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

Speaker: Casey Jao

Title: Mass-critical inverse Strichartz theorems for 1d Schr\”{o}dinger operators

Abstract: I will discuss refined Strichartz estimates at $L^2$ regularity for a family of Schrödinger equations in one space dimension. Existing results rely on sophisticated Fourier analysis in spacetime and are limited to the translation-invariant equation $i\partial_t u = -\tfrac{1}{2} \Delta u$. Motivated by applications to mass-critical NLS, I will describe a physical space approach that applies in the presence of potentials including (but not limited to) the harmonic oscillator. This is joint work with Rowan Killip and Monica Visan.

The Analysis and PDE Seminar will take place on Monday, November 21th, in room 740, Evans Hall, from 4:10-5:00 pm.

Title: Scattering below the ground state for the radial focusing NLS

Abstract: We consider scattering below the ground state for the radial cubic focusing NLS in three dimensions. Holmer and Roudenko originally proved this via concentration compactness and a localized virial estimate. We present a simplified proof that avoids the use of concentration compactness, relying instead on the radial Sobolev embedding and a virial/Morawetz hybrid. This is joint work with Ben Dodson.

The Analysis and PDE Seminar will take place on Monday, November 14th, in room 740, Evans Hall, from 4:10-5:00 pm.

Title: Finite depth gravity water waves In holomorphic coordinates

Abstract: In this article we consider irrotational gravity water waves with finite bottom. Our goal is two-fold. First, we represent the equations in holomorphic coordinates and discuss the local well-posedness of the problem in this context. Second, we consider the small data problem and establish cubic lifespan bounds for the solutions. Our results are uniform in the infinite depth limit, and match our earlier infinite depth paper.

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

Title: Global well-posedness for the energy critical Massive Maxwell-Klein-Gordon equation with small data

Abstract: We discuss the global well-posedness and modified scattering for the massive Maxwell-Klein-Gordon equation in the Coulomb gauge on $ R^{1+d}$ $(d \geq 4)$ for data with small critical Sobolev norm. This extends to $ m^2 > 0 $ the results of Krieger-Sterbenz-Tataru ($d=4,5 $) and Rodnianski-Tao ($ d \geq 6 $).

The proof is based on generalizing the global parametrix construction for the covariant wave operator and the functional framework from the massless case to the Klein-Gordon setting. The equation exhibits a trilinear cancelation structure identified by Machedon-Sterbenz. To treat it one needs sharp $ L^2 $ null form bounds, which are proved by estimating renormalized solutions in null frames spaces similar to the ones considered by Bejenaru-Herr.
To overcome logarithmic divergences we rely on an embedding property of $ \Box^{-1} $ in conjunction with endpoint Strichartz estimates in Lorentz spaces.

The Analysis and PDE Seminar will take place on Monday, October 24th, in room 740, Evans Hall, from 4:10-5:00 pm.

Title: The essential spectrum of the Neumann-Poincare operator

Using an idea of Poincare one can realize the Neumann-Poincare operator on a space of square integrable fields. In two variables this leads to a precise estimate of the essential spectrum, for domains with corners.

The Analysis and PDE Seminar will take place on Monday, September 26, in room 740, Evans Hall, from 4:10-5:00 pm.

Speaker: Satoshi Masaki

Title: Minimization problems on non-scattering solutions to NLS equation

Abstract: We consider global dynamics of focusing nonlinear Schrodinger equations. A first step in this direction is small data scattering which tells us that solutions around the zero solution asymptotically behave like free solutions. On the other hand, there exists non-scattering solutions such as standing waves and blowing-up solutions.

In this talk, we will seek threshold solutions between scattering solutions around zero and solutions with other behaviors, by introducing two minimization problems on non-scattering solutions. In particular, our main interest is the analysis of mass-subcritical case, in which the ground states are stable. The analysis of the minimization problems are based on concentration compactness/rigidity argument initiated by Kenig and Merle.

The Analysis and PDE Seminar will take place on Monday, September 19, in room 740, Evans Hall, from 4:10-5:00 pm.

Speaker: Marcelo Disconzi

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.

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.