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International Conference on Differential Geometry, Geometric Topology, and Global Analysis
This is the second conference covering the entire thematic breadth of the DFG Priority Programme SPP 2026 "Geometry at Infinity''. In plenary lectures, important developments in the research areas of the Priority Programme will be presented by internationally leading scientists. In addition, the members of the Priority Programme will report on their research results from the second funding period in parallel sessions.
Pseudo-Anosov flows on 3-manifolds are dynamical systems generalizing the behavior of geodesic flow on the unit tangent bundle of a hyperbolic surface. Like geodesic flows, they come with two transverse, invariant 2-dimensional foliations (possibly with some prong singularities) which meet along the 1-dimensional foliation by orbits. Because of various surgery techniques, there are many examples known, and their "topological" classification is an interesting and important problem both in low-dimensional geometric topology and dynamics. I will describe some of this framework, and then some joint work with T. Barthelmé, S. Frankel, S. Fenley and C. Bonatti, on describing the structure and classification of such flows and their associated foliatoins.
In 2011, V. Kapovitch and B. Wilking proved that in any given dimension $n$ there is a constant $C(n)$ such that the fundamental group of a closed $n$-dimensional Riemannian manifold with nonnegative Ricci curvature has a nilpotent subgroup with index less than $C(n)$. A natural question left in their work is whether one can strengthen this result by replacing "nilpotent" with "abelian". I will discuss joint work with E. Bruè and A. Naber where we answer Kapovitch and Wilking’s question in the negative.
I will present recent progress on finding homological characterisation of when a RFRS Poincaré-duality group virtually fibres, that is, admits a virtual epimorphism to the integers with kernel that is itself a Poincaré-duality group of one dimension less. Based on joint work with S. Fisher and G. Italiano.
Fine curve graphs have been introduced by Bowden, Hensel and Webb to study homeomorphism and diffeomorphism groups of closed surfaces. A main tool in their work is the fact that fine curve graphs can be approximated by curve graphs of surfaces with punctures. I will talk about joint work with Sebastian Hensel, where we study to which extent the boundary of the fine curve graph can be approximated via curve graphs of surfaces with punctures. If time permits, I will also show how fine curve graph techniques can be used to construct a parabolic isometry of a graph of curves of an infinite-type surface.
Coarse assembly maps are the main players in important conjectures like coarse Baum-Connes or coarse Farrell-Jones. By the descent principle, these are intimately connected to the usual Baum-Connes and Farrell-Jones conjectures. In this talk I discuss a potential proof of these conjectures for exact (also called Property A) spaces resp. groups. This is work in progress with Ulrich Bunke.
I will discuss the following scalar curvature rigidity result for the round sphere in the low regularity setting. Let M be a closed smooth connected spin manifold of dimension n, and let g be a Riemannian metric on M of Sobolev regularity W1,p, for p>n, whose distributional scalar curvature in the sense of Lee-LeFloch is bounded below by n(n−1). Let f:(M,g)→Sn be a Lipschitz continuous map, that is area-nonincreasing almost everywhere and has non-zero degree. Then f is a metric isometry. In this talk, I will focus on the odd-dimensional case, using the analysis of Lipschitz Dirac operators on manifolds with conical singularities.
This is based on joint work with Bernhard Hanke, Thomas Schick, and Lukas Schoenlinner
In this talk we present the homological version of $\Sigma$-theory for locally compact Hausdorff groups, the corresponding talk for the homotopical version will be given by José in the same parallel session. Both versions are connected by a Hurewicz-like theorem. They can be thought of as directional versions of type $\mathrm{CP}_m$ and type $\mathrm{C}_m$, respectively. And classical $\Sigma$-theory is recovered if we equip an abstract group with the discrete topology.We provide criteria for type $\mathrm{CP}_m$ and homological locally compact $\Sigma^m$. In the setting of an exact sequence of locally compact Hausdorff groups, we study in which way compactness properties of the kernel/extension/quotient can be derived from the other two groups in the sequence. This project is joint work with Kai-Uwe Bux and José Pedro Quintanilha.
