Inscribing Rectangles in Jordan Loops

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.

On the (in)stability of the identity map in optimal transportation

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.

Asymptotic spectra and Applications II

Jeroen Zuiddam
Member, School of Mathematics
October 15, 2019

In this second lecture in my series on asymptotic spectra we will focus on one application: the matrix multiplication problem. We will use the asymptotic spectrum of tensors to prove that a very general method (that includes the methods used to obtain the currently best algorithms) cannot give faster matrix multiplication algorithms. Keywords are Shannon entropy, representation theory and moment polytopes, but prior knowledge of these is not assumed.

The Continuing Search for Planet Nine

Matthew Holman
Harvard-Smithsonian Center for Astrophysics
October 15, 2019

Several lines of evidence, both theoretical and observational, indicate that additional planets in the outer solar system remain to be discovered.  Most recently, attention has been focused on 'Planet Nine',a few Earth mass planet with a moderate inclination at a distance of a few hundred AU (Trujillo & Sheppard 2014, Batygin & Brown 2016, Brown & Batygin 2016).  I will review this evidence for the existence of this body and will present a variety of approaches that are being used to find it.