topological Combinatorics'' was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference were (1) Federico Ardila and Tom Braden discussed recent exciting developments in the intersection theory of matroids; (2) Stavros Papadakis and Vasiliki Petrotou presented their proof of the Lefschetz property for spheres, and, more generally, for pseudomanifolds and cycles (this second part is joint with Karim Adiprasito); (3) Gaku Liu reported on his joint work with Spencer Backman that establishes the existence of a regular unimodular triangulation of an arbitrary matroid base polytope. http://publications.mfo.de/handle/mfo/4137 Mini-Workshop: Homological Aspects for TDLC-Groups This mini-workshop aimed at bringing together experts and early career researchers on finiteness conditions for discrete groups, and experts on varying aspects of locally compact groups to find a common framework to develop a systematic theory of homological finiteness conditions for totally disconnected locally compact groups. Whereas the homological theory of finiteness conditions of discrete groups is well developed and the structure theory of totally disconnected locally compact groups has seen some important breakthroughs in the last decade, the homological theory for (non-compact) totally disconnected locally compact groups is an emerging research area. Specific topics include finiteness conditions for locally compact groups, Mackey functors and Bredon cohomology for topological groups, connections to condensed mathematics, connections to $\ell^2$-invariants and $\Sigma$-invariants. Sun, 01 Jan 2023 00:00:00 GMT http://publications.mfo.de/handle/mfo/4137 2023-01-01T00:00:00Z This mini-workshop aimed at bringing together experts and early career researchers on finiteness conditions for discrete groups, and experts on varying aspects of locally compact groups to find a common framework to develop a systematic theory of homological finiteness conditions for totally disconnected locally compact groups. Whereas the homological theory of finiteness conditions of discrete groups is well developed and the structure theory of totally disconnected locally compact groups has seen some important breakthroughs in the last decade, the homological theory for (non-compact) totally disconnected locally compact groups is an emerging research area. Specific topics include finiteness conditions for locally compact groups, Mackey functors and Bredon cohomology for topological groups, connections to condensed mathematics, connections to $\ell^2$-invariants and $\Sigma$-invariants. http://publications.mfo.de/handle/mfo/4128 MFO-RIMS Tandem Workshop 2023: Arithmetic Homotopy and Galois Theory This report presents a general panorama of recent progress in the arithmetic-geometry theory of Galois and homotopy groups and its ramifications. While still relying on Grothendieck's original pillars, the present program has now evolved beyond the classical group-theoretic legacy to result in an autonomous project that exploits a new geometrization of the original insight and sketches new frontiers between homotopy geometry, homology geometry, and diophantine geometry. This panorama "closes the loop'' by including the last twenty-year progress of the Japanese arithmetic-geometry school via Ihara's program and Nakamura-Tamagawa-Mochizuki's anabelian approach, which brings its expertise in terms of algorithmic, combinatoric, and absolute reconstructions. These methods supplement and interact with those from the classical arithmetic of covers and Hurwitz spaces and the motivic and geometric Galois representations. This workshop has brought together the next generation of arithmetic homotopic Galois geometers, who, with the support of senior experts, are developing new techniques and principles for the exploration of the next research frontiers. Sun, 01 Jan 2023 00:00:00 GMT http://publications.mfo.de/handle/mfo/4128 2023-01-01T00:00:00Z This report presents a general panorama of recent progress in the arithmetic-geometry theory of Galois and homotopy groups and its ramifications. While still relying on Grothendieck's original pillars, the present program has now evolved beyond the classical group-theoretic legacy to result in an autonomous project that exploits a new geometrization of the original insight and sketches new frontiers between homotopy geometry, homology geometry, and diophantine geometry. This panorama "closes the loop'' by including the last twenty-year progress of the Japanese arithmetic-geometry school via Ihara's program and Nakamura-Tamagawa-Mochizuki's anabelian approach, which brings its expertise in terms of algorithmic, combinatoric, and absolute reconstructions. These methods supplement and interact with those from the classical arithmetic of covers and Hurwitz spaces and the motivic and geometric Galois representations. This workshop has brought together the next generation of arithmetic homotopic Galois geometers, who, with the support of senior experts, are developing new techniques and principles for the exploration of the next research frontiers. http://publications.mfo.de/handle/mfo/4106 