Ebru Toprak (UIUC and MSRI)

The next Analysis and PDE seminar will take place Monday, October 15, from 4-5pm in 740 Evans.

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.

Victor Vilaça Da Rocha (BCAM, MSRI)

The Analysis and PDE seminar will take place Monday Oct 1st in 740 Evans from 4-5pm.

Title: Construction of unstable quasi-periodic solutions for a system of coupled NLS equations.
Abstract: The systems of coupled NLS equations occur in some physical problems, in particular in nonlinear optics (coupling between two optical waveguides, pulses or polarized components…). From the mathematical point of view, the coupling effects can lead to truly nonlinear behaviors, such as the beating effect (solutions with Fourier modes exchanging energy) of Grébert, Paturel and Thomann (2013).
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.

Georgios Moschidis

The Analysis and PDE seminar will take place Monday Sept 17 in 740 Evans from 4-5pm.

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.

Laura Cladek (UCLA)

The Analysis and PDE seminar will take place Monday Sept 10 in 740 Evans from 4-5pm.

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.

Jan Dereziński (University of Warsaw)

The Analysis and PDE seminar will take place Monday, Aug 27, in 740 Evans from 4-5pm.

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.

Mihajlo Cekic (Max Planck Institute for Mathematics, Bonn)

The Analysis and PDE seminar will take place Monday, March 19, in 740 Evans from 4:10 to 5pm.

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.

Aleksandr Logunov (IAS)

The Analysis and PDE seminar will take place Monday, March 5, in 740 Evans from 4:10 to 5pm.

Title: Several questions on Laplace eigenfunctions

Abstract: Let $(M,g)$ be a compact Riemannian manifold without boundary. We are interested in asymptotic properties of Laplace eigenfunctions on $M$ as the eigenvalue $\lambda$ tends to infinity. The advances of the last few years will be discussed and a survey of interesting open questions will be given.

Alexis Drouot (Columbia University)

The Analysis and PDE seminar will take place Monday, February 12, in 740 Evans from 4:10 to 5 pm.

Title: Edge (resonant) states for 1D bi-periodic systems

Abstract: We study the bifurcation of Dirac points under the introduction of edges in certain periodic systems. For honeycomb Schrodinger operators, Fefferman, Lee-Thorp and Weinstein showed that if introducing the edge opens an essential spectral gap near the Dirac energy, then the perturbed operator has an edge-localized eigenstate. This state is associated to the topologically protected zero-mode of a Dirac operator, which emerges from a formal multiscale approach (one of the two scales being the size of the edge).

We consider 1D models where the introduction of a large edge does not necessarily open an essential spectral gap near the Dirac energy. We approach such systems with Fredholm analytic tools and show that Dirac points bifurcate to resonant states. When the edge perturbation happens to open an essential spectral gap, this improves a previous result of Fefferman–Lee-Thorp–Weinstein by (a) proving the validity of the multiscale approach; (b) relating each eigenvalue in the gap to an eigenvalue of the above Dirac operator; (c) deriving full expansions of the associated states.

Joint work with Michael Weinstein and Charles Fefferman.