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Lectures held at Université de Montréal

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Wednesday, May 8, 2024, at 3:30 pm
Room 6214, Pavillon André-Aisenstadt

(Lecture for a general mathematical audience - will be followed by a wine and cheese reception)

Ancient solutions to Geometric flows I Abstract of the Monday and Friday lecturesSome of the most important problems in partial differential equations are related to the understanding of singularities. This usually happens through a blow up procedure near the potential singularity which uses the scaling properties of the equation. In the case of a parabolic equation the blow up analysis often leads to special solutions which are defined for all time $- \infty < t \leq T$, for some $T \leq +\infty$. We refer to them as {\em ancient} solutions. The classification of such solutions often sheds new insight to the singularity analysis. In these two lectures we will discuss Uniqueness Theorems for ancient solutions geometric flows.In the first lecture we will give an overview of results including various examples of geometric flows that illustrated different features. Such flows are: Mean curvature flow, Ricci flow, Yamabe flow and Gauss curvature flow. In the second lecture, special emphasis will be given in the classification of non-collapsed solutions to Mean Curvature flow and the Ricci flow. We will discuss some of the techniques that have recently been developed to tackle such problems as well as applications and further research directions. gularity.. Ìý

Thursday, May 9, 2024, at 3:30 pm
Room 5340, Pavillon André-Aisenstadt

Type II smoothing in Mean curvature flow AbstractIn 1994 Velazquez constructed a family of smooth \( O(4)\times O(4)\) invariant solutions to Mean Curvature Flow that form a type-II singularity at the origin. Stolarski has recently shown that the Velazquez solutions have bounded Mean curvature at the singularity. We will discuss joint work with Angenent and Sesum where we show how to continue the flow past the Velázquez's singularity in a manner that the mean curvature remains bounded. Combined with the earlier results of Velazquez--Stolarski we therefore show the existence of a Mean curvature flow solution that is defined for time (-t_0, t_0), it has an isolated singularity at the origin at time t=0 and moreover, the Mean curvature remains uniformly bounded on this solution, even though the second fundamental form is unbounded near the singularity.

Friday, May 10, 2024, at 2:00 pm
Université de Montréal
Room 5340, Pavillon André-Aisenstadt

Ancient solutions to Geometric flows II
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Biography
Panagiota Daskalopoulos (Columbia University) is one of the leading experts in geometric flows. Her early work involved the understanding of regularity of natural flows, such as Gauss curvature, and often involving an application, such as wear of surfaces, porous media, flame propagation. Her recent work has focussed more on flows in a geometric context, with an emphasis on ‘ancient’ solutions, those existing on a semi-infinite backwards interval. She is a member of the American Academy of Arts and Sciences and was recently awarded the AMS’s Ruth Lyle Satter Prize.

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