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.

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.

Same room, 891, Evans Hall, from 4:10-5:00pm.

Speaker: Mihai Tohaneanu

Title: Global existence for quasilinear wave equations close to Schwarzschild

Abstract: We study the quasilinear wave equation $\Box_{g} u = 0$, where the metric $g$ depends on $u$ and equals the Schwarzschild metric when u is identically 0. Under a couple of extra assumptions on the metric $g$ near the trapped set and the light cone, we prove global existence of solutions. This is joint work with Hans Lindblad.

The Analysis and PDE Seminar will take place on Monday, March 28, 2016 from 4:10-5:00 pm in Evans Hall, room 891.

Speaker: Michael Singer (University College London, MSRI)

Title: Asymptotics of Partial Bergman Kernels for Divisors

See you all there!

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!

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!

The Analysis and PDE Seminar will take place on Monday on November 22nd 2015 from 4:10-5:00pm in Evans Hall, room 740.

Speaker: Jeremy Marzuola (UNC)

Title: Euler Equations on Rotating Surfaces

Abstract: In an appendix to a recent paper by Michael Taylor, he and I explored various questions related to stability of striated patterns for fluids on rotating spheres.  I will discuss these results and some open problems related to this study.

See you all there!

Analysis and PDE seminar which will take place in 740 Evans Hall on Nov 16th

Speaker: Marina Iliopoulou (University of Birmingham)

Title: Algebraic aspects of harmonic analysis

Abstract: When we want to understand a geometric picture, finding the zero set of a polynomial hiding in it can be very helpful: it can reveal structure and allow computations. Polynomial partitioning, developed by Guth and Katz, is a technique to find such a nice algebraic hypersurface. Polynomial partitioning has revolutionised discrete incidence geometry in the recent years, thanks to the fact that interaction of lines with algebraic hypersurfaces is well-understood. Recently, however, Guth discovered agreeable interaction between tubes and algebraic hypersurfaces, and thus used polynomial partitioning to improve on the 3-dim restriction problem. In this talk, we will present polynomial partitioning via a discrete analogue of the Kakeya problem, and discuss its potential to be extensively used in harmonic analysis.

Monday, Nov 9th 2015

Evans Hall, Room 740

Speaker: Baoping Liu, (Peking University)