Robert Haslhofer and Egor Shelukin are the winners of the 2020 André Aisenstadt Prize
This year, the André-Aisenstadt Prize recognizes the talent of two young Canadian mathematicians: Robert Haslhofer (Université de Toronto) and Egor Shelukin (Université de Montréal). The CRM International Scientific Advisory Committee met to select this year’s winner and was so impressed with their accomplishments that it recommended that both be awarded the prize. This is rarely done and is certainly an expression of high appreciation.
Robert Haslhofer (University of Toronto)
Mean curvature flow through neck-singularities
Abstract: A family of surfaces moves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow first arose as a model of evolving interfaces and has been extensively studied over the last 40 years.
In this talk, I will give an introduction and overview for a general mathematical audience. To gain some intuition we will first consider the one-dimensional case of evolving curves. We will then discuss Huisken’s classical result that the flow of convex surfaces always converges to a round point. On the other hand, if the initial surface is not convex we will see that the flow typically encounters singularities. Getting a hold of these singularities is crucial for most striking applications in geometry, topology and physics. Specifically, singularities can be either of neck-type or conical-type. We will discuss examples from the 90s, which show, both experimentally and theoretically, that flow through conical singularities is utterly non-unique.
In the last part of the talk, I will report on recent work with Kyeongsu Choi, Or Hershkovits and Brian White, where we proved that mean curvature flow through neck-singularities is unique. The key for this is a classification result for ancient asymptotically cylindrical flows that describes all possible blowup limits near a neck-singularity. In particular, this confirms the mean-convex neighborhood conjecture. Assuming Ilmanen’s multiplicity-one conjecture, we conclude that for embedded two-spheres mean curvature flow through singularities is well-posed.
Biography: Robert Haslhofer is a mathematician currently working as an Assistant Professor in the Department of Mathematics at the University of Toronto. He obtained his PhD in 2012 at the Swiss Federal Institute of Technology in Zurich. Haslhofer then was a Courant Instructor at the Courant Institute of Mathematical Sciences. Since 2015, he has been working as an Assistant Professor at the University of Toronto. His research interests are centered around Geometric Analysis, Differential Geometry, Partial Differential Equations, Calculus of Variations, Stochastic Analysis, General Relativity. His research is currently supported by NSERC Discovery Grant and a Sloan Research Fellowship.
Egor Shelukin (Université de Montréal)
Symmetry, barcodes, and Hamiltonian dynamics
Abstract: In the early 60s Arnol’d has conjectured that Hamiltonian diffeomorphisms, the motions of classical mechanics, often possess more fixed points than required by classical topological considerations. In the late 80s and early 90s Floer has developed a powerful theory to approach this conjecture, considering fixed points as critical points of a certain functional. Recently, in joint work with L. Polterovich, we observed that Floer theory filtered by the values of this functional fits into the framework of persistence modules and their barcodes, originating in data sciences. I will review these developments and their applications, which arise from a natural time-symmetry of Hamiltonians. This includes new constraints on one-parameter subgroups of Hamiltonian diffeomorphisms, as well as my recent solution of the Hofer-Zehnder periodic points conjecture. The latter combines barcodes with equivariant cohomological operations in Floer theory recently introduced by Seidel to form a new method with further consequences.
Biography: Egor Shelukhin is a mathematician currently working as an Assistant Professor in the Department of Mathematics and Statistics at the Université de Montréal. He obtained his PhD in 2012 at Tel Aviv University under the supervision of Leonid Polterovich. Shelukhin then was a CRM-ISM Postdoctoral Research Fellow in Mathematics at the Centre de recherches mathématiques (CRM) from 2012 to 2014. He spent the Spring semester of 2014 in the Hebrew University of Jerusalem and the Summer of 2015 at Université Lyon 1 Claude Bernard. Shelukhin was a Fellow at Institut Mittag Leffler in September 2015. From 2015 to 2017 he was a member at the School of Mathematics of the Institute for Advances Study, Princeton. His work is centered around Symplectic Topology, Contact Topology and Geometric Analysis.