On this site you find the list of projects supported by the DFG priority programme „Geometry at Infinity“ during the second funding period 2020-2023. The pages of the individual projects provide information on their research goals, publications, members, and activities.
The project focuses on studying geometric properties for sofic approximations of discrete groups and manifolds. It consists of three interacting themes: sofic boundary actions, study of analytic/geometric properties of groups using actions at infinity, and sofic manifolds.
The focus of this project lies on the interactions of special geometric concepts as, for example, constituted by lower curvature bounds (in particular in the Alexandrov sense), isometric group actions (like cohomogeneity one actions), or symmetric structures and the topology of the underlying manifolds as primarily captured by invariants from equivariant cohomology or rational homotopy theory.
Building on the results of the previous project "Index theory on Lorentzian manifolds'' we investigate boundary value problems and index theory for first-order operators in both the Riemannian and the Lorentzian setting. This encompasses the derivation of geometric index formulas for general first-order elliptic operators, relative index theory a la Gromov and Lawson, boundary value problems for non-compact and non-smooth boundary, higher index theory and Callias-type operators on the Riemannian side. On Lorentzian manifolds we investigate local index theory for Dirac-type and more general operators, initial-boundary value problems and the characteristic initial value problem for Dirac operators.
This project is concerned with diffeomorphism or homeomorphism groups of surfaces (of genus \(g \geq 1\). Closed surfaces \(S_g\) of genus \(g\geq 2\) are among the most basic, and fundamental objects in geometry and topology, and so is the study of their homeomorphism, diffeomorphism and mapping class groups.
We investigate higher geometric invariants (as introduced by Bieri and Renz) for arithmetic groups. Since arithmetic groups are lattices, we also want to develop the theory of higher geometric invariants for locally compact groups.
40Construction of Riemannian manifolds with scalar curvature constraints and applications to general relativity
In this project we will apply PDE techniques to construct families of non time-symmetric asymptotically flat Riemannian manifolds. These families of manifolds will then be used test and study the positive mass theorem and the Penrose inequality — two fundamental conjectures in general relativity. If time allows, the stability of these conjectures for the aforementioned families of manifolds will be studied.
The main objective of this project is to study the relation of geometric, asymptotic foliations of an asymptotically flat manifold to the asymptotic structure itself. Of main interest are physical and geometric invariants such as mass and center of mass. We wish to construct geometrically defined asymptotic center of mass coordinate systems to study the condition of being asymptotically flat from a purely invariant point of view.
The goal of this project will be to develop synthetic notions of Ricci flow in the setting of time-dependent metric measure spaces, to investigate the monotonicity of natural analogues of Perelman’s W-entropy and related quantities under these flows and corresponding rigidity results, as well as to develop geometric and analytic comparison results for non-smooth Ricci flows aiming for new insights beyond smooth manifolds.
47Self-adjointness of Laplace and Dirac operators on Lorentzian manifolds foliated by noncompact hypersurfaces
Geometric operators on spaces with fibred cusp or fibred boundary geometry are studied using tools from (singular) microlocal analysis. These include spectral boundary value problems on families of spaces degenerating to fibred boundary spaces, operators having non-constant rank normal operators and the microlocalization of left-invariant vector fields on the Lie group SLn.
50Probabilistic and spectral properties of weighted Riemannian manifolds with Kato bounded Bakry-Emery-Ricci curvature
The aim of this project is to study geometric, analytic and probabilistic aspects of noncompact weighted Riemannian manifolds, with a view towards studying the geometry of (molecular) Schrödinger operators.
Given a homotopy class of a map between two Riemannian manifolds, it is natural to look for representatives with optimal geometric properties. If the domain is a Riemann surface and the target is a suitable locally symmetric manifold of non-positive curvature, then such a harmonic map is unique, and the energy of these maps defines a function on Teichmueller space. The main goal of the project is to use this energy function to establish structural geometric properties of the locally symmetric manifold.
52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II
This project is concerned with the construction of invariants of compact G2 manifolds by gauge-theoretic means. Conjecturally, an invariant, which remains invariant along isotopies of G2 metrics, can be obtained by counting G2 instantons together with certain Seiberg-Witten monopoles on distinguished 3-submanifolds of the ambient G2 manifold. The focus of this project is on the interplay between G2 instantons and the Seiberg-Witten monopoles.
This project investigates to what extent the geometry at infinity of a Riemannian symmetric space of non-compact type and the cohomology of its compact dual symmetric space reflect each other. To this end, we investigate the comparison map between the continuous bounded cohomology of a semisimple Lie group (which contains information about the asymptotic geometry of the Riemannian symmetric space) and its continuous cohomology (which is is isomorphic to the cohomology of the compact dual symmetric space).
The goal of the project is to compute the Willmore energy of Lawson surfaces and find a pathway towards an implicit function theorem argument to confirm the Kusner conjecture for (symmetric) compact surfaces of high genus.
How much information on a residually finite group can one recover from its finite quotients? In this research project, we want to investigate to what extend L2-cohomological properties of groups, in particular lattices in Lie groups, are determined by the profinite completions.
The goal of the project is to paint a detailed picture of the rigidity landscape for groups of key importance in Geometric Group Theory, namely Chevalley groups, Mapping Class Groups, and outer automorphism groups. To do this, we will study the actions of these groups on infinite-dimensional Euclidean geometries. More specifically, we will investigate the presence of Kazhdan's property (T), and we will provide lower bounds for Kazhdan constants when they exist. This way we will study rigidity both from the qualitative and the quantitative point of view.
