http://publications.mfo.de/handle/mfo/3703 Oberwolfach Reports Volume 16 (2019) Wed, 08 Apr 2026 03:56:20 GMT 2026-04-08T03:56:20Z http://publications.mfo.de/handle/mfo/3818 Random Matrices Large complex systems tend to develop universal patterns that often represent their essential characteristics. For example, the cumulative effects of independent or weakly dependent random variables often yield the Gaussian universality class via the central limit theorem. For non-commutative random variables, e.g. matrices, the Gaussian behavior is often replaced by another universality class, commonly called random matrix statistics. Nearby eigenvalues are strongly correlated, and, remarkably, their correlation structure is universal, depending only on the symmetry type of the matrix. Even more surprisingly, this feature is not restricted to matrices; in fact Eugene Wigner, the pioneer of the field, discovered in the 1950s that distributions of the gaps between energy levels of complicated quantum systems universally follow the same random matrix statistics. This claim has never been rigorously proved for any realistic physical system but experimental data and extensive numerics leave no doubt as to its correctness. Since then random matrices have proved to be extremely useful phenomenological models in a wide range of applications beyond quantum physics that include number theory, statistics, neuroscience, population dynamics, wireless communication and mathematical finance. The ubiquity of random matrices in natural sciences is still a mystery, but recent years have witnessed a breakthrough in the mathematical description of the statistical structure of their spectrum. Random matrices and closely related areas such as log-gases have become an extremely active research area in probability 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. Tue, 01 Jan 2019 00:00:00 GMT http://publications.mfo.de/handle/mfo/3818 2019-01-01T00:00:00Z Large complex systems tend to develop universal patterns that often represent their essential characteristics. For example, the cumulative effects of independent or weakly dependent random variables often yield the Gaussian universality class via the central limit theorem. For non-commutative random variables, e.g. matrices, the Gaussian behavior is often replaced by another universality class, commonly called random matrix statistics. Nearby eigenvalues are strongly correlated, and, remarkably, their correlation structure is universal, depending only on the symmetry type of the matrix. Even more surprisingly, this feature is not restricted to matrices; in fact Eugene Wigner, the pioneer of the field, discovered in the 1950s that distributions of the gaps between energy levels of complicated quantum systems universally follow the same random matrix statistics. This claim has never been rigorously proved for any realistic physical system but experimental data and extensive numerics leave no doubt as to its correctness. Since then random matrices have proved to be extremely useful phenomenological models in a wide range of applications beyond quantum physics that include number theory, statistics, neuroscience, population dynamics, wireless communication and mathematical finance. The ubiquity of random matrices in natural sciences is still a mystery, but recent years have witnessed a breakthrough in the mathematical description of the statistical structure of their spectrum. Random matrices and closely related areas such as log-gases have become an extremely active research area in probability 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/3817 Heat Kernels, Stochastic Processes and Functional Inequalities The aims of the 2019 workshop 'Heat Kernels, Stochastic Processes and Functional Inequalities' were: (a) to provide a forum to review recent progresses in a wide array of areas of analysis (elliptic, subelliptic and parabolic differential equations, transport, functional inequalities), geometry (Riemannian and sub-Riemannian geometries, metric measure spaces, geometric analysis and curvature), and probability (Brownian motion, Dirichlet spaces, stochastic calculus and random media) that have natural common interests, and (b) to foster, encourage and develop further interactions and cross-fertilization between these different directions of research. Tue, 01 Jan 2019 00:00:00 GMT http://publications.mfo.de/handle/mfo/3817 2019-01-01T00:00:00Z The aims of the 2019 workshop 'Heat Kernels, Stochastic Processes and Functional Inequalities' were: (a) to provide a forum to review recent progresses in a wide array of areas of analysis (elliptic, subelliptic and parabolic differential equations, transport, functional inequalities), geometry (Riemannian and sub-Riemannian geometries, metric measure spaces, geometric analysis and curvature), and probability (Brownian motion, Dirichlet spaces, stochastic