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 19, in room 740, Evans Hall, from 4:10-5:00 pm.
Speaker: Marcelo Disconzi
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.
Title: Integrable systems, inverse scattering and conservation laws
Abstract: One common property of classical integrable systems (e.g. NLS, KdV) is that they have an infinite number of conservation laws, associated to the Sobolev spaces $H^n$. In this talk I will describe joint work with Herbert Koch aimed at finding a continuum of conservation laws for such systems.
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)
Place & Time : Evans Hall, room 740, Nov 2nd 2015, 4:10-5:00 pm.
Speaker: Richard B. Melrose (MIT)
Title: Differential operators undergoing adiabatic transitions
Abstract: I will describe a geometric type of degeneration of differential operators, which includes semiclassical and adiabatic limits. The most basic result for elliptic operators of this type is the inheritance of invertibility from the limiting operators. I will discuss this and applications of it, in particular in differential topology.
Organizers: Mihaela and Peter
Speaker: Mohammad Reza Pakzad
Title: Rigidity of weak solutions to Monge-Ampere equations
Abstract: In this talk, we will explore rigidity of the weak solutions to the Monge-Amp\`ere equation, by replacing the Hessian determinant by other weaker variants, without any a priori convexity assumptions. Some past and recent results and their proofs concerning rigid behaviour (e.g. convexity or developabilty) of Sobolev solutions in two and higher dimensions will be discussed. We will also study the rigidity of solutions with H\”older continuous derivatives. We will contrast these results with some some non-rigidity statements recently proved by the speaker and M. Lewicka using convex integration.
Speaker: Vedran Sohinger (ETH Zurich)
Title: The Gross-Pitaevskii hierarchy on periodic domains
Abstract: The Gross-Pitavskii hierarchy is a system of infinitely many linear PDEs which occurs in the derivation of the nonlinear Schrodinger equation from the dynamics of many-body quantum systems. We will study this problem in the periodic setting. Even though the hierarchy is linear, it is non closed, in the sense that the equation for the k-th density matrix in the system depends on the (k+1)-st density matrix. This structure poses its challenges in the study of the problem, in particular in the understanding of uniqueness of solutions. Moreover, by randomizing in the collision operator, it is possible to use probabilistic techniques in order to study related hierarchies at low regularities. I will present some recent results obtained on these problems, partly in joint work with Philip Gressman, Sebastian Herr, and Gigliola Staffilani.