Mapping class groups of infinite type surfaces, also called « big » mapping class groups, arise naturally in several dynamical contexts, such as two-dimensional dynamics, one-dimensional complex dynamics, « Artinization » of Thompson groups, etc.

In this talk, I will present recent objects and phenomena related to big mapping class groups. In particular, I will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. I will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). If time allows, I will explain how to use these simultaneous actions to construct nontrivial quasimorphisms on subgroups of big mapping class groups.
This is joint work with Alden Walker.

The Calabi theorem states that for every regular convex cone K in R^n, the Monge-Ampère equation log det F” = 2F/n has a unique convex solution on the interior of K which tends to +infty on the boundary of K. It turns out that this solution is self-concordant and logarithmically homogeneous, and thus is a barrier which can be used for conic optimization. We consider different aspects of this barrier:

affine spheres as level surfaces
metrization of the interior of K by the Hessian metric F”
primal-dual symmetry
interpretation as a minimal Lagrangian submanifold in a certain para-Kähler space form
complex-analytic structure on 3-dimensional cones.

Kitaev’s quantum double models provide a rich class of examples of two-dimensional lattice systems with topological order in the ground states and a spectrum described by anyonic elementary excitations. The infinite volume ground states of the abelian quantum double models come in a number of equivalence classes called superselection sectors. We prove that the superselection structure remains unchanged under uniformly small perturbations of the Hamiltonians. (joint work with Matthew Cha and Pieter Naaijkens)

The study of Anosov representations deals with discrete subgroups of Lie groups that have a nice limit set, meaning that they share the dynamical properties of limit sets in hyperbolic geometry. However, the geometry of these limits sets is different: while limit sets in hyperbolic geometry have a fractal nature (e.g. non-integer Hausdorff dimension), some Anosov groups have a more regular limit set (e.g. $C^1$ for Hitchin representations).
My talk will focus on quasi-Fuchsian subgroups of SO(n,2), and show that the situation is intermediate: their limit sets are Lipschitz submanifolds, but not $C^1$. I will discuss the two main steps of the proof. The first one classifies the possible Zariski closures of such groups. The second uses anti-de Sitter geometry in order to determine the limit cone of such a group with a $C^1$ limit set.
Based on joint work with Olivier Glorieux.

We discuss some quite general properties of infinite discrete groups G acting on compact spaces. The spaces we mainly have in mind are horoboundaries of metric spaces which admit an isometric action of G. As an application, we show that the mapping class group of a surface of finite type admits a proper action on some $L^p$-space.

The Teichmüller space of a compact 2-orbifold X can be defined as the space of faithful and discrete representations of the fundamental group $pi_1$(X) of X into PGL(2,R). It is a contractible space. For closed orientable surfaces, « higher analogues » of the Teichmüller space are, by definition, (unions of) connected components of representation varieties of $pi_1$(X) that consist entirely of discrete and faithful representations. There are two known families of such spaces, namely Hitchin representations and maximal representations, and conjectures on how to find others. In joint work with Daniele Alessandrini and Gye-Seon Lee, we show that the natural generalisation of Hitchin components to the orbifold case yields new examples of higher Teichmüller spaces: Hitchin representations of orbifold fundamental groups are discrete and faithful, and share many other properties of Hitchin representations of surface groups. However, we also uncover new phenomena, which are specific to the orbifold case.

The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the derivations' arguments. Let us explore the implications of this principle: in effect, we develop the calculus of variations on the infinite jet spaces for maps from sheaves of free associative algebras over commutative manifolds to the generators and then, quotients of free associative algebras over the linear relation of equivalence under cyclic shifts. In the frames of such variational noncommutative symplectic (super)geometry we prove the main properties of the Batalin-Vilkovisky Laplacian and variational Schouten bracket.

 

We show as by-product that the structures which arise in the variational Poisson geometry of infinite-dimensional integrable systems do actually not refer to the graded commutativity assumption.

The 8-vertex model and the XYZ spin chain have been found to emerge from gauge theories in various ways, such as 4d and 2d Nekrasov-Shatashvili correspondences, the action of surface operators on the supersymmetric indices of class-Sk theories, and correlators of line operators in 4d Chern-Simons theory. I will explain how string theory unifies these phenomena.

 

This is based on my work with Kevin Costello [arXiv:1810.01970].

 

A long-standing question of Hopf asks whether every self-map of absolute degree one of a closed oriented manifold is a homotopy equivalence. This question gave rise to several other problems, most notably whether the fundamental groups of aspherical manifolds are Hopfian, i.e. any surjective endomorphism is an isomorphism. Recall that the Borel conjecture states that any homotopy equivalence between two closed aspherical manifolds is homotopic to a homeomorphism. In this talk, we verify a strong version of Hopf’s problem for certain aspherical manifolds. Namely, we show that every self-map of non-zero degree of a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group (e.g. negatively curved manifold) is either homotopic to a homeomorphism or homotopic to a non-trivial covering and the bundle is trivial. Our main result is that a non-trivial circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group does not admit self-maps of absolute degree greater than one. This extends in all dimensions the case of circle bundles over closed hyperbolic surfaces (which was shown by Brooks and Goldman) and provides the first examples (beyond dimension three) of non-vanishing functorial semi-norms on the fundamental classes of circle bundles over aspherical manifolds with hyperbolic fundamental groups.