Arbeitsgemeinschaft: QFT and Stochastic PDEs Quantum field theory (QFT) is a fundamental framework for a wide range of phenomena is physics. The link between QFT and SPDE was first observed by the physicists Parisi and Wu (1981), known as Stochastic Quantisation. The study of solution theories and properties of solutions to these SPDEs derived from the Stochastic Quantisation procedure has stimulated substantial progress of the solution theory of singular SPDE, especially the invention of the theories of regularity structures and paracontrolled distributions in the last decade. Moreover, Stochastic Quantisation allows us to bring in more tools including PDE and stochastic analysis to study QFT. This Arbeitsgemeinschaft starts by covering some background material and then explores some of the advances made in recent years. The focus of this Arbeitsgemeinschaft is QFT models such as the $\Phi^4$, sine-Gordon and Yang--Mills models as examples to discuss stochastic quantisation and SPDE methods and their applications in these models. We introduce the key ideas, results and applications of regularity structure and paracontrolled distributions, construction of solutions of the SPDEs corresponding to these models, and use the PDE method to study some qualitative behaviors of these QFTs, and connections with the corresponding lattice or statistical physical models. We also discuss some other topics of QFT, such as Wilsonian renormalisation group, log-Sobolev inequalities and their implications, and various connections between these topics and SPDEs. Sun, 01 Jan 2023 00:00:00 GMT http://publications.mfo.de/handle/mfo/4106 2023-01-01T00:00:00Z Quantum field theory (QFT) is a fundamental framework for a wide range of phenomena is physics. The link between QFT and SPDE was first observed by the physicists Parisi and Wu (1981), known as Stochastic Quantisation. The study of solution theories and properties of solutions to these SPDEs derived from the Stochastic Quantisation procedure has stimulated substantial progress of the solution theory of singular SPDE, especially the invention of the theories of regularity structures and paracontrolled distributions in the last decade. Moreover, Stochastic Quantisation allows us to bring in more tools including PDE and stochastic analysis to study QFT. This Arbeitsgemeinschaft starts by covering some background material and then explores some of the advances made in recent years. The focus of this Arbeitsgemeinschaft is QFT models such as the $\Phi^4$, sine-Gordon and Yang--Mills models as examples to discuss stochastic quantisation and SPDE methods and their applications in these models. We introduce the key ideas, results and applications of regularity structure and paracontrolled distributions, construction of solutions of the SPDEs corresponding to these models, and use the PDE method to study some qualitative behaviors of these QFTs, and connections with the corresponding lattice or statistical physical models. We also discuss some other topics of QFT, such as Wilsonian renormalisation group, log-Sobolev inequalities and their implications, and various connections between these topics and SPDEs. http://publications.mfo.de/handle/mfo/4104 Mini-Workshop: Mathematics of Many-body Fermionic Systems Fermionic quantum systems are well described by the linear many-body Schrödinger equation. For interacting systems the full Schrödinger theory is extremely complicated and theoretical as well numerical investigations are not feasible. In practice, macroscopic properties of large systems can therefore only be accessed by means of approximate theories. The intention of this workshop was to showcase the most recent advances in the mathematical study of many-body interacting fermionic systems and to stimulate discussions among different research groups. Sun, 01 Jan 2023 00:00:00 GMT http://publications.mfo.de/handle/mfo/4104 2023-01-01T00:00:00Z Fermionic quantum systems are well described by the linear many-body Schrödinger equation. For interacting systems the full Schrödinger theory is extremely complicated and theoretical as well numerical investigations are not feasible. In practice, macroscopic properties of large systems can therefore only be accessed by means of approximate theories. The intention of this workshop was to showcase the most recent advances in the mathematical study of many-body interacting fermionic systems and to stimulate discussions among different research groups. http://publications.mfo.de/handle/mfo/4103 Variational Methods for Evolution Variational principles for evolutionary systems arise in many settings, both in those describing the physical world and in man-made algorithms for data science and optimization tasks. Variational principles are available for Hamiltonian systems in classical mechanics, gradient flows for dissipative systems, as well as in time-incremental minimization techniques for more general evolutionary problems. Additional challenges arise via the interplay of two or more functionals (e.g. a free energy and a dissipation potential), thus encompassing a large variety of