Mario Bonk, UCLA
Title: Quasiregular maps and non-linear potential theory
Abstract: Non-linear potential theory is closely related to the theory of quasiconformal and quasiregular maps in higher dimensions. For example, one can use potential-theoretic methods to prove Liouville's theorem for 1-quasiconformal or the Rickman-Picard theorem for quasiregular mappings. In my talk I will outline some of the basic ideas and discuss recent developments.
Katya Krupchyk, UC Irvine
Title: Inverse boundary problems for elliptic PDE
Abstract: We shall provide a general introduction to the field of inverse boundary problems for elliptic PDE, with the celebrated Calderon problem serving as a prototypical example. We shall also survey some of the more recent developments, including partial data inverse problems, inverse boundary problems on Riemannian manifolds, as well as inverse boundary problems for non-linear equations. In particular, we shall see that the presence of a nonlinearity may actually help, allowing one to solve inverse problems in situations where the corresponding linear counterpart is open. This talk is based on joint works with Yavar Kian, Tony Liimatainen, Mikko Salo, and Gunther Uhlmann.
Jason Metcalfe, UNC Chapel Hill
Title: Local energy in the presence of degenerate trapping
Abstract: Trapping is a known obstruction to local energy estimates for the wave equation and local smoothing estimates for the Schrödinger equation. When this trapping is sufficiently unstable, it is known that estimates with a logarithmic loss can be obtained. On the other hand, for very stable trapping, it is known that all but a logarithmic amount of local energy decay is lost. Until somewhat recently, explicit examples of scenarios where an algebraic loss (of regularity) was both necessary and sufficient for local energy decay had not be constructed. We will review what is known in these specific examples. We will also examine the relationship between the trapping and the existence of a boundary. In this highly symmetric case, a relatively simple proof showing a bifurcation in the behavior of local energy as the boundary passes through the trapping is available. This is related, e.g., to the instability of ultracompact neutrino stars.
Natasa Pavlovic, UT Austin
Title: Beyond binary interactions of particles
Abstract: In this talk we shall discuss dynamics of systems of particles that allow interactions beyond binary, and their behavior as the number of particles goes to infinity. In particular, an example of such a system of bosons leads to a quintic nonlinear Schrodinger equation, which we rigorously derived in a joint work with Thomas Chen. An example of a system of classical particles allowing instantaneous ternary interactions leads to a new kinetic equation that can be understood as a step towards modeling a dense gas in non-equilibrium. We call this equation a ternary Boltzmann equation and we rigorously derive it in a recent work with Ioakeim Ampatzoglou. Time permitting, we will also discuss the recent work with Ampatzoglou on a derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a dense hard-spheres gas, where particles undergo either binary or ternary instantaneous interactions, while preserving momentum and energy. An important challenge we overcome in deriving this equation is related to providing a mathematical framework that allows us to detect both binary and ternary interactions. Furthermore, this work introduces new algebraic and geometric techniques in order to eventually decouple binary and ternary interactions and understand the way they could succeed one another in time.