## Fixing GAN optimization through competitive gradient descent

Anima Anandkumar

Caltech

October 15, 2019

Amir Asadi, Dimitris Kalimeris

October 15, 2019

Sanjeev Arora

Princeton University; Distinguished Visiting Professor, School of Mathematics

October 15, 2019

Karolina Dziugaite Dziugaite

Simons Institute for the Theory of Computing

October 15, 2019

Ke Li

University of California, Berkeley

October 15, 2019

Aleksander Madry

Massachusetts Institute of Technology

October 15, 2019

Boaz Nadler

Weizmann Institute of Science; Member, School of Mathematics

October 14, 2019

Rich Schwartz

Brown University

October 14, 2019

I'll show a graphical user interface I wrote which explores the problem of inscribing rectangles in Jordan loops. The motivation behind this is the notorious Square Peg Conjecture of Toeplitz, from 1911.

I did not manage to solve this problem, but I did get the result that at most 4 points of any Jordan loop are vertices of inscribed rectangles. I will sketch a proof of this result, mostly through visual demos, and also I will explain two other theorems about inscribed rectangles which at least bear a resemblance to theorems in symplectic geometry.

Yash Jhaveri

Member, School of Mathematics

October 14, 2019

In the optimal transport problem, it is well-known that the geometry of the target domain plays a crucial role in the regularity of the optimal transport. In the quadratic cost case, for instance, Caffarelli showed that having a convex target domain is essential in guaranteeing the optimal transport’s continuity. In this talk, we shall explore how, quantitatively, important convexity is in producing continuous optimal transports.

Geoff Penington

Stanford University

October 14, 2019