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Lauréat 2020 du Prix Andre-Aisenstadt
CRM > Prix > Prix André-Aisenstadt > Lauréat > Robert Haslhofer (Université de Toronto) | Egor Shelukin (Université de Montréal)

Lauréats 2020 du prix de mathématiques André-Aisenstadt
Robert Haslhofer (Université de Toronto) | Egor Shelukin (Université de Montréal)

[ English ]

Le prix André-Aisenstadt reconnaît cette année le talent de deux jeunes mathématiciens canadiens. Le Comité scientifique international du CRM s'est réuni pour sélectionner le gagnant de cette année et il a été tellement impressionné par les réalisations de ces deux candidatures qu'il a recommandé que les deux reçoivent le prix. Ce qui est rare et c'est une expression de haute appréciation.

Les deux lauréats choisis par le Comité scientifique international du CRM sont :

Robert Haslhofer(Université de Toronto) Egor Shelukhin (Université de Montréal)


1re Conférence 22 janvier 2021 à 15 h

Robert Haslhofer (Université de Toronto), ex aequo

Mean curvature flow through neck-singularities

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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.


2e Conférence 5 février 2021 à 15 h

Egor Shelukhin (Université de Montréal), ex aequo

Symmetry, barcodes, and Hamiltonian dynamics

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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.