One step in the definition of Dirac-Ramond operators on the loop space of a manifold is structure of smooth projective representations of loop groups. The resulting Fr ́echet spaces and central extension can be analyzed in terms of Toeplitz operators on the circle.

The problem of finding and classifying constant curvature metrics with conical singularities has a long history bringing together several different areas of mathematics. This talk will focus on the particularly difficult spherical case where many new phenomena appear. When some of the cone angles are bigger than $2\pi$, uniqueness fails and existence is not guaranteed; smooth deformation is not always possible and the moduli space is expected to have singular strata. I will give a survey of several recent results regarding this singular uniformization problem, connecting microlocal techniques with complex analysis and synthetic geometry. Based on joint works with Rafe Mazzeo and Bin Xu.

I will report on recent and ongoing work on the analysis of Dirac-type operators on incomplete Riemannian manifold on which the metric undergoes structured degeneration. This includes joint work with Pierre Albin.

I first discuss a new perspective on fractional Laplacians, leading to a sharp boundary regularity theory for this problem. This is a counterpoint to Grubb’s more analytic approach. Joint work with Gimperlein and Louca. I will then discuss the beginnings of a general theory being developed with Albin and Marzuola of Dirichlet-to-Neumann operators on domains with stratified boundary.

899_27564_8043_10-Burq from MSRI / National Math Festival on Vimeo.

We consider the semiclassical Schrdinger equation inside a model strictly convex domain and we obtain dispersion estimates for the linear flow; we highlight a 1/4 loss compared to the flat case.

In the 80's, D. Ruelle, D. Bowen and others have introduced probabilistic and spectral methods in order to study deterministic chaos ("Ruelle resonances"). For example, a geodesic flow on a strictly negative curvature Riemannian manifold is chaotic:  each trajectory is strongly unstable and its behavior is unpredictable. A smooth probability distribution evolves also in a complicated way since it acquires higher and higher oscillations. Nevertheless this evolution is predictable in the sense of distributions and converges towards equilibrium. Following this approach and use of microlocal analysis, one obtains that long time fluctuations of classical probabilities are described by an effective quantum wave equation. This may be surprising because there is no added quantization procedure. Joint work with Masato Tsujii.

We present in this talk recent results obtained in collaboration with V ́eronique Fischer (University of Bath, UK) aiming at studying the spectral properties of sub-Laplacian operators. We develop a semi-classical approach compatible with the Lie group structure and we discuss the associated notion of semi-classical measures, together with their dispersive properties when related with solutions of a Schr ̈odinger equation in H-type groups.

In this talk I will discuss the recent developments in the inverse spectral theory of bounded planner domains with strictly convex smooth boundaries. I will first present a joint work with Steve Zelditch in which we prove ellipses of small eccentricity are spectrally unique among all smooth domains. I will then discuss an inverse spectral result for nearly circular domains with an axial symmetry. A linearized version of this problem was studied by De Simoi, Kaloshin, and Wei. The non-linear problem is more interesting and uses second variations of the length functions and also some techniques of Avila, De Simoi, and Kaloshin