'Quasi-isometric rigidity' in group theory is the slogan for questions of the following nature: let A be some class of groups (e.g. finitely presented groups). Suppose an abstract group H is quasi-isometric to a group in A: does it imply that H is in A? Such statements link the coarse geometry of a group with its algebraic structure. 

Much is known in the case A is some class of lattices in a given Lie group. I will present classical results and outline ideas in their proofs, emphasizing the geometric nature of the proofs. I will focus on one key ingredient, the quasi-flat rigidity, and discuss some geometric objects that come into play, such as neutered spaces, asymptotic cones and buildings. I will end the talk with recent developments and possible generalizations of these results and ideas.

Primitive elements are elements that are part of a basis for a free group. We present the classical Whitehead algorithm for the recognition of such elements, and discuss the ideas behind the proof. We also present a second algorithm, more recent and completely different in the approach.

What do random spanning trees, graph embeddings, random walks, simplices and graph curvature have in common? As you may have guessed from the title, they are indeed all intimately connected to the effective resistance on graphs! While originally invented as a tool to study electrical circuits, the effective resistance has proven time and again to be a graph characteristic with a variety of interesting and often surprising properties. Starting from a number of equivalent but complementary definitions of the effective resistance, we will take a stroll through some classical theorems (Rayleigh monotonicity, Foster's theorem), a few modern results (Klein's metricity, Fiedler's graph-simplex correspondence) and finally discuss number of recent developments (variance on graphs, discrete curvature and graph embeddings).

Topological data analysis has proven to be an effective tool in machine learning, supporting the analysis of neural networks, but also driving the development of new algorithms that make use of topological features. Graph classification is of particular interest here, since graphs are inherently amenable to a topological description in terms of their connected components and cycles. This talk will briefly summarise recent advances in topology-based graph classification, focussing equally on ’shallow’ and ‘deep’ approaches. Starting from an intuitive description of persistent homology, we will discuss how to incorporate topological features into the Weisfeiler–Lehman colour refinement scheme, thus obtaining a simple feature-based graph classification algorithm. We will then build a bridge to graph neural networks and demonstrate a topological variant of ‘readout’ functions, which can be learned in an end-to-end fashion. Care has been taken to make the talk accessible to an audience that might not have been exposed to machine learning or topological data analysis.
 

Unlike the classical Cauchy problem in general relativity, which has been well-understood since the pioneering work of Y. Choquet-Bruhat (1952), the initial boundary value problem for the Einstein equations still lacks a comprehensive treatment. In particular, there is no geometric description of the boundary data yet known, which makes the problem well-posed for general timelike boundaries. Various gauge-dependent results have been established. Timelike boundaries naturally arise in the study of massive bodies, numerics, AdS spacetimes. I will give an overview of the problem and then present recent joint work with Jacques Smulevici that treates the special case of a totally geodesic boundary.

We propose a modulated free energy which combines of the method previously developed by the speaker together with the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy  to cancel the more singular terms involving the divergence of the flow. This modulated free energy allows to treat singular interactions of gradient-flow type and allows potentials with large smooth part, small attractive singular part and large repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as Patlak-Keller-Segel system in the subcritical regimes, is obtained.

Artin groups are a family of groups generalizing braid groups. The Tits conjecture, which was proved by Crisp-Paris, states that squares of the standard generators generate an obvious right-angled Artin subgroup. In a joint work with Kevin Schreve, we consider a larger collection of elements, and conjecture that their sufficiently large powers generate an obvious right-angled Artin subgroup. In the case of the braid group, regarded as a mapping class group of a punctured disc, these elements correspond to Dehn twist around the loops enclosing multiple consecutive punctures. This alleged right-angled Artin group is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for some classes of Artin groups. We use our results to conclude that certain Artin groups contain hyperbolic surface subgroups, answering questions of Gordon, Long and Reid.

Operads are tools to encode operations satisfying algebro-homotopic relations. They have proved to be extremely useful tools, for instance for detecting spaces that are iterated loop spaces. However, in many natural examples, composition of operations is only associative up to homotopy and operads are too strict to captured these phenomena. This leads to the notion of infinity operads. While they are a well-established tool, there are few examples of infinity operads in the literature that are not the nerve of an actual operad. I will introduce new topological operad of bracketed trees that can be used to identify and construct natural examples of infinity operads. The key example for this talk will be the normalised cacti model for genus 0 surfaces.

Glueing surfaces along their boundaries defines composition laws that have been used to construct topological field theories and to compute the homology of the moduli space of Riemann surfaces. Normalised cacti are a graphical model for the moduli space of genus 0 oriented surfaces. They are endowed with a composition that corresponds to glueing surfaces along their boundaries, but this composition is not associative. By using the operad of bracketed trees, I will show that this operation is associative up to all higher homotopies and hence that normalised cacti form an infinity operad.