Räume und Operatoren

The tannery is a surface with one orbifold singularity. Its unit tangent bundle is foliated by closed geodesics.
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Forschungsschwerpunkt B

Bartels, Bellissard (bis 2021), Böhm, Courtney (2022-2023), Cuntz, Ebert, Echterhoff, Gardam (2022-2023), Gardella (bis 2021), Geffen (seit 2022), Hein (seit 2020), Holzegel (seit 2020), Joachim, Kerr (seit 2021), Kramer, de Laat, Lohkamp, Löwe, Ludwig (seit 2022), Santoro (seit 2022), Siffert (2020-2023), Weber (seit 2022), Weiss, Wiemeler (seit 2020), Wilking, Winter, Wulkenhaar, Zeidler.

Dieser Schwerpunkt ist in den Gebieten Differentialgeometrie, Topologie und Operatoralgebren verankert. Ziele in Riemannscher Geometrie sind beispielsweise der Beweis eines Durchmesser-Sphärensatzes für Riemannsche Mannigfaltigkeiten, sowie Bedingungen zu finden, unter denen Ricciflüsse auf gewissen Mannigfaltigkeiten keine Singularitäten entwickeln. Im Bereich Topologie werden wir mit Hilfe des sogenannten Funktorenkalküls Diffeomorphismengruppen und Räume von Einbettungen untersuchen und mit Konfigurationskategorien vergleichen. Weiter wollen wir den Raum der Metriken mit positiver Skalarkrümmung auf einer geschlossenen Spinmannigfaltigkeit und die Beziehung zur Fundamentalgruppe studieren. Durch Isomorphievermutungen, wie z.B. der Baum-Connes Vermutung über die K-Theorie von Gruppen-C*-Algebren, ergeben sich enge Beziehungen zum Bereich Operatoralgebren. Hier werden wir nukleare C*-Algebren durch K-Theorie klassifizieren und Anwendungen auf topologische dynamische Systeme weiterentwickeln; insbesondere werden wir eine Rigiditätstheorie für Gruppen-C*-Algebren bzw. für verschränkte Produkte entwickeln.

 

 

 

 

 

Weitere Forschungsprojekte von Mitgliedern des Forschungsschwerpunkts B

CRC 1442: Geometry: Deformation and Rigidity - Geometric evolution equations

Hamilton's Ricci flow is a (weakly parabolic) geometric evolution equation, which deforms a given Riemannian metric in its most natural direction. Over the last decades, it has been used to prove several significant conjectures in Riemannian geometry and topology (in dimension three). In this project we focus on Ricci flow in higher dimensions, in particular on heat flow methods, new Ricci flow invariant curvature conditions and the dynamical Alekseevskii conjecture.

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Project members: Burkhard Wilking, Christoph Böhm

CRC 1442: Geometry: Deformation and Rigidity - B01: Curvature and Symmetry

The question of how far geometric properties of a manifold determine its global topology is a classical problem in global differential geometry. Building on recent breakthroughs we investigate this problem for positively curved manifolds with torus symmetry. We also want to complete the classification of positively curved cohomogeneity one manifolds and obtain structure results for the fundamental groups of nonnegatively curved manifolds. Other goals include structure results for singular Riemannian foliations in nonnegative curvature and a differentiable diameter pinching theorem.

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Project members: Burkhard Wilking, Michael Wiemeler

CRC 1442: Geometry: Deformation and Rigidity - C03: K-theory of group algebras

We will study K-theory of group algebras via assembly maps. A key tool is the Farrell—Jones Conjecture for group rings and its extension to Hecke algebra. We will study in particular integral Hecke algebras, investigate Efimov’s continuous K-theory as an alternative to controlled algebra in the context of the Farrell-Jones conjecture, and study vanishing phenomena for high dimensional cohomology of arithmetic groups.

