**Title**: The roller coaster through Landau damping

**Abstract**: Of great interest is to address the final state conjecture for the dynamics of charged particles near spatially homogeneous equilibria in a plasma, where particles are transported by the self-consistent electric field generated by the meanfield Coulomb’s interaction. The long-range interaction generates waves that oscillate in time and disperse in space through the dispersion of a Schrodinger type equation, known as plasma oscillations or Langmuir waves. The classical notion of Landau damping refers to the damping of oscillations when particles travel at a resonant speed with the waves. The talk is to address this classical picture for the Vlasov-Poisson system with relativistic or bounded velocities. Based on a joint work with E. Grenier and I. Rodnianski.

**Title**: The linear stability of weakly charged and slowly rotating Kerr-Newman family of charged black holes

**Abstract**: I will discuss the linear stability of weakly charged and slowly rotating Kerr-Newman black holes under coupled gravitational and electromagnetic perturbations. We show that the solutions to the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman solution plus a pure gauge term. The proof uses tools from microlocal analysis and a detailed description of the resolvent of the Fourier transformed linearized Einstein-Maxwell operator at low frequencies.

**Title**: Dirac operators and topological insulators.

**Abstract**: I will discuss a 2×2 semiclassical Dirac equation that

emerges from the effective analysis of topological insulators, and

specifically focus on the evolution of coherent states initially

localized on the crossing set of the eigenvalues of the symbol.

Standard propagation of singularities results do not apply; instead,

we discover a surprising phenomenon. The dynamics breaks down in two

parts, one that immediately collapses, and one that propagates along a

seemingly novel quantum trajectory. This observation is consistent

with the bulk-edge correspondence, a principle that coarsely describes

features of transport in topological insulators. We illustrate our

result with various numerical simulations.

**Title**: A localized picture of the maximal development for shock forming solutions of the 3D compressible Euler equations

**Abstract**: It is well known that solutions to the inviscid Burgers’ equation form shock singularities in finite time, even when launched from smooth data. A far less documented fact, at least in the popular works on 1D hyperbolic conservation laws, is that shock singularities are intimately tied to a lack-of-uniqueness for the classical Burgers’ equation.

We prove that, locally, solutions to the Compressible Euler equations do not suffer from the same lack-of-uniqueness, even though they can be written as a coupled system of Burgers’ in isentropic plane-symmetry. Roughly, the saving grace is that Euler flow involves two speeds of propagation, and one of them “prevents” the mechanism driving the lack-of-uniqueness. Analytically, this is done by explicitly constructing a portion of the boundary of classical hyperbolic development for shock forming data. This boundary is a connected co-dimension 1 submanifold of Cartesian space, and we will discuss the delicate geo-analytic degeneracies and difficulties involved in its construction. This is joint work with Jared Speck.

]]>**Title**: Reversal in the stationary Prandtl equations

**Abstract**: We discuss a recent result in which we investigate reversal and recirculation for the stationary Prandtl equations. Reversal describes the solution after the Goldstein singularity, and is characterized by spatio-temporal regions in which $u>0$ and $u<0$. The classical point of view of regarding the Prandtl equations as an evolution completely breaks down. Instead, we view the problem as a quasilinear, mixed-type, free-boundary problem. Joint work with Nader Masmoudi.

**Title**: A complex WKB analysis for finite difference Schrödinger operators and adiabatic perturbations of periodic Schrödinger operators

**Abstract**: I will explain the basic ideas and constructions of the methods underlining the similarities between the two classes of equations.

**Title**: Vanishing viscosity and shock formation in Burgers equation

**Abstract:** I will talk about the shock formation problem for 1D Burgers equation in the presence of small viscosity. Although the vanishing viscosity problem till moments before the first shock and in presence of a fully developed shocks is very classical, little is known about the moment of shock formation. We develop a matched asymptotic expansion to describe the solution to the viscous Burgers equation (with small viscosity) to arbitrary order up to the first singularity time. The main feature of the work is the inner expansion that accommodates the viscous effects close to the shock location and match it to the usual outer expansion (in viscosity). We do not use the Cole-Hopf transform and hence we believe that this approach works for more general scalar 1D conservation laws. Time permitting, I will talk about generalizing to vanishing viscosity limit from compressible Navier–Stokes to compressible Euler equations. This is joint work with Cole Graham (Brown university).

**Title**: Wave trace and resonances of the Aharonov–Bohm Hamiltonian

**Abstract**: I want to discuss propagation of singularities of the

magnetic Hamiltonian with singular vector potentials, which is related

to the so-called Aharonov–Bohm effect. In addition, I shall discuss a

Duistermaat–Guillemin type trace formula, as well as some

applications to scattering resonances in this setting.

**Title**: The weak null condition on Kerr backgrounds

**Abstract**: Understanding global existence for systems satisfying the weak null condition plays a crucial role in the proof of stability of Minkowski in harmonic coordinates. In this talk I will present a proof of global existence for a semilinear system of equations on Kerr spacetimes satisfying the weak null condition. This is joint work with Hans Lindblad.

**Title**: On isospectral connections

**Abstract**: Kac’s celebrated inverse spectral question “Can one hear the shape of a drum?” consists in recovering a metric from the knowledge of the

spectrum of its Laplacian. I will discuss a very similar question on negatively-curved manifolds, where the word “metric” is now replaced by “connection” on a vector bundle. This problem turns out to be very rich and connects unexpectedly to two other a priori unrelated fields of

mathematics:

1) in dynamical systems: the study of the ergodic behaviour of partially hyperbolic flows obtained as isometric extensions of the geodesic flow (over negatively-curved Riemannian manifolds);

2) in algebraic geometry: the classification of non-trivial algebraic maps between spheres.

Using this relation, I will explain a positive answer to Kac’s inverse spectral problem for connections under a low rank assumption. Joint work with Mihajlo Cekić.