While the singularity formation of the flow by curvature of curves in the plane is well understood, much less is known about the flow of curves in higher codimension. In particular, closed embedded planar curves stay embedded, eventually become convex and shrink to a round point in finite time, but this is not necessarily the case for curves in codimension two and higher. However, it turns out that these curves in fact become asymptotically planar in the vicinity of a singularity; and combined with a bound on a suitably defined entropy of the inital curve we can show that such curves do become circular and thus shrink to a point. We will also survey some more recent developments on curve shortening flow in codimensions higher than one.
We study the behaviour of noncompact graphical solutions to mean curvature flow and present results obtained in this priority programme, in particular by Wolfgang Maurer.
I introduce recent developments in coarse homotopy theory that build on prior results computing the coarse homotopy groups of cones over finite simplicial complexes cX, which coincide with the usual homotopy groups of the underlying space X (Mitchener, Norouzizadeh, Schick, 2020). Extending these ideas, I present new results on the coarse homotopy groups of cones over compact metric spaces. These are related to the Čech homotopy groups of the underlying space X via a $\varprojlim^{1}$ sequence. In this talk, I will provide a brief outline of the proof, present some interesting examples, and discuss the implications of this work in bridging coarse homotopy theory and shape theory.
The features of a group being finitely generated or finitely presented are, respectively, the n=1 and n=2 cases of the finiteness property $F_n$. Towards the end of the last century, the question of when these finiteness conditions descend to subgroups of G led to the discovery of the sets $\Sigma^n(G)$ (and their homological counterparts $\Sigma^n(G;A)$, for A a Z[G]-module). Each set $\Sigma^n(G)$ is a collection of homomorphisms G --> R, refining property $F_n$ in the sense that G has type $F_n$ precisely if $\Sigma^n(G)$ contains the zero map. In the literature, $\Sigma$-sets are also often called BNSR-invariants, due to Bieri, Neumann, Strebel and Renz, who pioneered the theory.Another direction in which to generalize finiteness properties is to consider groups G with a locally compact Hausdorff topology. In that setting, Abels and Tiemeyer introduced the compactness properties $C_n$, and used a well known criterion of Brown to show that they specialize to $F_n$ for G discrete. In joint work with Kai-Uwe Bux and Elisa Hartmann, we have refined these properties $C_n$ to sets $\Sigma^n(G)$, with our definition recovering the classical Sigma sets in the discrete case. We have also generalized various results of classical Sigma-theory to the setting of locally compact groups. I will give an introduction to the theory of Sigma sets and explain some of these results.
We will discuss some details of the proof of a Lipschitz rigidity result that will be stated in the talk by Simone Cecchini that precedes this talk.
I will introduce an abstract functional analytic setup that allows the analysis of the Dirac operator on manifolds with cone-like singularities. This setup is based on the work of Brüning and Seeley. This will be used to derive an index formula for the twisted Lipschitz Dirac operator on the spherical suspension from an index result of Chou.
This is based on joint work with Simone Cecchini, Bernhard Hanke and Thomas Schick.
We consider hyperbolic spaces in the presence of a representation on the fundamental group. We report on the status of our project to study the spectral theory in this setting.
A metric flow (M, g(t))(0,T ) on a compact manifold M sat- isfying the Ricci-flow equation is said to have a metric space (X,d) as its initial condition if (X, d) is the Gromov-Hausdorff limit of (M, dg(t)) as t → 0. The question of what assumptions on (X,d) guarantee the existence of a Ricci flow (M, g(t)) with (X, d) as its initial condition, and how further regularity assumptions on (X, d) and (M, g(t)) can improve the convergence, is among the primary problems in this area. In this talk, I will present results on the existence and uniqueness of a solution to the Ricci flow, where the initial condition is a compact length space with bounded curvature, specifically a space that is both Alexandrov and CAT, and discuss how these conditions strengthen the convergence. This work is joint with Diego Corro and Adam Moreno.
Is the volume of a locally symmetric space determined by the profinite completion of its fundamental group?
