Members Seminar

Geometry of 2-dimensional Riemannian disks and spheres.

Regina Rotman
University of Toronto; Member, School of Mathematics
March 11, 2019

I will discuss some geometric inequalities that hold on
Riemannian 2-disks and 2-spheres. 

For example, I will prove that on any Riemannian 2-sphere there M exist
at least three simple periodic geodesics of length at most 20d, where d is the diameter of M, (joint with A. Nabutovsky, Y. Liokumovich).
This is a quantitative version of the well-known Lyusternik and Shnirelman theorem.

Positive geometries

Thomas Lam
University of Michigan; von Neumann Fellow, School of Mathematics
February 25, 2019

Positive geometries are real semialgebraic sets inside complex varieties characterized by the existence of a meromorphic top-form called the canonical form. The defining property of positive geometries and their canonical forms is that the residue structure of the canonical form matches the boundary structure of the positive geometry. A key example of a positive geometry is a projective polytope. 

The Sample Complexity of Multi-Reference Alignment

Philippe Rigollet
Massachusetts Institute of Technology; Visiting Professor, School of Mathematics
February 4, 2019
How should one estimate a signal, given only access to noisy versions of the signal corrupted by unknown cyclic shifts? This simple problem has surprisingly broad applications, in fields from aircraft radar imaging to structural biology with the ultimate goal of understanding the sample complexity of Cryo-EM. We describe how this model can be viewed as a multivariate Gaussian mixture model whose centers belong to an orbit of a group of orthogonal transformations.

Recent Progress on Zimmer's Conjecture

David Fisher
Indiana University, Bloomington; Member, School of Mathematics
December 3, 2018
Lattices in higher rank simple Lie groups are known to be extremely rigid. Examples of this are Margulis' superrigidity theorem, which shows they have very few linear represenations, and Margulis' arithmeticity theorem, which shows they are all constructed via number theory. Motivated by these and other results, in 1983 Zimmer made a number of conjectures about actions of these groups on compact manifolds and in a recent breakthrough with Brown and Hurtado we have proven many of them.

A tale of two conjectures: from Mahler to Viterbo.

Yaron Ostrover
Tel Aviv University; von Neumann Fellow, School of Mathematics
November 19, 2018
In this talk we explain how billiard dynamics can be used to relate a symplectic isoperimetric-type conjecture by Viterbo with an 80-years old open conjecture by Mahler regarding the volume product of convex bodies. The talk is based on a joint work with Shiri Artstein-Avidan and Roman Karasev.

New and old results in the classical theory of minimal and constant mean curvature surfaces in Euclidean 3-space R^3

Bill Meeks
University of Massachusetts Amherst
October 22, 2018
In this talk I will present a survey of some of the famous results and examples in the classical theory of minimal and constant mean curvature surfaces in R^3. The first examples of minimal surfaces were found by Euler (catenoid) around 1741, Muesner (helicoid) around 1746 and Riemann (Riemann minimal examples) around 1860. The classical examples of non-zero constant mean curvature surfaces are the Delaunay surfaces of revolution found in 1841, which include round spheres and cylinders.