Alternatively have a look at the program.

## Inverse localization of eigenfunctions and applications

We will discuss some situations where the following phenomenon can be shown to hold: let f be any solution of the Helmholtz equation (\Delta f+f=0) on the unit ball of Euclidean space; if one examines the eigenfunctions of the Laplace-Beltrami operator on certain Riemannian manifolds at increasingly high eigenvalues K2 and at small balls of radius 1/K, one ends up finding an eigenfunction that in rescaled coordinates approximates, with a given arbitrarily small precision, the function f.

## The twisted dynamical zeta functions of Ruelle and Selberg of locally symmetric spaces of rank 1 and applications

The twisted zeta functions of Ruelle and Selberg are dynamical zeta functions, which are represented by Euler-type products over the prime closed geodesics on a compact hyperbolic manifold. For locally symmetric manifolds, harmonic analysis provides a powerful tool, the Selberg trace formula, that one can use to prove the meromorphic continuation of the dynamical zeta functions. These zeta functions have been long studied by Fried, Bunke and Olbrich, Müller, Pfaff, Fedosova and Pohl under certain assumptions for the representation of the fundamental group.

## Propagators of the wave equation on low regularity spacetimes

In this talk, I will define the classical and non-classical of the wave equation.

Also, I will give some motivations about why low regularity scenarios are interesting.

Finally, I will focus on the classical ones and its relationship with Quantum Field Theory in spacetimes with limited regularity.

## Determinants of Laplacians on sectors and surfaces with conical singularities.

I will talk about current on going work with Julie Rowlett and Klaus Kristen. In previous work with J. Rowlett, we proved a formula for the derivative of the determinant of the Laplace operator on convex euclidean sectors with respect to the opening angle. However, while comparing our formula with an equivalent formula proved by K. Kirsten using different methods, we found a missing term. I will present this term and present the formula for angles of the form pi/n. I will introduce the terminology and present some motivation and tools for this work.

## Twisted Ruelle zeta function at zero

I will talk about work in progress with Gabriel Paternain. The talk will discuss the order at zero of the Ruelle zeta function for Anosov flows on three manifolds, twisted with a unitary connection. I will recall the microlocal approach to hyperbolic dynamics and discuss how to compute the dimensions of resonant states for differential forms. The final result depends on topological and dynamical data.

## On the Lagrange spectrum of moduli spaces of Riemann surfaces $M_{g,n}$

We consider the moduli space $M_{g,n}$ of Riemann surfaces of

genus g with n punctures endowed with the Teichmueller metric.

The Teichmueller geodesic flow is a non-uniformly hyperbolic

flow on $M_{g,n}$ (with respect to the Masur-Veech measure). The

Lagrange spectrum of $M_{g,n}$ is a closed subset of the positive

real numbers that measures how closed geodesics escape $(M_{g,n}$ is

not compact). The classical Lagrange spectrum corresponds to the

case of $M_{1,1}$ and is motivated by diophantine approximations.

## Lyapunov exponents and diffusion rates

A windtree model is an extremely simplified modelling of wind blowing in a planted forest. These models turn out to be very rich, and a lot of questions about them remain open.

Recently the speed of the wind in some of them have been linked with Lyapunov exponents introduced for translation surfaces in stata of quadratic differentials.

The speed in that matter being measured by a number called diffusion rate.

## Dynamics on pseudo-Riemannian manifolds

Much more than in Riemannian geometry, isometries and conformal transformations of pseudo-Riemannian manifolds can have rich dynamics. There are, accordingly, more interesting groups acting, more complicated orbit structures, and more flexible global geometries. I will survey a few topics in dynamics on pseudo-Riemannian manifolds and present some contributions of me and my collaborators.

## Horocycle integral and transfer operators

A well-known example of a contact Anosov flow is the geodesic flow on the unit tangent bundle of closed Riemannian manifolds with variable negative sectional curvature. On by the Ansosov flow induced stable submanifolds one defines the horocycle flow. This horocycle flow is uniquely ergodic. What is the speed of convergence to the Birkhoff average? In a paper of Livio Flaminio and Giovanni Forni (2003) they investigated this question for the geodesic flow in the case of constant negative curvature.

## The Surface Quasi-Geostrophic Equation on the sphere

The SQG equation models the evolution of buoyancy, or potential temperature, on the 2D horizontal boundaries of general 3D quasi- geostrophic equations. In this talk, we will present some regularity results for its dissipative analogue in the critical regime for the two dimensional sphere. The main techniques build over the work of Caffarelli-Vasseur based on the De Giorgi method and the nonlinear maximum principles due to Constantin-Vicol.

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