Author Archives: peterhintz

Semyon Dyatlov (MIT)

The Analysis and PDE Seminar will take place on Monday, September 12, in room 740, Evans Hall, from 4:10-5:00 pm.

Speaker: Semyon Dyatlov

Title: Resonances for open quantum maps

Abstract: Quantum maps are a popular model in physics: Symplectic relations on tori are quantized to produce families of $N\times N$ matrices and the high energy limit corresponds to the large $N$ limit. They share a lot of features with more complicated quantum systems but are easier to study numerically. We consider open quantum baker’s maps, whose underlying classical systems have a hole allowing energy escape. The eigenvalues of the resulting matrices lie inside the unit disk and are a model for scattering resonances of more general chaotic quantum systems. However in the setting of quantum maps we obtain results which are far beyond what is known in scattering theory.

We establish a spectral gap (that is, the spectral radius of the matrix is separated from 1 as $N\to\infty$) for all the systems considered. The proof relies on the notion of fractal uncertainty principle and uses the fine structure of the trapped sets, which in our case are given by Cantor sets, together with simple tools from harmonic analysis, algebra, combinatorics, and number theory. We also obtain a fractal Weyl upper bound for the number of eigenvalues in annuli. These results are illustrated by numerical experiments which also suggest some conjectures.

This talk is based on joint work with Long Jin.

Svetlana Jitomirskaya (UC Irvine)

The Analysis and PDE Seminar will take place on Monday, May 2, in room 891, Evans Hall, from 4:10-5:00 pm.

Speaker: Svetlana Jitomirskaya

Title: Very small denominators and sharp arithmetic spectral transitions

Abstract: We will discuss two popular discrete quasiperiodic models: the Maryland model and the almost Mathieu operator, both coming from physics. In the regime of positive Lyapunov exponents, spectral properties differ for Diophantine and Liouville frequencies. We will address the question of the location and nature of the corresponding transition, presenting sharp and constructive arithmetic results for both models, that solve some longstanding conjectures. Close to the transition regime, eigenfunctions decay at the non-Lyapunov rate, and we will also present a sharp description of the eigenfunction profile and also of the non-uniformly hyperbolic dynamics of the corresponding transfer-matrix cocycle. The talk is based on works joint with W. Liu.

Peter Hintz (UC Berkeley)

The Analysis and PDE Seminar will take place on Monday, April 18, in room 891, Evans Hall, from 4:10-5:00 pm.

Speaker: Peter Hintz

Title: Finite codimension solvability of quasilinear wave equations

Abstract: I will describe a general framework, applicable on de Sitter and Kerr-de Sitter spacetimes, which allows one to solve quasilinear wave equations globally for restricted initial data even if the linearized operator has exponentially growing modes. As an application, I will revisit the nonlinear stability of de Sitter space in the context of general relativity. This is work in progress with András Vasy.

Boris Hanin (MIT)

The Analysis and PDE Seminar will take place on Monday, February 8, 2016, from 4:10-5:00 pm in Evans Hall, room 891.

Speaker: Boris Hanin (MIT)

Title: Scaling Limit of Spectral Projector for the Laplacian on a Compact Riemannian Manifold

Abstract: Let (M,g) be a compact smooth Riemannian manifold. I will give some new off-diagonal estimates for the remainder in the pointwise Weyl Law. A corollary is that, when rescaled around a non self-focal point, the kernel of the spectral projector of the Laplacian onto the frequency interval (lambda,lambda+1] has a universal scaling limit as lambda goes to infinity (depending only on the dimension of M). This is joint work with Y. Canzani.

See you all there!

Jeffrey Galkowski (Stanford)

The Analysis and PDE Seminar will take place on Monday, December 7th 2015, from 4:10-5:00 pm in Evans Hall, room 740.

Speaker: Jeffrey Galkowski (Stanford)

Title: Resonance Free Regions and Average Smoothing Times

Abstract: We give a quantitative version of Vainberg’s method relating pole free regions to propagation of singularities. In particular, we show that there is a logarithmic resonance free region near the real axis of size \tau with polynomial bounds on the resolvent if and only if the wave propagator gains derivatives at rate \tau. Next we show that if there exist singularities in the wave trace at times tending to infinity which smooth at rate \tau, then there are resonances in logarithmic strips whose width is given by \tau. As our main application of these results, we give generically optimal bounds on the size of resonance free regions in scattering on geometrically nontrapping manifolds with conic points and exteriors of nontrapping polygonal domains.

See you all there!