School of Mathematics

Geometric PDE - Optimal transportation and nonlinear elliptic PDE

Neil Trudinger
Institute for Advanced Study
November 11, 2008


In these lectures we will describe the relationship between optimal transportation and nonlinear elliptic PDE of Monge-Ampere type, focusing on recent advances in characterizing costs and domains for which the Monge-Kantorovich problem has smooth diffeomorphism solutions.

Background references.

L.C Evans, PDE and Monge-Kantorovich mass transfer. Current developments in Mathematics, 1997. Int. Press, Boston, (1999).

The "P vs. NP" Problem: Efficient Computation, Internet Security, and the Limits of Human Knowledge

Avi Wigderson
Institute for Advanced Study
October 24, 2008

The "P vs. NP" problem is a central outstanding problem of computer science and mathematics.  In this talk, Professor Wigderson attempts to describe its technical, scientific, and philosophical content, its status, and the implications of its two possible resolutions.

Geometric PDE - Variational techniques for the prescribed Q-curvature equation

Andrea Malchiodi
Institute for Advanced Study
October 21, 2008


After recalling the definition of Q-curvature and some applications, we will address the question of prescribing it through a conformal deformation of the metric. We will address some compactness issues, treated via blow-up analysis, and then study the problem, which has variational structure, using a Morse-theoretical approach.

Geometric PDE - Fully Nonlinear Equations in Conformal Geometry

Matthew Gursky
Institute for Advanced Study
October 7, 2008

The goal of this course to provide an introduction to Monge-Ampere-type equations in conformal geometry and their applications.

The plan of the course is the following: After providing some background material in conformal geometry, I will describe the k-Yamabe problem, a fully nonlinear version of the Yamabe problem, and discuss the associated ellipticity condition and its geometric consequences.