calculus and random media) that have natural common interests, and (b) to foster, encourage and develop further interactions and cross-fertilization between these different directions of research. http://publications.mfo.de/handle/mfo/3816 Mini-Workshop: Seshadri Constants Seshadri constants were defined by Demailly around 30 years ago using the ampleness criterion of Seshadri. Demailly was interested in studying problems related to separation of jets of line bundles on projective varieties, specifically in the context of the well-known Fujita Conjecture. However, Seshadri constants turned out to be objects of fundamental importance in the study of positivity of linear series and many other areas. Consequently, in the past three decades, they have become a central object of study in numerous directions in algebraic geometry and commutative algebra. In this mini-workshop, we studied some of the most interesting current research problems concerning Seshadri constants. We expect that this exploration will help focus research on some of the most important questions in this area in the years to come. Tue, 01 Jan 2019 00:00:00 GMT http://publications.mfo.de/handle/mfo/3816 2019-01-01T00:00:00Z Seshadri constants were defined by Demailly around 30 years ago using the ampleness criterion of Seshadri. Demailly was interested in studying problems related to separation of jets of line bundles on projective varieties, specifically in the context of the well-known Fujita Conjecture. However, Seshadri constants turned out to be objects of fundamental importance in the study of positivity of linear series and many other areas. Consequently, in the past three decades, they have become a central object of study in numerous directions in algebraic geometry and commutative algebra. In this mini-workshop, we studied some of the most interesting current research problems concerning Seshadri constants. We expect that this exploration will help focus research on some of the most important questions in this area in the years to come. http://publications.mfo.de/handle/mfo/3815 Mini-Workshop: Rank One Groups and Exceptional Algebraic Groups Rank one groups are a class of doubly transitive groups that are natural generalizations of the groups $ \operatorname{SL}_2(k) $. The most interesting examples arise from exceptional algebraic groups of relative rank one. This class of groups is, in turn, intimately related to structurable algebras. The goal of the mini-workshop was to bring together experts on these topics in order to make progress towards a better understanding of the structure of rank one groups. Tue, 01 Jan 2019 00:00:00 GMT http://publications.mfo.de/handle/mfo/3815 2019-01-01T00:00:00Z Rank one groups are a class of doubly transitive groups that are natural generalizations of the groups $ \operatorname{SL}_2(k) $. The most interesting examples arise from exceptional algebraic groups of relative rank one. This class of groups is, in turn, intimately related to structurable algebras. The goal of the mini-workshop was to bring together experts on these topics in order to make progress towards a better understanding of the structure of rank one groups. http://publications.mfo.de/handle/mfo/3814 Analytic Number Theory Analytic number theory is a subject which is central to modern mathematics. There are many important unsolved problems which have stimulated a large amount of activity by many talented researchers. At least two of the Millennium Problems can be considered to be in this area. Moreover in recent years there has been very substantial progress on a number of these questions. Tue, 01 Jan 2019 00:00:00 GMT http://publications.mfo.de/handle/mfo/3814 2019-01-01T00:00:00Z Analytic number theory is a subject which is central to modern mathematics. There are many important unsolved problems which have stimulated a large amount of activity by many talented researchers. At least two of the Millennium Problems can be considered to be in this area. Moreover in recent years there has been very substantial progress on a number of these questions. http://publications.mfo.de/handle/mfo/3813 Subfactors and Applications The theory of subfactors connects diverse topics in mathematics and mathematical physics such as tensor categories, vertex operator algebras, quantum groups, quantum topology, free probability, quantum field theory, conformal field theory, statistical mechanics, condensed matter physics and, of course, operator algebras. We invited an international group of researchers from these areas and many fruitful interactions took place during the workshop. Tue, 01 Jan 2019 00:00:00 GMT http://publications.mfo.de/handle/mfo/3813 2019-01-01T00:00:00Z The theory of subfactors connects diverse topics in mathematics and mathematical physics such as tensor categories, vertex operator algebras, quantum groups, quantum topology, free