This project investigates the interaction between a complex or split real Kac-Moody symmetric space and its twin building at infinity. The aims are to use Galois descent in order to introduce and study almost split real Kac-Moody symmetric spaces and to establish Kostant convexity, which in turn would imply that the causal structure on a Kac-Moody symmetric space does not allow time travel.
We will study the metric structure and the large-scale geometry of Riemannian symmetric spaces and Euclidean buildings from a unified viewpoint. Our uniform approach will have applications to S-arithmetic groups, to Kostant-type convexity and to automorphisms.
The project will be devoted to treatment of spectral geometric questions, index theory and geometric flows using the currently available microlocal methods on simple edge spaces with constant indicial roots.
The goal of this project is to associate a meaningful spectrum of dynamical resonances to a reasonable class of non-compact locally symmetric spaces of higher rank. We focus on spaces arising from Anosov representations, which provide a generalization of convex cocompact spaces.
In this project we study the asymptotic behavior of singularities in constraint geometric flows and Morse theoretical questions. For the latter, we focus on Willmore surfaces. An important task in this setting is to understand and use the relation of Willmore surfaces to Minimal surfaces.
The goal of this project is to explore analytic aspects of the geometry of moduli spaces of parabolic Higgs bundles in genus 0, such as their asymptotic hyperkähler geometry at infinity, the topology of their tautological classes, as well as the dependence of the latter on variations of parabolic weights and wall-crossing. These problems will be approached in terms of explicitly constructed geometric models, and together they will constitute an alternative and complementary perspective to the standard methods to understand the complex geometry of these moduli spaces.
In this project we will study the spectral properties of noncompact Riemannian locally symmetric spaces with twists of non-expanding cusp monodromy. This class of twists reaches far beyond the set of representations that are unitary at cusps, and it constitutes a frontier until which we might currently expect a twisted spectral theory. Our investigations will focus on hyperbolic spaces (thus, Riemannian locally symmetric spaces of noncompact type and of rank 1) but we will also carry out first steps towards an extension of the expected results to locally symmetric spaces of higher rank as well as to non locally symmetric spaces with hyperbolic ends.
The project aims at studying geometric properties of the locally symmetric spaces associated to Anosov and maximal representations; both generalizing the classical structure of Teichmüller theory and discovering new higher rank phenomena.
In the study of locally compact groups and their representations, the p-adic groups, or more generally totally disconnected groups, form an important special case, complementary to the much studied Lie groups. Important aspects of the representation theory of such a totally disconnected group G are governed by the K-theory of their reduced C*-algebra. The Baum-Connes conjecture (known in many cases) identifies this with the G-equivariant K-homology of the universal proper G-space. The goal of the project is a geometric description of equivariant K-homology for a proper G-space X, where G is a totally disconnected locally compact group. We aim for a cycle model based on spaces generalizing Bruhat-Tits buildings, containing additional index theoretic information. We then plan to construct in a geometric way a Chern character isomorphism to a computable equivariant homology. Secondly and as one building block for this Chern character, we plan to develop a new and particularly convenient model for the classifying space of G-equivariant K-theory for such a totally disconnected group G. We will use this to construct in a geometric way a Chern character for equivariant K-theory, and ultimately, a geometric construction of bivariant equivariant K-theory and a bivariant equivariant Chern character. A comparison with previous, non-geometric constructions (in particular for compact and discrete groups) will be carried out. This opens the way for applications in representation theory of p-adic groups and a deeper K-theoretic understanding of (discrete) arithmetic groups and their proper actions via the use of their non-Archimedean completions.
This project studies various problems in nearly G2 geometry. There are three main directions proposed. First the stability of nearly G2 metrics among Einstein metrics shall be considered. Then the question whether or not nearly G2 structures are rigid is to be studied. This should include the study of the deformation theory for the second Einstein metric on 7-dimensional 3-Sasaki manifolds and finally the problem of the existence of deformations for associative submanifolds in nearly G2 geometry.
The aim of this project is to further investigate the types of finite time singularities that occur for the Ricci flow in four dimensions in the real case, and higher dimensions in the Kaehler case, when the scalar curvature is bounded in the L^p norm.
The project is related to the analysis of forming singularities under the Lagrangian mean curvature flow. Of particular interest are type-II singularities and singularities of Lagrangian spheres and product of spheres.
The project aims at completing our picture of the asymptotic geometry of the moduli space of Higgs bundles with its hyperkähler metric. The methods and expected results will crucially be used to answer questions concerning the asymptotic structure of the nonabelian Hodge correspondence and its relationship with representation varieties. In a related but different direction, the subclass of parabolic Higgs bundle moduli spaces which give rise to four-dimensional hyperkähler manifolds is given particular attention, where one goal is to gain a better understanding of various spaces of gravitational instantons from this point of view.
Alexandrov spaces are a generalization of complete Riemannian manifolds with a lower sectional curvature bound. However, they may exhibit behaviors different from Riemannian manifolds due to the topological and metric singularities that they carry. It is then of fundamental importance to investigate whether one can extend a given property in the Riemannian setting to the Alexandrov setting. In this project we address some of these properties and explore how Alexandrov spaces behave with respect to each property. In the one direction, we focus our attention on the topological features. The primary objective here is to understand how far the topology of Alexandrov spaces are from that of the smooth manifolds. In the other direction, we examine Alexandrov spaces with positive curvature in the presence of symmetry and the goal is to classify them. To this end, we need to find obstructions and recognition tools, which, in particular, rely upon our understanding of topological behaviors of Alexandrov spaces investigated in the former direction.
We consider nonlocal boundary value problems associated with group actions on manifolds. Our goal: finding ellipticity conditions for these operators that entail the Fredholm property in adapted Sobolev spaces and establishing index formulae. Moreover, we intend to study semiclassical asymptotics for these operators.