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Project members: Arthur Bartels

CRC 1442: Geometry: Deformation and Rigidity - B04: Harmonic maps and symmetry

Many important geometric partial differential equations are Euler–Lagrange equations of natural functionals. Amongst the most prominent examples are harmonic and biharmonic maps between Riemannian manifolds (and their generalisations), Einstein manifolds and minimal submanifolds. Since commonly it is extremely difficult to obtain general structure results concerning existence, index and uniqueness, it is natural to examine these partial differential equations under symmetry assumptions.

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Project members: Christoph Böhm, Anna Siffert

CRC 1442: Geometry: Deformation and Rigidity - D03: Integrability

We investigate blobbed topological recursion for the general Kontsevich matrix model, as well as the behaviour of Baker–Akhiezer spinor kernels for deformations of the spectral curve and for the quartic Kontsevich model. We study relations between spin structures and square roots of Strebel differentials, respectively between topological recursion and free probability. We examine factorisation super-line bundles on infinite-dimensional Grassmannians and motivic characteristic classes for intersection cohomology sheaves of Schubert varieties.

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Project members: Jörg Schürmann, Raimar Wulkenhaar, Yifei Zhao

CRC 1442: Geometry: Deformation and Rigidity - D01: Amenable dynamics via C*-algebras

We study Cartan pairs of nuclear C*-algebras through their completely positive approximations. We are particularly interested in Cartan pairs for which the ambient C*-algebra is classifiable by K-theory data, and we explore first steps to classify such pairs themselves, at least under suitable additional conditions.

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Project members: Wilhelm Winter

CRC 1442: Geometry: Deformation and Rigidity - D05: C*-algebras, groups, and dynamics: beyond amenability

Our project will explore the regularity properties of non-nuclear C*-algebras, with a particular emphasis on stable rank one and strict comparison. We focus on two main classes of examples: C*-algebras associated with non-amenable groups and crossed product C*-algebras arising from non-amenable actions on compact Hausdorff spaces. We intend to leverage dynamical tools, including dynamical comparison and the structure of topological full groups.

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Project members: Wilhelm Winter, David Kerr

CRC 1442: Geometry: Deformation and Rigidity - B03: Moduli spaces of metrics of positive curvature

We will develop a family version of coarse index theory which encompasses all existing index invariants for the understanding of spaces of positive scalar curvature (psc) metrics—the higher family index and index difference—as well as new ones such as family rho-invariants. This will enable the detection of new non-trivial elements in homotopy groups of certain moduli spaces of psc metrics. We will also further study the concordance space of psc metrics together with appropriate index maps.

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Project members: Johannes Ebert, Rudolf Zeidler

CRC 1442: Geometry: Deformation and Rigidity - B05: Scalar curvature between Kähler and spin

This project aims to connect recent developments in Kähler geometry and spin geometry related to lower scalar curvature bounds and the Positive Mass Theorem. We would like to sharpen the Cecchini-Zeidler bandwidth inequality in the case of Kähler metrics and to find new proofs and extensions of the Positive Mass Theorem. One setting of interest is the case of spin^c manifolds equipped with almost-Kähler metrics, particularly in real dimension 4.

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Project members: Rudolf Zeidler, Hans-Joachim Hein

CRC 1442: Geometry: Deformation and Rigidity - B06: Einstein 4-manifolds with two commuting Killing vectors

We will investigate the existence, rigidity and classification of 4-dimensional Lorentzian and Riemannian Einstein metrics with two commuting Killing vectors. Our goal is to address open questions in the study of black hole uniqueness and gravitational instantons. In the Ricci-flat case, the problem reduces to the analysis of axisymmetric harmonic maps from R^3 to the hyperbolic plane. In the case of negative Ricci curvature, a detailed understanding of the conformal boundary value problem for asymptotically hyperbolic Einstein metrics is required.

