Halting problems for sandpiles and abelian networks

Lionel Levine
Cornell University; von Neumann Fellow
March 12, 2019

Will this procedure be finite or infinite? If finite, how long can it last? Bjorner, Lovasz, and Shor asked these questions in 1991 about the following procedure, which goes by the name “abelian sandpile”: Given a configuration of chips on the vertices of a finite directed graph, choose (however you like) a vertex with at least as many chips as out-neighbors, and send one chip from that vertex to each of its out-neighbors. Repeat, until there is no such vertex. 

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.