applications in the modeling of materials and fluids, in biology, and in multi-agent systems. Variational principles and associated evolutions are also at the core of the modern approaches to machine learning tasks, since many of them are posed as minimizing an objective functional that models the problem. The discrete and random nature of these problems and the need for accurate computation in high dimension present a set of challenges that require new mathematical insights. Variational methods for evolution allow for the usage of the rich toolbox provided by the calculus of variations, metric-space geometry, partial differential equations, and other branches of applied analysis. The variational methods for evolution have seen a rapid growth over the last two decades. This workshop continued the successful line of meetings (2011, 2014, 2017, and 2020), while evolving and branching into new directions. We have brought together a wide scope of mathematical researchers from calculus of variations, partial differential equations, numerical analysis, and stochastics, as well as researchers from data science and machine learning, to exchange ideas, foster interaction, develop new avenues, and generally bring these communities closer together. Sun, 01 Jan 2023 00:00:00 GMT http://publications.mfo.de/handle/mfo/4103 2023-01-01T00:00:00Z Variational principles for evolutionary systems arise in many settings, both in those describing the physical world and in man-made algorithms for data science and optimization tasks. Variational principles are available for Hamiltonian systems in classical mechanics, gradient flows for dissipative systems, as well as in time-incremental minimization techniques for more general evolutionary problems. Additional challenges arise via the interplay of two or more functionals (e.g. a free energy and a dissipation potential), thus encompassing a large variety of applications in the modeling of materials and fluids, in biology, and in multi-agent systems. Variational principles and associated evolutions are also at the core of the modern approaches to machine learning tasks, since many of them are posed as minimizing an objective functional that models the problem. The discrete and random nature of these problems and the need for accurate computation in high dimension present a set of challenges that require new mathematical insights. Variational methods for evolution allow for the usage of the rich toolbox provided by the calculus of variations, metric-space geometry, partial differential equations, and other branches of applied analysis. The variational methods for evolution have seen a rapid growth over the last two decades. This workshop continued the successful line of meetings (2011, 2014, 2017, and 2020), while evolving and branching into new directions. We have brought together a wide scope of mathematical researchers from calculus of variations, partial differential equations, numerical analysis, and stochastics, as well as researchers from data science and machine learning, to exchange ideas, foster interaction, develop new avenues, and generally bring these communities closer together. http://publications.mfo.de/handle/mfo/4102 Tensor-Triangular Geometry and Interactions The workshop brought together experts in a rapidly growing field of tensor triangular geometry highlighting applications to and techniques coming from homotopy theory, algebraic geometry, modular representation theory, motivic homotopy theory and noncommutative algebra. Sun, 01 Jan 2023 00:00:00 GMT http://publications.mfo.de/handle/mfo/4102 2023-01-01T00:00:00Z The workshop brought together experts in a rapidly growing field of tensor triangular geometry highlighting applications to and techniques coming from homotopy theory, algebraic geometry, modular representation theory, motivic homotopy theory and noncommutative algebra. http://publications.mfo.de/handle/mfo/4101 Control Methods in Hyperbolic Partial Differential Equations Control of hyperbolic partial differential equations (PDEs) is a truly interdisciplinary area of research in applied mathematics nurtured by challenging problems arising in most modern applications ranging from road traffic, gas pipeline management, blood circulation, to opinion dynamics and socio-economical models, as well as in environmental and biological issues, or more recently in the analysis of deep learning and machine learning methods. The topic has gained an increasing attraction of researchers in the last twenty years due to fundamental theoretical as well as numerical advances achieved in the field of nonlinear hyperbolic PDEs. The hyperbolic and the control of PDEs communities, while pursuing separate interests in their respective range of action with a different focus and, often, with a different array of technical tools, do share a substantial body of common knowledge and background. We think the time is right and the momentum is propitious to bring those communities together at a joint workshop, to mutually stimulate each other and interact with one another, for a marked advancement of this area of research on