We show that this is the case in higher rank: the covolume of an irreducible lattice in a higher rank semisimple Lie group with the congruence subgroup property is determined by the profinite completion. It is an open question whether a similar result holds for hyperbolic 3-manifolds. We explain how our methods generalize to another rank-one situation: octonionic hyperbolic congruence manifolds.
(This is based on joint work with Holger Kammeyer and Ralf Köhl)
For differential operators preserved by the action of a group $G$, the notion of index generalises to the $G$-index. A prototype arises for Dirac operators on a compact Riemannian spin manifold admitting an isometry group $G$ and the resulting theorem is known as the Atiyah-Segal-Singer index theorem. In this talk, I shall discuss a generalization of this theorem for a $G$-equivariant Dirac operator on a globally hyperbolic spin spacetime with spacelike boundaries subject to the Atiyah-Patodi-Singer boundary condition. Our analysis is based on the singularity structure of Feynman parametrices instead of the heat-kernel proof due to Berlin and Vergne. (Joint work with C. Bär and L. Ronge).
In this talk, we show that localised, weighted curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed $n$-dimensional Kähler Ricci flow always hold. These integral estimates improve and extend the integral curvature estimates shown in an earlier paper by the speaker. If $M^4$ is closed and four dimensional, and the spatial $L^p$ norm of the scalar curvature is uniformly bounded for some $p>2$, for $t\in [0,T),$ $T< \infty$, then we show:a) a uniform bound on the spatial $L^2$ norm of the Riemannian curvature tensor for $t\in [0,T)$,b) uniform non-expanding and non-inflating estimates for $t\in [0,T)$,c) convergence to an orbifold as $t \to T$,d) existence of an extension of the flow to times $t\in [0,T+\sigma)$ for some $\sigma>0$ using the orbifold Ricci flow.
This is joint work with Jiawei Liu.
Many common ad hoc definitions of bivariant K-theory for non-separable C*-algebras have some kind of drawback, usually that one cannot expect the long exact sequences to hold in full generality. I will present a way to define E-theory for non-separable C*-algebras without such disadvantages. There is indication that cycles of this new model might arise naturally in index theory on infinite dimensional manifolds.
In this talk I will introduce the notion of an apartment system of a complete CAT(0) space. This setting allows us to simultaneously study the geometry of Euclidean buildings and symmetric spaces of non-compact type and give uniform proofs of well-known properties of these spaces.
The Fredholm properties of a Lorentzian Dirac operator depend on the choice of boundary conditions that one imposes. The most relevant ones, APS boundary conditions, are well understood, but for others it is not even clear if the operator is Fredholm. We investigate how far APS boundary conditions may be perturbed whithout destroying the Fredholm properties of the Dirac operator with those boundary conditions.
I'll explain how to associate to any Anosov subgroup a family of non-empty domains of discontinuity in such a way that in the torsion-free case each quotient manifold carries an Axiom A flow. These manifolds possess convenient geometric properties and the Axiom A flows extend Sambarino's refraction flows. In particular, this makes it possible to associate spectra of dynamical resonances to the Anosov subgroup, which was the main goal of my SPP project. This is joint work with Daniel Monclair and Andrew Sanders.
Let $\Gamma$ be a discrete group which acts on a manifold $N$. Suppose $M\subseteq N$ is a submanifold which is not $\Gamma$-invariant and has a boundary. There are partial shifts $U_g$ for $g\in\Gamma$ on $L^2(M)$ defined by $$ U_g\varphi(x)=\begin{cases} \varphi(g^{-1}\cdot x) &\text{if }g^{-1}\cdot x\in M,\\ 0 &\text{else.} \end{cases}$$In this talk, I will describe an algebra of operators generated by boundary value problems on $M$ and the partial shifts $U_g$ for $g\in\Gamma$ (under suitable assumptions on the action). As in the classical Boutet de Monvel calculus there are two principal symbol maps: one associated with the interior and one with the boundary. Here, they take values in crossed product algebras of corresponding partial group actions. I will discuss how one can classify the stable homotopy classes of elliptic operators over the considered algebra in terms of $K$-theory.
The talk is based on joint work with Anton Savin and Elmar Schrohe.
In the talk I will give an overview over some existence results for parametrized surfaces of minimal area in metric spaces and some applications of this existence.
