Spaces and Operators

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

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

In Area B we will study spaces and in particular manifolds, dynamical systems, geometric structures on manifolds, symmetries and automorphisms of manifolds. A central topic is the classification theory of separable nuclear C*-algebras. We will apply and advance a wide range of methods and tools, such as the Ricci flow, heat flow methods, Gromov-Hausdorff convergence of sequences of manifolds, singular foliations, topological and algebraic K-theory, surgery theory, cobordism categories, index theory, notions of dimension and regularity for dynamical systems and C*-algebras, notions of positive and negative curvature, and the isomorphism conjectures of Baum-Connes and Farrell-Jones. We will establish a differentiable diameter sphere theorem and study manifolds of positive curvature and torus actions on them. We will study the topology of spaces of positive scalar curvature metrics on manifolds and the topology of groups of homeomorphisms and diffeomorphisms of manifolds. For nuclear C*-algebras we will attack the UCT problem and quasi-diagonality.

  • B1. Smooth, singular and rigid spaces in geometry.

    Many interesting classes of Riemannian manifolds are precompact in the Gromov–Hausdorff topology. The closure of such a class usually contains singular metric spaces. Understanding the phenomena that occur when passing from the smooth to the singular object is often a first step to prove structure and finiteness results. In some instances one knows or expects to define a smooth Ricci flow coming out of the singular objects. If one were to establish uniqueness of the flow, the differentiable stability conjecture would follow. If a dimension drop occurs from the smooth to the singular object, one often knows or expects that the collapse happens along singular Riemannian foliations or orbits of isometric group actions.

    Rigidity aspects of isometric group actions and singular foliations are another focus in this project. For example, we plan to establish rigidity of quasi-isometries of CAT(0) spaces, as well as rigidity of limits of Type III Ricci flow solutions and of positively curved manifolds with low-dimensional torus actions. We will also investigate area-minimising hypersurfaces by means of a canonical conformal completion of the hypersurface away from its singular set.

  • B2. Topology.

    We will analyse geometric structures from a topological point of view. In particular, we will study manifolds, their diffeomorphisms and embeddings, and positive scalar curvature metrics on them. Via surgery theory and index theory many of the resulting questions are related to topological K-theory of group C*-algebras and to algebraic K-theory and L-theory of group rings.

    Index theory provides a map from the space of positive scalar curvature metrics to the K-theory of the reduced C*-algebra of the fundamental group. We will develop new tools such as parameterised coarse index theory and combine them with cobordism categories and parameterised surgery theory to study this map. In particular, we aim at rationally realising all K-theory classes by families of positive scalar curvature metrics. Important topics and tools are the isomorphism conjectures of Farrell–Jones and Baum–Connes about the structure of the K-groups that appear. We will extend the scope of these conjectures as well as the available techniques, for example from geometric group theory and controlled topology. We are interested in the K-theory of Hecke algebras of reductive p-adic Lie groups and in the topological K-theory of rapid decay completions of complex group rings. Via dimension conditions and index theory techniques we will exploit connections to operator algebras and coarse geometry. Via assembly maps in algebraic K- and L-theory we will develop index-theoretic tools to understand tangential structures of manifolds. We will use these tools to analyse the smooth structure space of manifolds and study diffeomorphism groups. Configuration categories will be used to study spaces of embeddings of manifolds.

  • B3. Operator algebras & mathematical physics.

    The development of operator algebras was largely motivated by physics since they provide the right mathematical framework for quantum mechanics. Since then, operator algebras have turned into a subject of their own. We will pursue the many fascinating connections to (functional) analysis, algebra, topology, group theory and logic, and eventually connect back to mathematical physics via random matrices and non-commutative geometry.


Further research projects of Research Area B members

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Project members: David Kerr

SFB 1442: Geometrie: Deformation und Rigidität - B07: Analytische und typologische Torsion

In this project we study analytic and topological torsion using analytic, topological and arithmetic tools and ideas: In the first subproject we will generalise the Cheeger-Müller comparison theorem of torsion to spaces with wedge singularities. In the second subproject we initiate the study of higher analytic and topological torsion for singular fibrations. In the third subproject we will establish a formula for special values of certain Ruelle zeta functions in analogy with conjectured formulas in arithmetic geometry.

Project members: Christopher Deninger, Ursula Ludwig

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.

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.