a broad spectrum of control ranging from theoretical to numerical problems and covering also the emerging challenges involving the interplay between (topics of) control and learning. In order to also attract young scientists to this striving field we focus on selected lectures, in-depth discussions, transfer of information from senior to young researchers, and vice versa, and invited plenary talks. Sun, 01 Jan 2023 00:00:00 GMT http://publications.mfo.de/handle/mfo/4101 2023-01-01T00:00:00Z Control of hyperbolic partial differential equations (PDEs) is a truly interdisciplinary area of research in applied mathematics nurtured by challenging problems arising in most modern applications ranging from road traffic, gas pipeline management, blood circulation, to opinion dynamics and socio-economical models, as well as in environmental and biological issues, or more recently in the analysis of deep learning and machine learning methods. The topic has gained an increasing attraction of researchers in the last twenty years due to fundamental theoretical as well as numerical advances achieved in the field of nonlinear hyperbolic PDEs. The hyperbolic and the control of PDEs communities, while pursuing separate interests in their respective range of action with a different focus and, often, with a different array of technical tools, do share a substantial body of common knowledge and background. We think the time is right and the momentum is propitious to bring those communities together at a joint workshop, to mutually stimulate each other and interact with one another, for a marked advancement of this area of research on a broad spectrum of control ranging from theoretical to numerical problems and covering also the emerging challenges involving the interplay between (topics of) control and learning. In order to also attract young scientists to this striving field we focus on selected lectures, in-depth discussions, transfer of information from senior to young researchers, and vice versa, and invited plenary talks. http://publications.mfo.de/handle/mfo/4100 Mini-Workshop: Positivity and Inequalities in Convex and Complex Geometry The workshop convened researchers from algebraic geometry, convex geometry, and complex geometry to explore themes arising from the Alexandrov-Fenchel and Brunn-Minkowski inequalities. It featured three introductory talks delving into the basics of Lorentzian polynomials, valuations in convex geometry, and plurisubharmonic functions, that served as a foundation for the subsequent research talks. As anticipated, significant overlap emerged among the varied perspectives within these three areas, evident in the presentations and ensuing discussions. Sun, 01 Jan 2023 00:00:00 GMT http://publications.mfo.de/handle/mfo/4100 2023-01-01T00:00:00Z The workshop convened researchers from algebraic geometry, convex geometry, and complex geometry to explore themes arising from the Alexandrov-Fenchel and Brunn-Minkowski inequalities. It featured three introductory talks delving into the basics of Lorentzian polynomials, valuations in convex geometry, and plurisubharmonic functions, that served as a foundation for the subsequent research talks. As anticipated, significant overlap emerged among the varied perspectives within these three areas, evident in the presentations and ensuing discussions. http://publications.mfo.de/handle/mfo/4099 Mini-Workshop: Combinatorial and Algebraic Structures in Rough Analysis and Related Fields Recent years have seen an explosion of algebraic methods to study singular stochastic and rough dynamics. These include developments in geometric rough path theory based on the algebra of words, the introduction of decorated trees in regularity structures, and the recent approach to singular stochastic partial differential equations based on multi-indices. These developments have furthermore led to important links with numerical analysis, machine learning, stochastic quantisation, and the study of symmetries of physical systems. The aim of this mini-workshop was to bring together experts working on these fields using algebraic structures that appear in rough dynamics. The goal was to facilitate the exchange of ideas and encourage further connections to be established. Sun, 01 Jan 2023 00:00:00 GMT http://publications.mfo.de/handle/mfo/4099 2023-01-01T00:00:00Z Recent years have seen an explosion of algebraic methods to study singular stochastic and rough dynamics. These include developments in geometric rough path theory based on the algebra of words, the introduction of decorated trees in regularity structures, and the recent approach to singular stochastic partial differential equations based on multi-indices. These developments have furthermore led to important links with numerical analysis, machine learning, stochastic quantisation, and the study of symmetries of physical systems. The aim of this mini-workshop was to bring together experts working on these fields using algebraic structures that appear in rough dynamics. The goal was to facilitate the exchange of ideas and encourage further connections to be established.