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.

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

In this talk I will argue that after certain timescale (which scales with the system size as L^{d+2}) dynamics of a local observable becomes universal and it can be described by a random matrix.

 

This talks is based on https://arxiv.org/abs/1804.08626 and other recent works.

Mathematicians have long understood periodic trajectories on the square billiard table, which occur when the slope of the trajectory is rational. In this talk, I will explain my joint work with Samuel Lelièvre on periodic trajectories on the regular pentagon, describing their geometry, symbolic dynamics, and group structure. The periodic trajectories are very beautiful, and some of them exhibit a surprising « dense but not equidistributed » behavior.

In my talk, I will report on my ongoing collaborating project together with Yifeng Liu, Liang Xiao, Wei Zhang, and Xinwen Zhu, which concerns the rank 0 case of the Beilinson-Bloch-Kato conjecture on the relation between L-functions and Selmer groups for certain Rankin–Selberg motives for GL(n) x GL(n+1). I will state the main results with some examples coming from elliptic curves, sketch the strategy of the proof, and then focus on the key geometric ingredients, namely the semi-stable reduction of unitary Shimura varieties of type U(1,n) at non-quasi-split places.

In order to analyze mathematical and physical systems it is often necessary to assume some form of order, e.g. perfect symmetry or complete randomness. Fortunately, nature seems to be biased towards such forms of order as well. During the second half of the 20th century, a new paradigm of « aperiodic order » was suggested. Instances of aperiodic order were discovered in different areas of mathematics, such as harmonic analysis and diophantine approximation (Meyer), tiling theory (Wang, Penrose) and additive combinatorics (Freiman, Erdös-Szemeredi); after some initial resistance is has now been accepted that aperiodic order also exists in nature in the form of quasicrystals.

Together with Michael Björklund we have proposed a general mathematical framework for the study of aperiodic structures in metric spaces, based on the notion of an « approximate lattices ». Roughly speaking, approximate lattices generalize lattices in the same way that approximate subgroups (in the sense of Tao) generalize subgroups. Approximate lattices in Euclidean space are essentially the « harmonious sets » of Meyer (a.k.a. mathematical quasi-crystals), but there are interesting examples in other geometries, such as symmetric spaces, Bruhat-Tits buildings or nilpotent Lie groups. It turns out that with every approximate lattice one can associate a dynamical system, which replaces the homogeneous space associated with a lattice – thus the study of approximate lattices can be considered as « geometric group theory enriched over dynamical systems ».

In this talk I will (1) give an overview over the basic framework of approximate lattices and geometric approximate group theory; (2) illustrate the framework by formulating Meyer’s theory of harmonious sets in this language; (3) time permitting, discuss some recent structure theory of approximate lattices in nilpotent Lie groups and applications to Bragg peaks in the Schrödinger spectrum of magnetic quasicrystals.

Based on joint works with Michael Björklund (Chalmers), Matthew Cordes (ETH), Felix Pogorzelski (Leipzig) and Vera Tonić (Rijeka).

I will consider the asymptotic behavior of eigenfunctions of the Laplacian on a compact locally symmetric manifold M « in the level aspect », that is as the volume of M tends to infinity. I will formulate a precise conjecture of « Berry type », and describe partial results obtained in a joint work with Miklos Abert and Étienne Le Masson.

A classification will be given of all separable minimal hypersurfaces in ${mathbb R}^{n geq 3}$.Rn≥3

An important conjecture in quantum mechanics is that non-interacting, disordered 3d quantum systems should exhibit a localization-delocalization transition as a function of the disorder strength. From a mathematical viewpoint, a lot is known about the localized phase for strong disorder, much less about the delocalized phase at weak disorder. In this talk I will present results for a hierarchical supersymmetric model for a class of 3d quantum systems, called Weyl semimetals. We use rigorous renormalization group methods to compute the correlation functions of the system. In particular, I will report a result about the algebraic decay of the averaged two-point correlation function, compatible with delocalization. Our method is based on a rigorous implementation of RG, reminiscent of the Gawedzki-Kupiainen block spin transformations; the main technical novelty is the multiscale analysis of Gaussian measures with purely imaginary covariances.

Joint work with Luca Fresta and Marcello Porta.

In 1996 Rinat Kashaev introduced a new way to write the star-triangle move (or Yang-Baxter equations) of the Ising model as a single polynomial relation. In 2013 Richard Kenyon and Robin Pemantle understood that this relation could be seen as a kind of spatial recurrence. I will show how the iteration of Kashaev’s recurrence can be related the combinatorics of a loop model with two colors, that was introduced for different reasons by Ole Warnaar and Bernard Nienhuis in 1993. It is also possible to couple this loop model with known relatives of the Ising model: a dimer model and a six-vertex model. Finally I will show a few limit shape phenomena.

I will show that for any integer N, there are only finitely many cuspidal algebraic automorphic representations of GL_m over Q whose Artin conductor is N and whose « weights » are in the interval {0,…,23} (with m varying). Via the conjectural yoga between geometric Galois representations (or motives) and algebraic automorphic forms, this statement may be viewed as a generalization of the classical Hermite-Minkowski theorem in algebraic number theory. I will also discuss variants of these results when the base field Q is replaced by an arbitrary number field.