probability, quantum field theory, conformal field theory, statistical mechanics, condensed matter physics and, of course, operator algebras. We invited an international group of researchers from these areas and many fruitful interactions took place during the workshop. http://publications.mfo.de/handle/mfo/3812 Arbeitsgemeinschaft: Zimmer's Conjecture The aim of this Arbeitsgemeinschaft was to understand the recent progress on Zimmer's conjecture in [1,2]. The week focuses on the cocompact case from [1]. Tue, 01 Jan 2019 00:00:00 GMT http://publications.mfo.de/handle/mfo/3812 2019-01-01T00:00:00Z The aim of this Arbeitsgemeinschaft was to understand the recent progress on Zimmer's conjecture in [1,2]. The week focuses on the cocompact case from [1]. http://publications.mfo.de/handle/mfo/3811 Mini-Workshop: Self-adjoint Extensions in New Settings The main focus of the workshop is on the analysis of boundary value problems for differential and difference operators in some non-classical geometric settings, such as fractal graphs, sub-Riemannian manifolds or non-elliptic transmission problems. Taking into account their importance in modern mathematical analysis, we aim at developing suitable tools in the operator theory to deal with the new problem settings. Tue, 01 Jan 2019 00:00:00 GMT http://publications.mfo.de/handle/mfo/3811 2019-01-01T00:00:00Z The main focus of the workshop is on the analysis of boundary value problems for differential and difference operators in some non-classical geometric settings, such as fractal graphs, sub-Riemannian manifolds or non-elliptic transmission problems. Taking into account their importance in modern mathematical analysis, we aim at developing suitable tools in the operator theory to deal with the new problem settings. http://publications.mfo.de/handle/mfo/3810 Mini-Workshop: Degeneration Techniques in Representation Theory Modern Representation Theory has numerous applications in many mathematical areas such as algebraic geometry, combinatorics, convex geometry, mathematical physics, probability. Many of the object and problems of interest show up in a family. Degeneration techniques allow to study the properties of the whole family instead of concentrating on a single member. This idea has many incarnations in modern mathematics, including Newton-Okounkov bodies, tropical geometry, PBW degenerations, Hessenberg varieties. During the mini-workshop Degeneration Techniques in Representation Theory various sides of the existing applications of the degenerations techniques were discussed and several new possible directions were reported. Tue, 01 Jan 2019 00:00:00 GMT http://publications.mfo.de/handle/mfo/3810 2019-01-01T00:00:00Z Modern Representation Theory has numerous applications in many mathematical areas such as algebraic geometry, combinatorics, convex geometry, mathematical physics, probability. Many of the object and problems of interest show up in a family. Degeneration techniques allow to study the properties of the whole family instead of concentrating on a single member. This idea has many incarnations in modern mathematics, including Newton-Okounkov bodies, tropical geometry, PBW degenerations, Hessenberg varieties. During the mini-workshop Degeneration Techniques in Representation Theory various sides of the existing applications of the degenerations techniques were discussed and several new possible directions were reported. http://publications.mfo.de/handle/mfo/3809 Mini-Workshop: Operator Algebraic Quantum Groups This mini-workshop brought together a rich and varied cross-section of young and active researchers working on operator algebraic aspects of quantum group theory. The primary goals of this meeting were to highlight the state-of-the-art results on the subject and to trigger new research by advertising some of the main open directions in operator algebraic quantum group theory: classification problems for C$^\ast$- and von Neumann algebras, relations to free/non-commutative probability, applications in quantum information theory, and the creation of new quantum groups and potential classification results for subclasses of quantum groups. Tue, 01 Jan 2019 00:00:00 GMT http://publications.mfo.de/handle/mfo/3809 2019-01-01T00:00:00Z This mini-workshop brought together a rich and varied cross-section of young and active researchers working on operator algebraic aspects of quantum group theory. The primary goals of this meeting were to highlight the state-of-the-art results on the subject and to trigger new research by advertising some of the main open directions in operator algebraic quantum group theory: classification problems for C$^\ast$- and von Neumann algebras, relations to free/non-commutative probability, applications in quantum information theory, and the creation of new quantum groups and potential classification results for subclasses of quantum groups.