Title: Rough control for Schr\”odinger operators on 2-tori.

Abstract: I will explain how the results of Bourgain, Burq and the

speaker ’13 can be used to obtain control and observability by rough

functions and sets on 2-tori. We show that for the time dependent

Schrödinger equation, any set of positive measure can be used for

observability and controllability.

For non-empty open sets this follows from the results of Haraux ’89

and Jaffard ’90, while for sufficiently long times and rational tori

this can be deduced from the results of Jakobson ’97.

title: The Marked Length Spectrum of Anosov manifolds

Abstract: We discuss new results on the geometric problem of determining a Riemannian metric with negative curvature on a closed manifold from the lengths of its periodic geodesics. We obtain local rigidity results in all dimensions using combination of dynamical system results with microlocal analysis. Joint work with Thibault Lefeuvre.

]]>Title: Wave maps on (1+2)-dimensional curved spacetimes

Abstract: I will discuss recent joint work, with Cristian Gavrus and Daniel Tataru, in which we consider wave maps on a (1+2)-dimensional nonsmooth background. Our main result asserts that in this variable-coefficient context, the wave maps system is wellposed at almost-critical regularity.

]]>Title: The effect of threshold energy obstructions on the $L^1 \to L^\infty$

dispersive estimates for some Schrödinger type equations

Abstract: In this talk, I will discuss the differential equation $iu_t

= Hu, H := H_0 + V$ , where $V$ is a decaying potential and $H_0$ is a

Laplacian related operator. In particular, I will focus on when $H_0$

is Laplacian, Bilaplacian and Dirac operators. I will discuss how the

threshold energy obstructions, eigenvalues and resonances, effect the

$L^1 \to L^\infty$ behavior of $e^{itH} P_{ac} (H)$. The threshold

obstructions are known as the distributional solutions of $H\psi = 0$

in certain dimension dependent spaces. Due to its unwanted effects on

the dispersive estimates, its absence have been assumed in many

work. I will mention our previous results on Dirac operator and recent

results on Bilaplacian operator under different assumptions on

threshold energy obstructions.

In this talk, I will use the coupling between two NLS equations on the 1D torus to construct a family of linearly unstable tori, and therefore unstable quasi-periodic solutions. The idea is to take profit of the Hamiltonian structure of the system via the construction of a Birkhoff normal form and the application of a KAM theorem. In particular, we will see of this surprising behavior (this is the first example of unstable tori for a 1D PDE) is strongly related to the existence of beating solutions.

This is a work in collaboration with Benoît Grébert.

]]>Title: A proof of the instability of AdS spacetime for the Einstein–massless Vlasov system.

Abstract: The AdS instability conjecture is a conjecture about the initial value problem for the Einstein vacuum equations with a negative cosmological constant. It states that there exist arbitrarily small perturbations to the initial data of the AdS spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions on conformal infinity, lead to the formation of black holes after sufficiently long time. In the recent years, a vast amount of numerical and heuristic works have been dedicated to the study of this conjecture, focusing mainly on the simpler setting of the spherically symmetric Einstein–scalar field system.

In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein–massless Vlasov system. The construction of the unstable family of initial data will require working in a low regularity setting, carefully designing a family of initial configurations of localised Vlasov beams and estimating the exchange of energy taking place between interacting beams over long period of times. Time permitting, I will briefly discuss how the main ideas of the proof can be extended to more general matter fields, including the Einstein–scalar field system.

Title: Quantitative additive energy estimates for regular sets and connections to discretized sum-product theorems

Abstract: We prove new quantitative additive energy estimates for a large class of porous measures which include, for example, all Hausdorff measures of Ahlfors-David subsets of the real line of dimension strictly between 0 and 1. We are able to obtain improved quantitative results over existing additive energy bounds for Ahlfors-David sets by avoiding the use of inverse theorems in additive combinatorics and instead opting for a more direct approach which involves the use of concentration of measure inequalities. We discuss some connections with Bourgain’s sum-product theorem.

]]>Title: Balanced geometric Weyl quantization with applications to QFT on curved spacetimes

Abstract: First I will describe a new pseudodifferential calculus for (pseudo-)Riemannian spaces, which in our opinion (my, D.Siemssen’s and A.Latosiński’s) is the most appropriate way to study operators on such a manifold. I will briefly describe its applications to computations of the asymptotics the heat kernel and Green’s operator on Riemannian manifolds. Then I will discuss analogous applications to Lorentzian manifolds, relevant for QFT on curved spaces. I will mention an intriguing question of the self-adjointness of the Klein-Gordon operator. I will describe the construction of the (distinguished) Feynman propagator on asymptotically static spacetimes. I will show how our pseudodifferential calculus can be used to compute the full asymptotics around the diagonal of various inverses and bisolutions of the Klein-Gordon operator.

]]>Title: TBA

Abstract: TBA

]]>Title: Calderon problem for Yang-Mills connections

Abstract: In the classical Calderon conjecture we want to recover a metric on a compact manifold, up to diffeomorphism fixing the boundary, from the Dirichlet-to-Neumann (DN) map of the metric Laplacian. This problem is routinely motivated by applications and is open in dimension 3 and higher. In this talk, we fix the metric and consider the DN map of the connection Laplacian for Yang-Mills connections. We sketch the proof of uniqueness up to gauges fixing the boundary for smooth line bundles. The proof uses new techniques, involving unique continuation principles for degenerate elliptic equations and an analysis of the zero set of solutions to an elliptic PDE. Time permitting, we will also discuss some (counter) examples concerning zero sets of determinants of matrix solutions to elliptic PDE.

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