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].

I will describe a remarkable symmetric monoidal category associated to a reductive group G, which acts centrally on any G-category. This construction quantizes the universal centralizer group scheme, together with its action on Hamiltonian G-spaces used by Ngo in his proof of the Fundamental Lemma. The category and its central action appear most naturally in a Langlands dual incarnation, which is phrased in terms of convolution on the affine Grassmannian, via work of Bezrukavnikov, Finkelberg and Mirkovic. This is joint work with David Ben-Zvi.

The purpose of the talk is to explain a result in collaboration with B. Pozzetti and A. Wienhard expressing the Hausdorff dimension of certain attractors as a critical exponent. This class of attractors consists of limit sets of Anosov representations in PGLd (hence of non-conformal nature) that verify an extra open condition. If time permits, we will discuss implications of the formula to the geometry of the Hitchin component.

Mapping class groups associated to surfaces whose fundamental group is not finitely generated are called « big », and their study is linked to classical problems in dynamics. In this talk, we discuss some of the basic properties of big mapping class groups, their simplicial actions, and how these can be used to prove that big mapping class groups « detect » surfaces or (if time allows) that the space of non-trivial quasimorphisms of a big mapping class group is infinite dimensional.

We give a new interpretation of Artin conductors of characters in the framework of theory of motives with modulus. It gives a unified way to understand Artin conductors of characters and irregularities of line bundle with integrable connections as well as overconvergent F-isocrystals of rank 1. It also gives rise to new conductors, for example, for G-torsors with G a finite flat group scheme, which specializes to the classical Artin conductor in case G = Z/nZ. We also give a motivic proof of a theorem of Kato and Matsuda on the determination of Artin conductors along divisors on smooth schemes by its restrictions to curves. Its proof is based on a motivic version of a theorem of Gabber-Katz. This is a joint work with Kay Rülling.

In joint work with François Maucourant, we study the dynamics of unipotent flows on frame bundles of hyperbolic manifolds of infinite volume. We prove that they are topologically transitive, and that the natural invariant measure, the so-called « Burger-Roblin measure », is ergodic, as soon as the geodesic flow admits a finite measure of maximal entropy, and this entropy is strictly greater than the codimension of the unipotent flow inside the maximal unipotent flow. The latter result generalises a theorem of Mohammadi and Oh.

In the talk, I will present the main ideas of this work.

 

We introduce Feynman categories and show that they naturally define bi-algebras. In good circumstances these bi-algebras have Hopf quotients. Corresponding to several levels of sophistication and decoration (both terms have technical definitions), we recover the Hopf algebras of Goncharov and Brown from number theory, a Hopf algebra of Baues used in the analysis of double loop spaces and the various Hopf algebras of Connes-Kreimer used in QFT as examples of the general theory. Co-actions also appear naturally in this context as we will explain.

We will first define the moduli space of algebraic-group-valued representations of finitely presented groups. Then we will briefly describe how non-commutative rings influence the structure of the coordinate ring of these moduli spaces. Lastly, we will illustrate this general relationship by constructing minimal generating sets of the coordinate rings of these moduli spaces in some specific examples.

Minkowski space of dimension 2+1 is the Lorentzian analogue of Euclidean 3-space. It is well-known that there exists an isometric embedding of the hyperbolic plane in Minkowski space, which is the analogue of the embedding of the round sphere in Euclidean space. However, differently from the Euclidean case, the embedding of the hyperbolic plane is not unique up to global isometries. In this talk I will discuss several results on the classification of these embeddings, and explain how this problem is related to Monge-Ampère equations, harmonic maps, and Teichmüller theory. This is joint work with Francesco Bonsante and Peter Smillie.

Hitchin representations are an important class of representations of fundamental groups of closed hyperbolic surfaces into PSL(n,R), at the heart of higher Teichmüller theory. Given such a representation j, there is a unique j-equivariant harmonic map from the universal cover of the hyperbolic surface to the symmetric space of PSL(n,R). We show that its energy density is at least 1 and that rigidity holds. In particular, we show that given a Hitchin representation, every equivariant minimal immersion from the hyperbolic plane into the symmetric space of PSL(n,R) is distance-increasing. Moreover, equality holds at one point if and only if it holds everywhere and the given Hitchin representation j is an n-Fuchsian representation.

I shall report about a new direct construction of Nori motives, discovered independently by Barbieri-Viale and Prest on the one hand, by Joyal and myself on the other hand. Unlike previous constructions, one uses only standard constructions in category theory, like Frey free abelian category on a given additive category, and Serre’s construction of quotient categories.