Mathematical Physics and Harmonic Analysis Seminar
Summer 2020
Date: | May 29, 2020 |
Time: | 1:50pm |
Location: | Zoom |
Speaker: | Christoph Fischbacher, UC Irvine |
Title: | Logarithmic lower bounds for the entanglement entropy of droplet states for the XXZ model on the ring |
Abstract: | We study the free XXZ quantum spin model defined on a ring of size L and show that the bipartite entanglement entropy of eigenstates belonging to the first energy band above the vacuum ground state satisfy a logarithmically corrected area law. This is joint work with Ruth Schulte (LMU). |
Date: | July 31, 2020 |
Time: | 1:50pm |
Location: | Zoom |
Speaker: | Peter Kuchment, TAMU |
Title: | Spectral properties of periodically perforated spaces |
Abstract: | We study spectra of Schr\"odinger operators with periodic potentials in R^n with periodic perforations. We prove analytic dependence on the shape of the perforation and absolute continuity of the spectrum. |
Date: | August 7, 2020 |
Time: | 3:30pm |
Location: | Zoom |
Speaker: | Casey Rodriguez, MIT |
Title: | The Radiative Uniqueness Conjecture for Bubbling Wave Maps |
Abstract: | We will discuss the finite time breakdown of solutions to a canonical example of a geometric wave equation: energy critical wave maps. Breakthrough works of Krieger-Schlag–Tataru, Rodnianski-Sterbenz and Raphael–Rodnianski produced examples of wave maps that develop singularities in finite time. These solutions break down by concentrating energy at a point in space (via bubbling a harmonic map) but have a regular limit, away from the singular point, as time approaches the final time of existence. The regular limit is referred to as the radiation. This mechanism of breakdown occurs in many other PDE including energy critical wave equations, Schrödinger maps and Yang-Mills equations. A basic question is the following: Can we give a precise description of all bubbling singularities for wave maps with the goal of finding the natural unique continuation of such solutions past the singularity? In this talk, we will discuss recent work (joint with J. Jendrej and A. Lawrie) which is the first to directly and explicitly connect the radiative component to the bubbling dynamics by constructing and classifying bubbling solutions with a simple form of prescribed radiation. Our results serve as an important first step in formulating and proving the following Radiative Uniqueness Conjecture for a large class of wave maps: every bubbling solution is uniquely characterized by its radiation, and thus, every bubbling solution can be uniquely continued past blow-up time while conserving energy. |