Project members: Hendrik Weber

Quantization, Singularities and Holomorphic Dynamics

The goal of the team brought together in this project is to bundle our expertise with the aim of making important contributions to a number of fundamental problems and conjectures in Quantization, Holomorphic Dynamics and Foliation Theory. We will exhibit and exploit the deep ties between these areas and bring them to bear on diverse open questions. Our goal is to provide fresh perspectives and novel problem-solving strategies to encompass these fields and in the long-term to foster in the wider research community a stronger unification of these parts of mathematics.Using as a cornerstone the development of the theory of currents in the complex setting and of the Bergman/Szegö kernel (including L^2 methods) and their systematic exploitation in the study of a number of topics, we address the following interrelated questions: commutation of quantization and reduction on Kähler spaces and Cauchy-Riemann manifolds; hamiltonian actions; quantization of the space of Kähler potentials and of adapted complex structures; Bergman kernel asymptotics, analytic torsion, Newlander-Nirenberg theorem on complex spaces; singularities and accumulation points of a leaf of a holomorphic foliation, especially with non-hyperbolic singularities; unique ergodicity for singular holomorphic foliations; quantitative counting of dynamical phenomena for holomorphic dynamical systems, both in the phase and parameter spaces; equidistribution of zeros of random holomorphic sections.

Project members: Ursula Ludwig

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).
Project members: Christoph Böhm, Arnulf Jentzen

RTG 2149: Strong and Weak Interactions - from Hadrons to Dark Matter The Research Training Group (Graduiertenkolleg) 2149 "Strong and Weak Interactions - from Hadrons to Dark Matter" funded by the Deutsche Forschungsgemeinschaft focuses on the close collaboration of theoretical and experimental nuclear, particle and astroparticle physicists further supported by a mathematician and a computer scientist. This explicit cooperation is of essence for the PhD topics of our Research Training Group.Scientifically this Research Training Group addresses questions at the forefront of our present knowledge of particle physics. In strong interactions we investigate questions of high complexity, such as the parton distributions in nuclear matter, the transition of the hot quark-gluon plasma into hadrons, or features of meson decays and spectroscopy. In weak interactions we pursue questions, which are by definition more speculative and which go beyond the Standard Model of particle physics, particularly with regard to the nature of dark matter. We will confront theoretical predictions with direct searches for cold dark matter particles or for heavy neutrinos as well as with new particle searches at the LHC.The pillars of our qualification programme are individual supervision and mentoring by one senior experimentalist and one senior theorist, topical lectures in physics and related fields (e.g. advanced computation), peer-to-peer training through active participation in two research groups, dedicated training in soft skills, and the promotion of research experience in the international community. We envisage early career steps through a transfer of responsibilities and international visibility with stays at external partner institutions. An important goal of this Research Training Group is to train a new generation of scientists, who are not only successful specialists in their fields, but who have a broader training both in theoretical and experimental nuclear, particle and astroparticle physics. online
Project members: Raimar Wulkenhaar

Amenability, Approximation and Reconstruction Algebras of operators on Hilbert spaces were originally introduced as the right framework for the mathematical description of quantum mechanics. In modern mathematics the scope has much broadened due to the highly versatile nature of operator algebras. They are particularly useful in the analysis of groups and their actions. Amenability is a finiteness property which occurs in many different contexts and which can be characterised in many different ways. We will analyse amenability in terms of approximation properties, in the frameworks of abstract C*-algebras, of topological dynamical systems, and of discrete groups. Such approximation properties will serve as bridging devices between these setups, and they will be used to systematically recover geometric information about the underlying structures. When passing from groups, and more generally from dynamical systems, to operator algebras, one loses information, but one gains new tools to isolate and analyse pertinent properties of the underlying structure. We will mostly be interested in the topological setting, and in the associated C*-algebras. Amenability of groups or of dynamical systems then translates into the completely positive approximation property. Systems of completely positive approximations store all the essential data about a C*-algebra, and sometimes one can arrange the systems so that one can directly read of such information. For transformation group C*-algebras, one can achieve this by using approximation properties of the underlying dynamics. To some extent one can even go back, and extract dynamical approximation properties from completely positive approximations of the C*-algebra. This interplay between approximation properties in topological dynamics and in noncommutative topology carries a surprisingly rich structure. It connects directly to the heart of the classification problem for nuclear C*-algebras on the one hand, and to central open questions on amenable dynamics on the other. online
Project members: Wilhelm Winter