In his seminal work from 1979,
Joseph J. Kohn invented
his theory of multiplier ideal sheaves
connecting a priori estimates for the d-bar problem
with local boundary invariants
constructed in purely algebraic way.
I will explain the origin and motivation of the problem,
and how Kohn's algorithm reduces it
to a problem in local geometry
of the boundary of a domain.
I then present my recent work with Sung Yeon Kim
based on the technique of jet vanishing orders,
and show how it can be used to
control the effectivity of multipliers in Kohn's algorithm,
subsequently leading to precise a priori estimates.
Abreu type equations are fully nonlinear, fourth order, geometric PDEs which can be viewed as systems of two second order PDEs, one is a Monge-Ampere equation and the other one is a linearized Monge-Ampere equation. They first arise in the constant scalar curvature problem in differential geometry and in the affine maximal surface equation in affine geometry. This talk discusses the solvability and convergence properties of second boundary value problems of singular, fourth order equations of Abreu type arising from approximation of convex functionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint. These functionals arise in different scientific disciplines such as Newton's problem of minimal resistance in physics and the Rochet-Choné model of monopolist's problem in economics. This talk explains how minimizers of the 2D Rochet-Choné model can be approximated by solutions of singular Abreu equations.
Mean field game theory is the study of the limit of Nash
equilibria of differential games when the number of players tends to infinity. It
was introduced by J.-M. Lasry and P.-L. Lions, and independently by P.
Caines, M. Huang and R. Malhame. A fundamental object in the theory is the
master equation, which fully characterizes the limit equilibrium. In this
talk, we will introduce Mean field game and master equations on graphs. We will
construct solutions to both equations and link them to the solution to
a Hamilton-Jacobi equation on graph.
We discuss some recent work on stability of solutions to linear and nonlinear wave equations on asymptotically flat space times. This is joint work with Jason Metcalfe, Jesus Oliver, and Daniel Tataru.