Within pure mathematics, my research is multi-disciplinary: I made contributions to geometry, algebra, dynamical systems, topology and functional analysis. In one sentence:
My research applies geometric, analytic and dynamical tools to the study of groups.
Concretely, the motivation of my work is often a question on algebraic or arithmetic groups, on "abstract" groups, on fundamental groups of manifolds, on transformation groups such as homeomorphisms of spaces of interest, or on the structure theory of locally compact groups.
However, many of my contributions consist in developing new methods and techniques from other mathematical areas in order to solve the initial problem. At times, my work then turns entirely to the topic that was intended as a tool, thus producing new results in geometry or in dynamical systems.
As a consequence, my articles spread over several diverse mathematical areas, but they are all connected by a few important themes. It is indeed the interconnection of the different areas through common themes that allowed me to solve some open problems, see the problem page.
These unifying themes are organised into three threads below:
Fixed-point theorems and amenability
Bounded cohomology and rigidity
Geometry in non-positive curvature and groups
1. Fixed-point theorems and amenability.
Fixed points in topological vector spaces are a fundamental tool in many areas of mathematics, notably through von Neumann's group-theoretical notion of amenability. The latter has become central in group theory, operator algebras, dynamical systems, and beyond.
My Annals paper with Kate Juschenko shows that infinite simple groups can be amenable (and finitely generated). In the opposite direction, my PNAS paper (and this follow-up) gives what is still the simplest solution to the "von Neumann Problem" by proving that the group of all piecewise projective homeomorphisms of the line is non-amenable even though it contains no non-abelian free group.
I believe that the interest of this work lies in its new approach, using ideas from ergodic theory or even descriptive set theory to solve group-theoretical problems. This is an approach that I take also in three papers on Dixmier's unitarizability problem: first, second, third.
Fixed-point theorems are of course important beyond amenability: my Acta paper with Bader-Furman-Gelander has initiated the study of Kazhdan-like fixed point theorems in general Banach spaces, currently a very active field. Related to this is my Inventiones paper with Bader-Gelander where we establish a fixed-point theorem that solved the 1960s "Derivation Problem" while also implying a new, much simpler proof of a celebrated 1983 theorem of Haagerup: every C*-algebra is weakly amenable.
I complement my work on amenability with numerous applications of this notion to geometry and analysis. For instance, this Math. Annalen paper uses the fixed-point principle to establish that every Gelfand pair admits an Iwasawa decomposition, thus leading to a complete classification of Gelfand pairs within the non-positively curved groups.
This relies among others on joint work with Pierre-Emmanuel Caprace establishing a classification of non-positively curved spaces with cocompact amenable isometry groups.
2. Bounded cohomology and rigidity.
Bounded cohomology was born of two unrelated parents: geometry (Gromov, 1980s) and Banach algebras (Johnson, 1970s). It has since then become a powerful, albeit still mysterious, tool for the topology of manifolds, representation theory and rigidity.
My early work with Marc Burger and my Lecture Notes provide the foundations of its study for locally compact groups. This led to a number of applications, for instance to ergodic theory in this Annals paper with Yehuda Shalom, and later to rigidity theory.
A defining feature of my work with cohomology is that I view it as an open structure from which I reach to a variety of tools, which I sometimes need to develop for that purpose. For instance, my Crelle paper requires a new measure-theoretical analogue of the Borel-Serre use of Tits buildings for algebraic and arithmetic groups, or this Fund. Math paper leads me to refine lifting theorems dating back to von Neumann. A third instance, obtained recently in a joint Mem. EMS with Bharatram-Glebsky-Lubotzky, is our use of non-standard analysis to extend cohomological tools toward representation stability questions inspired by the 1982 work of Kazhdan.
I recently came back to the fundamentals of bounded cohomology: this Inventiones paper with Sam Nariman applies to the homeomorphism groups of the most familiar manifolds: circle, disc, spheres. We finally obtain the very first complete computations of non-trivial bounded cohomology. That work uses this GAFA paper, which introduced a new way to use ergodic theory to study group cohomology. Next, with Fournier-Facio-Kuers-Nariman, we managed to handle also the diffeomorphism group of Rn — which has a notoriously difficult cohomology, still unknown in general.
One of the main motivations in Gromov's treatment of bounded cohomology is: all fancy technique aside, it can encode numerical bounds on topological invariants such as the Euler class or minimal volume. A fundamental conjecture of Dupont and Gromov (1970s-1980s) is that all characteristic classes admit such a bound. This conjecture is still open, but the p-adic analogue was solved in my most recent Annals paper.
3. Geometry in non-positive curvature and groups.
My JAMS paper demonstrates that the metric theory of non-positive curvature (CAT(0) spaces) can be used to simplify, extend and generalize some of the landmark results of Margulis, namely some cases of his superrigidity and arithmeticity theorems. While Margulis's work masterfully settled the problem in the world of algebraic groups, my approach is designed to work with products of completely general locally compact groups. This relies among others on an apparently unrelated geometric splitting principle: an infinite-dimensional generalization of the classical splitting of Gromoll-Wolf/Lawson-Yau, which I establish in the same paper.
This led me to join forces with Pierre-Emmanuel Caprace to launch a long haul study of the interplay between CAT(0) geometry and groups. The resulting string of publications, starting with this one, leads to some rather definitive structural and classification results. For instance, we establish several results completely characterizing semisimple groups (Lie or non-Archimedean) in purely metric terms.
This joint work with Caprace, due to the breath of our goals, has forced us to take side-roads into the general theory of locally compact groups. In particular, we establish decomposition theorems which, together with Willis's groundbreaking work on totally disconnected groups, play some role in the revival of the structure theory of locally compact groups, a central topic but one that had witnessed almost no progress since the solution of Hilbert's fifth problem half a century ago (we edited the volume New directions in locally compact groups).
In fact, after the solution of Hilbert's fifth problem last mid-century, a common view was that locally compact groups were "understood"; but this meant to simply disregard the totally disconnected case. However, the importance of these totally disconnected groups can hardly be overstated: given any abstract mathematical structure that enjoys a local finiteness condition, its automorphism group admits a totally disconnected locally compact topology.