Schedule, Notes/Handouts & Videos

We review results on the concentration properties of quasimodes for Schrödinger operators whose underlying Hamiltonian flow is completely integrable. In particulrar, we will present joint work with Víctor Arnaiz (Madrid) and Gabriel Riviere (Nantes).

I will state the index theorem for twisted Dirac operators on globally hyperbolic spacetimes with compact Cauchy surfaces. These operators are hyperbolic but still Fredholm on appropriate function spaces. I will give a local proof of the index theorem based on a local construction of the Feynman parametrix. (joint work with C. Baer)

In this talk, we discuss the geodesic beam approach to eigenfunction concentration and give applications to the study of the norms of eigenfunctions for p larger than the critical exponent. In particular, we will give quantitative improvements to these estimates which rely only on dynamical assumptions and describe the concentration behavior of saturating examples.

For self-adjoint pseudodifferential operators of order 0, Colin de Verdiere and Saint-Raymond introduced natural dynamical conditions (motivated by the study of internal waves in fluids) guaranteeing absolute continuity of the spectrum. I will present an alternative approach to obtaining such results based on Melrose's radial propagation estimates from scattering theory (joint work with S Dyatlov). I will then explain how an adaptation of the Helffer–Sjoestrand theory of scattering resonances shows that in a complex neighbourhood of the continuous spectrum viscosity eigenvalues have limits as viscosity goes to 0. Here the viscosity eigenvalues are the eigen- values of the original operator to which an anti-self-adjoint elliptic 2nd order operator is added (joint work with J Galkowski).

We consider inverse boundary value problems for general real principal type differential operators. The first results state that the Cauchy data set uniquely determines the scattering relation of the operator and bicharacteristic ray transforms of lower order coefficients. We also give two different boundary determination methods for general operators, and prove global uniqueness results for determining coefficients in nonlinear real principal type equations. The article presents a unified approach for treating inverse boundary problems for transport and wave equations, and highlights the role of propagation of singularities in the solution of related inverse problems. This is joint work with Lauri Oksanen (UCL), Plamen Stefanov (Purdue) and Gunther Uhlmann (Washington / IAS HKUST).

In this talk we shall discuss recent progress for partial data inverse boundary problems for semilinear elliptic PDE. It turns out that the presence of a nonlinearity allows one to solve inverse problems in situations where the corresponding linear counterpart is open. In the first part of the talk, we shall also discuss some previous work on partial data inverse boundary problems for linear elliptic PDE, focusing on the case of coefficients of low regularity. This talk is based on joint work with Gunther Uhlmann.

I will talk about joint work with Guillarmou, Rivière, Shen and ongoing work with Chaubet on the relation between Ruelle zeta functions of contact Anosov flows and various forms of torsions. The goal of these works is to understand the Fried conjecture for smooth Anosov flows beyond constant negative curvature.

The smooth degree d complex curves of are Riemann surfaces of the same genus   . If we equip them with the restriction of the ambient metric and choose them at random, what can be say about the length of their systole? I will explain that the probability that the systole is of the order  is bounded from below by a uniform positive constant. This gives an partial analogous result to Mirzakhani's theorem on random hyperbolic curves. If I have time, I will explain that in higher dimensions, these probabilistic arguments provide a new deterministic result about Lagrangian submanifolds and the topology of complex projective hypersurfaces.