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.