A central theme in Kähler geometry is the search for canonical Kähler metrics, such as Kähler-Einstein metrics, constant scalar curvature Kähler metrics, extremal metrics, etc…
The Mabuchi functional M was introduced by Mabuchi in the 80's in relation to the existence of such canonical metrics on a compact Kähler manifold. The critical points of M are indeed the special metrics we look for.
Since then, the properties of the Mabuchi functional (such as convexity, properness) have been intensively studied. In this talk I will give a panorama on the state of the art and on the recent breakthroughs in the field.
I will present results obtained within the project about the topology of the moduli space of metrics satisfying a lower curvature bound. The focus will be on manifolds that are known to admit metrics of nonnegative/positive sectional or Ricci curvature. I will describe different classes of manifolds where the respective moduli space has non-trivial topology, e.g., has infinitely many connected components or non-trivial higher invariants. I will also try to explain some of the key ideas used in the proofs, as well as the limitations one faces when working with these curvature bounds.
We show that in dimension every dimension at least four, there exist infinitely many closed manifolds that admit an Einstein metric with negative sectional curvature, but that do not admit any locally symmetric metric. These are the first examples of such manifolds in dimensions at least five. If time permits, we quickly mention other results obtained in Project 51.
I will introduce the notion of a flat extension of a connection on a principal bundle. Roughly speaking, a connection admits a flat extension if it arises as the pull-back of a component of a Maurer–Cartan form. For trivial bundles over closed oriented 3-manifolds, I will relate the existence of certain flat extensions to the vanishing of the Chern–Simons invariant associated to the connection. Joint work with Andreas Cap & Keegan Flood.
In this talk I will present contributions to the complex differential geometry of moduli spaces of parabolic Higgs bundles on compact Riemann surfaces, that were obtained during the course of my project. Such results possess a unifying theme, namely, the interplay between the natural families of hyper-Kähler structures that such moduli spaces possess, and the families of stability conditions that are necessary to construct them. The former are uniquely determined by the Kähler structure of a special sub-moduli space of half dimension. Of special interest is the case when the chosen Riemann surface has genus 0, for which the relation admits a more explicit form.
In this talk we are going to present two collapsing procedures for singular Riemannian foliations, while maintaining some control on the sectional curvature and diameter of the ambient manifold. In the first one we consider a regular Riemannian foliation by flat tori on a compact manifold, and show that we can collapse the foliated manifold to the orbit space while preserving sectional curvature and diameter bounds. From this construction we obtain a rigidity result, proving that such a foliation on a simply-connected manifold is given by a smooth torus action.
We also present a deformation procedure for foliations induced by Lie groupoids, which extends the classical construction of Cheeger deformations. We also give an explicit description of the sectional curvature of the deformed metrics in the case of this generalized Cheeger deformation proceedure.
We consider Riemannian manifolds with boundary where the boundary exhibits singularities of fibred cusp type, or are conformal to these. A simple example is the complement of two touching balls in $\mathbb{R}^n$. This type of singularity (at the touching point), in case $n=2$, is often called an incomplete cusp (or horn). Other examples, conformal to these types of spaces and with 'singularity' at infinity, are fundamental domains of Fuchsian groups and uniformly fattened infinite cones in $\mathbb{R}^n$.
The Dirichlet-Neumann (DN) operator on a Riemannian manifold with boundary maps Dirichlet boundary data of harmonic functions to their Neumann data. This operator is well studied in the smooth compact case, for example it is known that it is a pseudodifferential operator (PsiDO), and its spectrum (the Steklov eigenvalues) has been studied intensively, as well as the inverse problem for it.
We show that the DN operator for fibred cusp singularities are in a PsiDO calculus adapted to the geometry, the so-called phi-calculus. This yields a precise description of their integral kernels near the singularities. In the talk I will introduce the necessary background on the phi-calculus, and also discuss some of the spectral properties of the DN operator in this setting.
This is joint work with K. Fritzsch und E. Schrohe.
