Macroscopically minimal hypersurfaces

Hannah Alpert
Ohio State University
March 12, 2019

A decades-old application of the second variation formula
proves that if the scalar curvature of a closed 3--manifold is bounded
below by that of the product of the hyperbolic plane with the line,
then every 2--sided stable minimal surface has area at least that of
the hyperbolic surface of the same genus. We can prove a coarser
analogue of this statement, taking the appropriate notions of
macroscopic scalar curvature and macroscopic minimizing hypersurface
from Guth's 2010 proof of the systolic inequality for the

Near log-convexity of measured heat in (discrete) time and consequences

Mert Saglam
University of Washington
March 11, 2019

We answer a 1982 conjecture of Erd&‌#337;s and Simonovits about the growth of number of $k$-walks in a graph, which incidentally was studied earlier by Blakley and Dixon in 1966. We prove this conjecture in a more general setup than the earlier treatment, furthermore, through a refinement and strengthening of this inequality, we resolve two related open questions in complexity theory: the communication complexity of the $k$-Hamming distance is $\Omega(k \log k)$ and that consequently any property tester for k-linearity requires $\Omega(k \log k)$.

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.

Filling metric spaces

Alexander Nabutovsky
University of Toronto; Member, School of Mathematics
March 8, 2019

Abstract:  Uryson k-width of a metric space X measures how close X is to being k-dimensional. Several years ago Larry Guth proved that if M is a closed n-dimensional manifold, and the volume of each ball of radius 1 in M does not exceed a certain small

constant e(n), then the Uryson (n-1)-width of M is less than 1. This result is a significant generalization of the famous Gromov's inequality relating the volume and the filling radius that plays a central role in systolic geometry.

 

Normalized harmonic map flow

Michael Struwe
ETH Zürich
March 6, 2019

Abstract: Finding non-constant harmonic 3-spheres for a closed target manifold N is a prototype of a super-critical variational problem. In fact, the 
direct method fails, as the infimum of the Dirichlet energy in any homotopy class of maps from the 3-sphere to any closed N is zero; moreover, the 
harmonic map heat flow may blow up in finite time, and even the identity map from the 3-sphere to itself is not stable under this flow.