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