We show that every quaternion-Kähler manifold of negative scalar curvature is stable as an Einstein manifold and therefore scalar curvature rigid. In particular, this implies that every irreducible nonpositive Einstein manifold of special holonomy is stable. In contrast, we demonstrate that there exist quaternion-Kähler manifolds of positive scalar curvature which are not scalar curvature rigid even though they are semi-stable.
We give a geometric approach for boundary value problems of the Laplacian with Dirichlet (or mixed) boundary conditions on domains with singularities. In two dimensions these singularities also include cusps. Our approach is by blowing up the singularities via a conformal change to translate the boundary problem to one on a noncompact manifold with boundary that is of bounded geometry and of finite width. This gives a natural geometric interpretation for the weights that appear and for the additional conditions needed to obtain well-posedness results.
This is joint work with Bernd Ammann and Victor Nistor.
I will survey recent results on the asymptotic geometry of Hitchin’s hyperkähler metric on the moduli space of Higgs bundles. I will then show how to obtain gravitational instantons from 4-dimensional Hitchin systems. Finally I will explain ongoing work with Fredrickson, Mazzeo and Swoboda on a particular case of Boalch’s modularity conjecture.
The aim of the talk is to explain a construction of certain harmonic maps into two copies of hyperbolic 3-space considered as half spheres in $S^3$ with the equatorial $S^2$ as the boundary at infinity for both hyperbolic spaces. We will also explain how these maps are related to certain sections of the Deligne–Hitchin moduli space of flat $\lambda-SL(2, C)$ connections and how this aids understanding the geometry of this space, which has many interesting properties, such as a natural hyperkähler structure.
For the Neumann heat flow on nonconvex Riemannian domains $D\subset M$, we provide sharp gradient estimates and transport estimates with a novel $\sqrt t$-dependence, for instance, $$\text{Lip}( P^D_tf)\le e^{2S \, \sqrt{t/\pi}+\mathcal{O}(t)}\cdot \text{Lip} (f),$$ and we provide an equivalent characterization of the lower bound $S$ on the second fundamental form of the boundary in terms of these quantitative estimates.
The Atiyah-Singer index theorem is a paramount result which depicts an interplay between Analysis, Topology and Geometry. Although its classical formulation takes place in the context of compact manifolds without boundary, subsequent developments have tried to extend it to broader classes of spaces. One of several lenses through which to analyse the problem, the so-called heat kernel method, seems to offer a way of looking for an analogue in singular spaces, for which there are many natural and relevant examples (algebraic varieties, moduli spaces, etc.).
In this talk, I would like to briefly discuss how this method can be adapted for a certain class of singular manifolds whose boundary is the total space of a fibration. The key ingredient will be a blow-up analysis à la Melrose of the asymptotics of the heat kernel.
The Mean Curvature Flow describes the evolution of a family of embedded surfaces in Euclidean space that move in the direction of the mean curvature vector. It is the gradient flow of the area functional and a natural analog of the heat equation for an evolving surface. Initially, this flow tends to smooth out geometries over brief time-intervals. However, due to its inherent non-linearity, the Mean Curvature Flow equation frequently leads to the formation of singularities. The analysis of such singularities is a central goal in the field.A long-standing conjecture addressing this goal has been the Multiplicity One Conjecture. Roughly speaking, the conjecture asserts that singularities along the flow cannot form by an "accumulation of several parallel sheets”. In recent joint work with Bruce Kleiner, we resolved this conjecture for surfaces in $R^3$. This had several applications. First, combining our work with previous results, we obtain that the problem of evolving embedded 2-spheres via the Mean Curvature Flow equation is well-posed within a natural class of singular solutions. Second, we remove an additional condition in recent work of Chodosh-Choi-Mantoulidis-Schulze to show that the Mean Curvature Flow starting from any generically chosen embedded surface only incurs cylindrical or spherical singularities. Third, our approach offers a new regularity theory for solutions of general Mean Curvature Flows that flow through singularities.
This talk is based on joint work with Bruce Kleiner.
I will give an overview of rigidity results that have played an important role in scalar curvature geometry in recent years. This includes metric inequalities under lower scalar curvature bounds as well as phenomena involving Riemannian metrics with (point) singularities.
