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In the 2025 Fall semester, the seminars will be held both online and in-person (CU216). Online talks are usually on Thursdays starting at 4:30 PM, and in-person talks will be on Mondays at 4:00 PM.
Organizers: Huai-Dong Cao, Andrew Harder, Ao Sun, Xiaofeng Sun.
If you are interested in participating in the seminar, please email Ao (aos223 at lehigh dot edu).
Monday 9/1/25 (in-person, CU216, 4 PM)
Speaker: Ao Sun (Lehigh University)
Title: Geometry of cylindrical singularities of mean curvature flow
Abstract: The cylindrical singularities are prevalent but complicated in geometric flows. We used the dynamical information to classify the cylindrical singularities of mean curvature flow into three types: nondegenerate, degenerate, and partially nondegenerate. We further prove that
Nondegenerate singularities are isolated in spacetime, and passing through them can be represented by Morse surgeries topologically.
Degenerate singularities are contained in C^{2,\alpha} submanifolds, and if they are indeed submanifolds, the second fundamental form is determined by the asymptotic information.
Partially nondegenerate singularities are lower-dimensional strata.
There are several new ideas and techniques for deriving the results, including a non-concentration estimate for mean curvature flow, a new asymptotic expansion near the singularities, and a relative version of these tools. This talk is based on several joint works with Zhihan Wang and Jinxin Xue.
Thursday 9/4/25 (Online, Zoom, 4:30 PM)
Speaker: Sven Hirsch (Columbia)
Title: Rigidity of scalar curvature
Abstract: I discuss several rigidity questions for scalar curvature. This includes the resolution of two questions concerning PSC fill-ins by Gromov and Miao, and a geometric characterization of pp-waves.
Monday 9/8/25 (in-person, CU216, 4 PM)
Speaker: Guanhua Shao (Rutgers) and Jiahua Zou (Rutgers)
Title: Self-shrinkers with any number of ends in R^3 by stacking R^2 & Self-expanders of positive genus
Abstract: For each half-integer J and large enough integer m, we use gluing PDE methods to construct a self-shrinker with 2(J + 1) ends and genus 2J(m + 1). The shrinker resembles the stacking of 2J + 1 levels of the hyperplane in the 3-dimensional Euclidean space with 2Jm catenoidal bridges connecting each adjacent level. The construction is based on the Linearised Doubling (LD) methodology which was first introduced by Kapouleas in the construction of minimal surface doublings of the equator 2-sphere in the 3-sphere.
We also construct self-expanders of positive genus that has the same asymptotic cones as the shrinkers above when J = 1/2 and m become sufficiently larger. As a result, we are able to construct a mean curvature flow whose genus strictly decreases at a singular time (half of the genus is consumed at the singular time). We also construct a sequence of self-expanders of unbounded genus asymptotic to the same rotationally symmetric cone.
Thursday 9/25/25 (Online, Zoom, 4:30 PM)
Speaker: Joshua Daniels-Holgate (Queen Mary University of London)
Title: TBD
Abstract: TBD
Monday 9/29/25 (in-person, CU216, 4 PM)
Speaker: Eric Chen (University of Illinois Urbana-Champaign)
Title: TBD
Abstract: TBD
Monday 10/6/25 (in-person, CU216, 4 PM)
Speaker: Hongyi Liu (Princeton University)
Title: TBD
Abstract: TBD
Monday 10/20/25 (in-person, CU216, 4 PM)
Speaker: Jared Marx-Kuo (Rice University)
Title: TBD
Abstract: TBD
Monday 10/27/25 (in-person, CU216, 4 PM)
Speaker: Xingyu Zhu (Michigan State University)
Title: TBD
Abstract: TBD
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