# Overview

##### [ Français ]

**Yevgeny Liokumovich, University of Toronto**

## Watch the videoconference of Yevgeny Liokumovich given on September 18.

## First lecture - September 18, 2020, 3 p.m.

This lecture is aimed at a general mathematical audience.

## Measuring size and complexity of Riemannian manifolds

Given a Riemannian manifold how hard is it to slice it into pieces of smaller dimension? More precisely, we can ask how large in terms of their volume, diameter and topological complexity the slices have to be? These questions give rise to various notions of width of a manifold, which turn out to be closely related to questions about minimal surfaces. I will describe some recent results relating widths to other notions of size of a Riemannian manifold. The talk will be partly based on separate joint works with A. Nabutovsky, R. Rotman and B. Lishak, with G.R. Chambers, and with D. Ketover and A. Song.

## Second lecture - September 21 2020, 3 p.m.

## Watch the videoconference of Yevgeny Liokumovich given on September 21

## Minimal surfaces and quantitative topology

To each cohomology class of the space of codimension 1 cycles, Min-Max theory associates a minimal hypersurface with some integer multiplicity. Volumes of these minimal hypersurfaces are called "widths" or "volume spectrum" of the manifold. Gromov conjectured that like eigenvalues of the Laplacian, the volume spectrum has asymptotic behaviour described by a Weyl law. I will discuss the proof of this conjecture for hypersurfaces and some recent progress for the Weyl law in higher codimension. The talk will be based on separate joint works with F.C. Marques and A. Neves and with L. Guth.

**Biographical note**:
Yevgeny Liokumovich received his Ph.D. in 2015 at the University of Toronto under the supervision of A. Nabutovsky and R. Rotman. After a postdoc at MIT and the Institute for Advanced Study, he returned to Toronto in 2019 as an Assistant Professor. Yevgeny Liokumovich has obtained several major results in geometric analysis, including a solution of Gromov’s
conjecture on the Weyl law for the volume spectrum in a recent joint work with F.C. Marques and A. Neves.

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**Antoine Song, University of California, Berkeley**

**Videoconferences on Zoom (registration is free but required via the register tab)**

## First lecture - September 23, 2020, 3 p.m.

## Watch the videoconference of Antoine Song given on September 23.

## Complexities of minimal hypersurfaces

Let M be a closed Riemannian 3-manifold. By min-max theory, one can construct a sequence of minimal surfaces embedded in M which are the geometric analogue of eigenfunctions of the Laplacian. Motivated by well known questions about eigenfunctions, one can ask about the complexity of the minimal surfaces in that sequence. There are several natural measures of complexity: genus, area, or Morse index. I will talk about the interaction between these quantities, which has been thoroughly studied since the classical work of R. Schoen and S.-T. Yau, and I will present new quantitative estimates that are relevant for the question previously mentioned.

## Second lecture - September 25, 2020, 3 p.m.

This lecture is aimed at a general mathematical audience.

## Watch the videoconference of Antoine Song given on September 25.

## Abundance of minimal hypersurfaces

Minimal hypersurfaces are higher dimensional analogues of geodesics. In the early 80's, S.-T. Yau conjectured that in any closed Riemannian 3-manifold, there is infinitely many minimal surfaces. I will introduce the problem and give an account of the recent series of work by many people, which led to the understanding that minimal hypersurfaces abound in closed Riemannian manifolds. In particular, Yau's conjecture is true and for generic metrics, much stronger properties, like equidistribution, hold. I will give some ideas from the proofs, which borrow tools from analysis, geometric measure theory and topology. This talk is partially based on joint work with F.C. Marques and A. Neves.

**Biographical note**:
Antoine Song received his Ph.D. in 2019 from Princeton University under the supervision of F.C. Marques. He has made several spectacular advances in the theory of minimal surfaces. In particular, in his Ph.D. thesis, he presented a complete solution of Yau’s conjecture on the existence of infinitely many minimal hypersurfaces in closed manifolds. Currently, Antoine Song is a Clay Research Fellow working at the University of California, Berkeley.