## Landscapes of St. Gregory: Towards an Ecoarthistory of Medieval Italy

## Local aspects of Venkatesh's thesis.

The thesis of Akshay Venkatesh obtains a ``Beyond Endoscopy'' proof of stable functorial transfer from tori to ${\rm SL}(2)$, by means of the Kuznetsov formula. In this talk, I will show that there is a local statement that underlies this work; namely, there is a local transfer operator taking orbital measures for the Kuznetsov formula to test measures on the torus. The global comparison of trace formulas is then obtained as a Poisson summation formula for this transfer operator.

## Halting problems for sandpiles and abelian networks

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

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

We answer a 1982 conjecture of Erdős 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.

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.

## Ricci flows with Rough Initial Data

## Filling metric spaces

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.

## Entropy of manifolds and of their fundamental group

Abstract : It is a joint work with G. Courtois, S. Gallot and A.Sambusetti. We prove a compactness theorem for metric spaces with anupper bound on the entropy and other conditions that will be discussed.Several finiteness results will be drawn. It is a work in progress.