The dominant energy condition (DEC) is a curvature condition for Lorentzian manifolds that is of great physical importance. The same name also denotes a condition for initial data sets -- the pairs of induced metric and second fundamental form on spacelike hypersurfaces -- that is induced by the spacetime DEC. In this talk, I want to elaborate on the connection between those two conditions and illustrate how the study of the initial data DEC can lead to consequences about DEC spacetimes.
In this talk we give an alternate existence proof of Lawson surfaces $\xi_{1,g}$ for $g\geq 3$ using complex analytic methods. When computing the Taylor approximation at $g= \infty$ we discover a pattern resulting in the coefficients of the involved expansions being alternating multiple zeta values (MZVs), a generalised notion of Riemann's zeta values to multiple integer variables. When specialising to the Taylor expansion of the area, we find for example that the third order coefficient is $\frac{9}{4}\zeta(4)$ (the first and second order tern are log(2) and 0, respectively). As a corollary of these higher order expansions, we obtain that the area of $\xi_{1,g}$ is monotonically increasing in their genus for all $g\geq0$. This is joint work with Steven Charlton, Sebastian Heller and Martin Traizet.
The notion of Kostant Convexity refers to a wide range of similar and related theorems in different areas of mathematics. In its original form, it is a classical result in Lie theory that was established by B. Kostant in 1973. It states that if G=KAU is an Iwasawa decomposition of a semisimple Lie group, then for any fixed a in A, the set of A-components of all elements in the left coset aK coincides with the convex hull of the Weyl group orbit of that chosen element. While this statement is completely algebraic, the theorem can be reformulated in many ways and has been transferred to various settings, e.g. symplectic geometry or the theory of buildings. In this talk, I will present a natural generalization of Kostant’s Convexity Theorem to the class of split real and complex Kac-Moody groups and some of its applications to their algebraic structure. Most notably, the result can be used to show that in the non-spherical case, the causal pre-order on Kac-Moody symmetric spaces introduced by Freyn, Hartnick, Horn and Köhl is a partial order.
Totally disconnected groups play an important role for arithmetic questions, in particular groups like $Sl_n(\mathbb{Q}_p)$. They have nice actions on their Bruhat-Tits building (with the special property that the orbits are discrete). Among the important invariants of such actions are the equivariant K-theory and equivariant K-homology, closely related also to the representation theory of these groups. For example, the equivariant K-homology of the Bruhat-Tits building features as the left hand side of the Baum-Connes conjecture for $Sl_n(\mathbb{Q}_p)$. Chern characters are an important tool for the computation of these equivariant K-theory groups. We present a new and very geometric construction of an equivariant Chern character, taking values in a suitably defined Alexander-Spanier cohomology. It takes values in an equivariant version of Alexander-Spanier cohomology. As a main result we prove that this Chern character is an isomorphism after tensor product with the complex numbers.
The goal of our project is to use ideas and methods from geometric group theory to study homeomorphism groups of surfaces. More concretely, we study the fine curve graph: a Gromov hyperbolic space on which the homeomorphism group acts in an interesting way.
In this talk, I will survey recent progress in this program, including a description of (part of) the Gromov boundary — which in turn allows to certify that most relative pseudo-Anosov homeomorphisms have positive stable commutator length — and connections between geometric properties of the graph to classical invariants in surface dynamics, namely rotation sets.
Asymptotically Euclidean initial data sets (IDS) in General Relativity model instants in time for isolated systems. In this talk, we show that an IDS is asymptotically Euclidean if it admits a cover by closed hypersurfaces of constant spacetime mean curvature (STCMC), provided these hypersurfaces satisfy certain geometric estimates, some weak foliation properties, and each
We present a new trace formula which connects heat semi-groups of a Dirac-Schrödinger operator with its operator potential. The trace formula induces a non-Fredholm extension of the classical Callias index theorem to the Witten index. We conclude with an invariant description of both sides of the generalized index formula in terms of (higher order) spectral shift functions.
