Title: Complex geometry and optimal transport
Speaker: Gabriel Khan (University of Michigan)
Abstract: In this talk, we consider the regularity theory of optimal transport and relate it to the curvature of certain Kahler metrics. The solutions of optimal transport problems are induced from solutions to associated Monge-Ampere equations. From the work of Ma, Trudinger, Wang, and Loeper, two assumptions must be made in order to establish $C^2$-regularity. Firstly, the support of the associated measures must be relatively c-convex and secondly, the so-called MTW tensor must be non-negative definite. In joint work with J. Zhang, we study cost functions of the form $c(x,y)= \psi (x-y)$ for a convex function $\psi$. In this setting, we construct an associated Kahler manifold whose orthogonal bisectional curvature is exactly the MTW tensor for the cost $c$. Furthermore, we show that relative c-convexity is geodesic convexity with respect to a particular connection that is not the Levi-Civita connection. We discuss two applications of this work. Firstly, our approach can be used to construct a complete Kahler metric with non-negative orthogonal bisectional curvature that is not biholomorphic to $C^2$. Secondly, we relate our work to some recent work by Pal and Wong on functional portfolio design. In their work, they study the problem of finding investment strategies that “beat the market” and show that such strategies can be induced by the solution of certain optimal transport problems. Roughly speaking, these results shows that their strategies are regular in the sense that small changes in market conditions induce small changes in the investment portfolio.
