A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!

A well-known conjecture of Margulis predicts the existence of a uniform lower bound on the systole of any irreducible arithmetic locally symmetric space. In joint work with F. Thilmany, we proved that this conjecture is equivalent to a weak version of the Lehmer conjecture, a well-known problem from Diophantine geometry.
In joint work with M. Fraczyk, we recently established a uniform lower bound for simple Lie groups of higher rank conditional on a uniform lower bound on Salem numbers, a much weaker – but still open – problem. I will discuss these results and some tools used in the proofs and present additional results which highlight the structure of the bottom of the length spectrum.

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!

I will review some old and new theorems on orbit closures and classification of invariant measures. We will start with the homogeneous space setting where the theory is most mature. We will then survey some recent developments outside of the homogeneous setting. 

 I will review some old and new theorems on orbit closures and classification of invariant measures. We will start with the homogeneous space setting where the theory is most mature. We will then survey some recent developments outside of the homogeneous setting. 

 I will review some old and new theorems on orbit closures and classification of invariant measures. We will start with the homogeneous space setting where the theory is most mature. We will then survey some recent developments outside of the homogeneous setting. 

Upper bound on the number of resonances for even asymptotically hyperbolic manifolds real-analytic at infinity.
Abstract I will explain how tools of real-analytic microlocal analysis can be used to prove a polynomial upper bound on the number of resonances for an asymptotically hyperbolic manifold with real-analytic ends (after recalling the definition of those). The proof is based on an adaptation of Vasy’s method, introducing an analytic Fourier-Bros-Iagolnitzer transform in the spirit of the work of Helffer and Sjöstrand.This strategy has similarities with my previous work with Yannick Bonthonneau on real-analytic Anosov flows, and also gives a bound on the number of quasi-normal frequencies for Schwarzschild-de Sitter black holes.

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!

 In the 1950s Phil Anderson made a prediction about the effect of random impurities on the conductivity properties of a crystal. Mathematically, these questions amount to the study of solutions of differential or difference equations and the associated spectral theory of self-adjoint operators obtained from an ergodic process. With the arrival of quasicrystals, in addition to random models, nonrandom lattice models such as those generated by irrational rotations or skew-rotations on tori have been studied over the past 30 years. 

By now, an extensive mathematical theory has developed around Anderson’s predictions, with several questions remaining open. This talk will attempt to survey certain aspects of the field, with an emphasis on the theory of SL(2,R) cocycles with an irrational or  Diophantine  rotation on the circle as base dynamics. In this setting, Artur Avila discovered about a decade ago that the Lyapunov exponent is piecewise affine in the imaginary direction after complexification of the circle. In fact, the slopes of these affine functions are integer valued. This is easy to see in the uniformly hyperbolic case, which is equivalent to energies falling into the gaps of the spectrum, due to the winding number of the complexified Lyapunov exponent. Remarkably, this property persists also in the non-uniformly hyperbolic case, i.e., on the spectrum of the Schrödinger operator. This requires a delicate continuity property of the Lyapunov exponent in both energy and frequency. Avila built his global theory (Acta Math. 2015) on this quantization property. I will present some recent results with Rui HAN (Louisiana) connecting Avila’s notion of  acceleration (the slope of the complexified Lyapunov exponent in the imaginary direction) to the number of zeros of the determinants of  finite volume Hamiltonians relative to the complex toral variable. This connection allows one to answer questions arising in the supercritical case of Avila’s global theory concerning the measure of the second stratum, Anderson localization on this stratum, as well as settle a conjecture on the Hölder regularity of the integrated density of states.

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!

In a recent joint work with KyeongRo Kim and Hongtaek Jung, we show that a group of circle homeomorphisms is a 3-manifold group if it preserves a veering pair of invariant laminations. The proof has two parts - the topological part and dynamical part. We will try to explain both aspects. In some sense this is a partial converse to Thurston’s universal circle theorem.