Oberwolfach Publications
http://publications.mfo.de:80
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Wed, 20 Jan 2021 01:27:12 GMT2021-01-20T01:27:12ZOberwolfach Publicationshttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:80
Boundary Conditions for Scalar Curvature
http://publications.mfo.de/handle/mfo/3824
Boundary Conditions for Scalar Curvature
Bär, Christian; Hanke, Bernhard
Based on the Atiyah-Patodi-Singer index formula, we construct an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite $K$-area. We also characterize the extremal case. Next we show a general deformation principle for boundary conditions of metrics with lower scalar curvature bounds. This implies that the relaxation of boundary conditions often induces weak homotopy equivalences of spaces of such metrics. This can be used to refine the smoothing of codimension-one singularites à la Miao and the deformation of boundary conditions à la Brendle-Marques-Neves, among others. Finally, we construct compact manifolds for which the spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.
Mon, 04 Jan 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38242021-01-04T00:00:00ZBär, ChristianHanke, BernhardBased on the Atiyah-Patodi-Singer index formula, we construct an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite $K$-area. We also characterize the extremal case. Next we show a general deformation principle for boundary conditions of metrics with lower scalar curvature bounds. This implies that the relaxation of boundary conditions often induces weak homotopy equivalences of spaces of such metrics. This can be used to refine the smoothing of codimension-one singularites à la Miao and the deformation of boundary conditions à la Brendle-Marques-Neves, among others. Finally, we construct compact manifolds for which the spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.Dynamics of Gravitational Collapse in the Axisymmetric Einstein-Vlasov System
http://publications.mfo.de/handle/mfo/3820
Dynamics of Gravitational Collapse in the Axisymmetric Einstein-Vlasov System
Ames, Ellery; Andréasson, Håkan; Rinne, Oliver
We numerically investigate the dynamcis near black hole formation of solutions to the Einstein-Vlasov system in axisymmetry. Our results are obtained using a particle-in-cell and finite difference code based on the (2+1)+1 formulation of the Einstein field equations in axisymmetry. Solutions are launched from generic type initial data and exhibit type I critical behaviour. In particular we find lifetime scaling in solutions containing black holes, and support that the critical solutions are stationary. Our results contain examples of solutions that form black holes, perform damped oscillations, and appear to disperse. We prove that complete dispersal of the solution implies that it has nonpositive binding energy.
Tue, 15 Dec 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38202020-12-15T00:00:00ZAmes, ElleryAndréasson, HåkanRinne, OliverWe numerically investigate the dynamcis near black hole formation of solutions to the Einstein-Vlasov system in axisymmetry. Our results are obtained using a particle-in-cell and finite difference code based on the (2+1)+1 formulation of the Einstein field equations in axisymmetry. Solutions are launched from generic type initial data and exhibit type I critical behaviour. In particular we find lifetime scaling in solutions containing black holes, and support that the critical solutions are stationary. Our results contain examples of solutions that form black holes, perform damped oscillations, and appear to disperse. We prove that complete dispersal of the solution implies that it has nonpositive binding energy.Jahresbericht | Annual Report - 2019
http://publications.mfo.de/handle/mfo/3819
Jahresbericht | Annual Report - 2019
Mathematisches Forschungsinstitut Oberwolfach
Wed, 01 Jan 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38192020-01-01T00:00:00ZMathematisches Forschungsinstitut OberwolfachRandom Matrices
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 GMThttp://publications.mfo.de/handle/mfo/38182019-01-01T00:00:00ZLarge 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.Heat Kernels, Stochastic Processes and Functional Inequalities
http://publications.mfo.de/handle/mfo/3817
Heat Kernels, Stochastic Processes and Functional Inequalities
The aims of the 2019 workshop \emph{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 GMThttp://publications.mfo.de/handle/mfo/38172019-01-01T00:00:00ZThe aims of the 2019 workshop \emph{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.Mini-Workshop: Seshadri Constants
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 GMThttp://publications.mfo.de/handle/mfo/38162019-01-01T00:00:00ZSeshadri 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.Mini-Workshop: Rank One Groups and Exceptional Algebraic Groups
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 GMThttp://publications.mfo.de/handle/mfo/38152019-01-01T00:00:00ZRank 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.Analytic Number Theory
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 GMThttp://publications.mfo.de/handle/mfo/38142019-01-01T00:00:00ZAnalytic 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.Subfactors and Applications
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 GMThttp://publications.mfo.de/handle/mfo/38132019-01-01T00:00:00ZThe 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.Arbeitsgemeinschaft: Zimmer's Conjecture
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 GMThttp://publications.mfo.de/handle/mfo/38122019-01-01T00:00:00ZThe 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].