Oberwolfach Publications
http://publications.mfo.de:80
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Fri, 05 Mar 2021 22:36:05 GMT2021-03-05T22:36:05ZOberwolfach Publicationshttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:80
$C^*$-algebras: structure and classification
http://publications.mfo.de/handle/mfo/3841
$C^*$-algebras: structure and classification
Kerr, David
The theory of $C^*$-algebras traces its origins back to
the development of quantum mechanics and it has
evolved into a large and highly active field of mathematics.
Much of the progress over the last couple
of decades has been driven by an ambitious program
of classification launched by George A. Elliott in the
1980s, and just recently this project has succeeded
in achieving one of its central goals in an unexpectedly
dramatic fashion. This Snapshot aims to recount
some of the fundamental ideas at play.
Tue, 23 Feb 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38412021-02-23T00:00:00ZKerr, DavidThe theory of $C^*$-algebras traces its origins back to
the development of quantum mechanics and it has
evolved into a large and highly active field of mathematics.
Much of the progress over the last couple
of decades has been driven by an ambitious program
of classification launched by George A. Elliott in the
1980s, and just recently this project has succeeded
in achieving one of its central goals in an unexpectedly
dramatic fashion. This Snapshot aims to recount
some of the fundamental ideas at play.From the dollar game to the Riemann-Roch Theorem
http://publications.mfo.de/handle/mfo/3840
From the dollar game to the Riemann-Roch Theorem
Lamboglia, Sara; Ulirsch, Martin
What is the dollar game? What can you do to win
it? Can you always win it? In this snapshot you
will find answers to these questions as well as several
of the mathematical surprises that lurk in the background,
including a new perspective on a century-old theorem.
Tue, 23 Feb 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38402021-02-23T00:00:00ZLamboglia, SaraUlirsch, MartinWhat is the dollar game? What can you do to win
it? Can you always win it? In this snapshot you
will find answers to these questions as well as several
of the mathematical surprises that lurk in the background,
including a new perspective on a century-old theorem.Quantum symmetry
http://publications.mfo.de/handle/mfo/3831
Quantum symmetry
Caspers, Martijn
The symmetry of objects plays a crucial role in many
branches of mathematics and physics. It allowed, for
example, the early prediction of the existence of new
small particles. “Quantum symmetry” concerns a
generalized notion of symmetry. It is an abstract
way of characterizing the symmetry of a much richer
class of mathematical and physical objects. In this
snapshot we explain how quantum symmetry emerges
as matrix symmetries using a famous example: Mermin’s
magic square. It shows that quantum symmetries
can solve problems that lie beyond the reach of
classical symmetries, showing that quantum symmetries
play a central role in modern mathematics.
Thu, 31 Dec 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38312020-12-31T00:00:00ZCaspers, MartijnThe symmetry of objects plays a crucial role in many
branches of mathematics and physics. It allowed, for
example, the early prediction of the existence of new
small particles. “Quantum symmetry” concerns a
generalized notion of symmetry. It is an abstract
way of characterizing the symmetry of a much richer
class of mathematical and physical objects. In this
snapshot we explain how quantum symmetry emerges
as matrix symmetries using a famous example: Mermin’s
magic square. It shows that quantum symmetries
can solve problems that lie beyond the reach of
classical symmetries, showing that quantum symmetries
play a central role in modern mathematics.Amorphic Complexity of Group Actions with Applications to Quasicrystals
http://publications.mfo.de/handle/mfo/3830
Amorphic Complexity of Group Actions with Applications to Quasicrystals
Fuhrmann, Gabriel; Gröger, Maik; Jäger, Tobias; Kwietniak, Dominik
In this article, we define amorphic complexity for actions of locally compact $\sigma$-compact amenable groups on compact metric spaces. Amorphic complexity, originally introduced for $\mathbb Z$-actions, is a topological invariant which measures the complexity of dynamical systems in the regime of zero entropy. We show that it is tailor-made to study strictly ergodic group actions with discrete spectrum and continuous eigenfunctions. This class of actions includes, in particular, Delone dynamical systems related to regular model sets obtained via Meyer's cut and project method. We provide sharp upper bounds on amorphic complexity of such systems. In doing so, we observe an intimate relationship between amorphic complexity and fractal geometry.
Tue, 02 Feb 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38302021-02-02T00:00:00ZFuhrmann, GabrielGröger, MaikJäger, TobiasKwietniak, DominikIn this article, we define amorphic complexity for actions of locally compact $\sigma$-compact amenable groups on compact metric spaces. Amorphic complexity, originally introduced for $\mathbb Z$-actions, is a topological invariant which measures the complexity of dynamical systems in the regime of zero entropy. We show that it is tailor-made to study strictly ergodic group actions with discrete spectrum and continuous eigenfunctions. This class of actions includes, in particular, Delone dynamical systems related to regular model sets obtained via Meyer's cut and project method. We provide sharp upper bounds on amorphic complexity of such systems. In doing so, we observe an intimate relationship between amorphic complexity and fractal geometry.Lifting Spectral Triples to Noncommutative Principal Bundles
http://publications.mfo.de/handle/mfo/3827
Lifting Spectral Triples to Noncommutative Principal Bundles
Schwieger, Kay; Wagner, Stefan
Given a free action of a compact Lie group $G$ on a unital C*-algebra $\mathcal{A}$ and a spectral triple on the corresponding fixed point algebra $\mathcal{A}^G$, we present a systematic and in-depth construction of a
spectral triple on $\mathcal{A}$ that is build upon the geometry of $\mathcal{A}^G$ and $G$. We compare our construction with a selection of established examples.
Mon, 11 Jan 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38272021-01-11T00:00:00ZSchwieger, KayWagner, StefanGiven a free action of a compact Lie group $G$ on a unital C*-algebra $\mathcal{A}$ and a spectral triple on the corresponding fixed point algebra $\mathcal{A}^G$, we present a systematic and in-depth construction of a
spectral triple on $\mathcal{A}$ that is build upon the geometry of $\mathcal{A}^G$ and $G$. We compare our construction with a selection of established examples.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.