We construct asymptotic foliations of asymtotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature (STCMC). For a surface in an ambient spacetime, the spacetime mean curvature is defined as the (Lorentzian) length of the co-dimension 2 mean curvature vector. Asymtotic foliations of asymptotically flat spacelike hypersurfaces by STCMC surfaces have previously been constructed by Cederbaum-Sakovich.
Our construction is motivated by the approach of Huisken-Yau for the Riemannian setting in employing a geometric flow. We show that initial data within a sufficient a-priori class converges exponentially to an STCMC surface under area preserving null mean curvature flow. We further show that the resulting STCMC surfaces form an asymptotic foliation. This is joint work with Klaus Kröncke.
The SL_3 Hitchin component is a moduli space of deformations of Fuchsian groups inside SL_3(R). This is an example of a higher rank Teichmüller space and can be seen from an analytic, algebraic, or dynamical perspective. For each of these points of view, there is a natural notion of "divergence to infinity". In this talk, we will discuss the relationships between these notions of divergence and show that they are often incompatible.
It is a classical result that one can obtain a sequence of expander graphs from a residually finite group with property (T), e.g. $SL(n, \mathbb{Z})$ for $n \geq 3$. Willet and Yu showed that these sequences of graphs even satisfy the stronger notion of geometric property (T), a property recently introduced by them to construct counterexamples to the coarse Baum-Connes conjecture. Going from residually finite groups to the more general sofic groups, another recent result by Kun says that sofic approximations of property (T) groups are up to a vanishing proportion of error terms a union of expander graphs. Building further on work of Alekseev and Finn-Sell and on work of Winkel we prove that sofic property (T) groups always admit a sofic approximation satisfying geometric property (T), which also characterizes groups with property (T) among the sofic groups. The tools used in the proof also apply to the case of general bounded degree graph sequences away from groups.
We discuss the renormalized analytic torsion of complete manifolds with fibred boundary metrics, also referred to as φ-metrics. We establish invariance of the torsion under suitable deformations of the metric, and establish a gluing formula. As an application, we relate the analytic torsions for complete φ- and incomplete wedge-metrics.
A central result within mathematical relativity is the well-posedness of the initial value formulation of GR for initial data satisfying the Einstein constraint equations. In this context, isolated gravitational systems are modelled by asymptotically Eu- clidean initial data sets and there are physically reasonable expectations about the asymptotic behaviour of these systems. It turns out that a rigorous mathemat- ical understanding of the conditions that guarantee these expectations is still an open problem. This is crucial for canonically conserved quantities carrying physical information to be well-defined. In the case of the ADM energy and linear momen- tum, precise geometric criteria making them well-defined are well-known, but for the ADM centre of mass and angular momentum this is not the case and ad-hoc asymptotic conditions tend to be demanded. In this talk, we will report on some recent results on regularity problems and asymptotic analysis of geometric partial differential equations which allow one to partially characterise, in pure geometric terms, those initial data sets which indeed obey these expected asymptotic proper- ties. In particular, we shall address a Riemannian version of a conjecture posed by Cederbaum-Sakovich concerning constant mean curvature asymptotic foliations in AE manifolds and establish geometric criteria guaranteeing the convergence of the ADM centre of mass.
I will introduce the problem of black hole stability in the context of a negative cosmological constant, describing some of the geometric and analytic challenges characteristic of this problem. In the second part, I will discuss recent work with Olivier Graf (Grenoble) establishing linear stability of Schwarzschild-anti de Sitter (AdS) black holes to gravitational perturbations. This is the statement that solutions to the linearisation of the Einstein equations $\textrm{Ric} = -\frac{3}{\ell^2} g$ around a Schwarzschild-AdS metric arising from regular initial data and with standard Dirichlet boundary conditions imposed at the conformal boundary (inherited from fixing the conformal class of the non-linear metric) remain globally uniformly bounded on the black hole exterior and in fact decay inverse logarithmically to a linearised Kerr-AdS metric.
Spacetimes with compact directions which have special holonomy, such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss two recent works showing the nonlinear stability of spacetimes which are the cartesian product of Minkowski space with a compact special holonomy space. I will not assume any prior knowledge of the Cauchy formulation of general relativity, and I will also explain how these results are related to claims of Penrose and Witten.