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Project members: Hans-Joachim Hein, Gustav Holzegel

CRC 1442: Geometry: Deformation and Rigidity - D04: Entropy, orbit equivalence, and dynamical tilings

This project aims to advance the theory of rigidity and classification for Bernoulli actions of general groups with respect to orbit equivalence and its quantitative strengthenings. One overarching problem is to determine the extent to which the boundary between rigid and flexible behaviour is reflected in geometric or analytic properties of the group, and specifically, whether such properties intervene in questions of entropy invariance under Shannon orbit equivalence.

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Project members: David Kerr

Comparison and rigidity for scalar curvature

Questions involving the scalar curvature bridge many areas inside mathematics including geometric analysis, differential geometry and algebraic topology, and they are naturally related to the mathematical description of general relativity.

There are two main flavours of methods to probe the geometry of scalar curvature: One goes back to Lichnerowicz and uses various versions of index theory for the Dirac equation on spinors. The other is broadly based on minimal hypersurfaces and was initiated by Schoen and Yau. On both types of methods there has been tremendous progress over recent years sparked by novel quantitative comparison and rigidity questions due to Gromov and by on-going attempts to arrive at a deeper geometric understanding of lower scalar curvature bounds.

In this proposal we view established landmark results, such as the non-existence of positive scalar curvature on the torus, together with the more recent quantitative problems from a conceptually unified standpoint, where a comparison principle for scalar and mean curvature along maps between Riemannian manifolds plays the central role.

Guided by this point of view, we aim to develop fundamentally new tools to study scalar curvature that bridge long-standing gaps in between the existing techniques. This includes a far-reaching generalization of the Dirac operator approach expanding upon techniques pioneered by the PI, and novel applications of Bochner-type methods. We will also study analogous comparison problems on domains with singular boundary motivated by a first synthetic characterization of lower scalar curvature bounds in terms of polyhedral domains, and by the general quest for extending the study of scalar curvature beyond smooth manifolds. At the same time, we will treat subtle almost rigidity questions corresponding to the rigidity aspect of our comparison principle.

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Project members: Rudolf Zeidler

Global Estimates for non-linear stochastic PDEs

Semi-linear stochastic partial differential equations: global solutions’ behaviours
Partial differential equations are fundamental to describing processes in which one variable is dependent on two or more others – most situations in real life. Stochastic partial differential equations (SPDEs) describe physical systems subject to random effects. In the description of scaling limits of interacting particle systems and in quantum field theories analysis, the randomness is due to fluctuations related to noise terms on all length scales. The presence of a non-linear term can lead to divergencies. Funded by the European Research Council, the GE4SPDE project will describe the global behaviour of solutions of some of the most prominent examples of semi-linear SPDEs, building on the systematic treatment of the renormalisation procedure used to deal with these divergencies.

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Project members: Hendrik Weber

Dynamical systems and irregular gradient flows The central goal of this project is to study asymptotic properties for gradient flows (GFs) and related dynamical systems. In particular, we intend to establish a priori bounds and related regularity properties for solutions of GFs, we intend to study the behaviour of GFs near unstable critical regions, we intend to derive lower and upper bounds for attracting regions, and we intend to establish convergence speeds towards global attrators. Special attention will be given to GFs with irregularities (discontinuities) in the gradient and for such GFs we also intend to reveal sufficient conditions for existence, uniqueness, and flow properties in dependence of the given potential. We intend to accomplish the above goals by extending techniques and concepts from differential geometry to describe and study attracting and critical regions, by using tools from convex analysis such as subdifferentials and other generalized derivatives, as well as by employing concepts from real algebraic geometry to describe domains of attraction. In particular, we intend to generalize the center-stable manifold theorem from the theory of dynamical systems to the considered non-smooth setting. Beside finite dimensional GFs, we also study GFs in their associated infinite dimensional limits. The considered irregular GFs and related dynamical systems naturally arise, for example, in the context of molecular dynamics (to model the configuration of atoms along temporal evoluation) and machine learning (to model the training process of artificial neural networks).
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Project members: Christoph Böhm, Arnulf Jentzen