An Exposition on Geodesic Convexity

I arrived at the Institute for Advanced Study to attend a workshop on Optimization, Complexity and Invariant Theory organized by Avi Wigderson. The workshop has an incredibly diverse agenda, also reflected in the participants.

I will talk about Geodesic Convexity: Sometimes, functions that are non-convex in the Euclidean space turn out to be convex if one introduces a suitable metric (or differential structure) and redefines convexity with respect to the induced straight lines — geodesics. Such a function is called geodesically convex. Unlike standard convexity,  when does a function have this property and how to optimize it, is not well-understood. My talk will focus on introducing geodesic convexity and show that the problem of computing the Brascamp-Lieb constant has a succinct geodesically convex formulation.

And, accompanying my talk are these self-contained and expository notes on differentiation on manifolds, geodesics, and geodesic convexity that are prepared with the help of my student Ozan Yildiz.