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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.
Monday 9/15/25 (in-person, CU216, 4 PM)
Speaker: Jingwen Chen (UPenn)
Title: Morse theory for the area functional
Abstract: Morse theory is a powerful tool for analyzing the topology of a manifold by studying the critical points of a smooth function. The theory constructs Morse homology from the space of gradient flow trajectories, which provides a topological invariant that is isomorphic to singular homology. In this talk, we will apply Morse theory to the area functional, focusing on low area min-max critical points and the existence of trajectories connecting them. This is based on joint work with Pedro Gaspar.
Monday 9/22/25 (in-person, CU216, 4 PM)
Speaker: Xingzhe Li (Cornell)
Title: Embedded Minimal Tori in Three-spheres
Abstract: In this talk, we introduce the strong Morse inequalities for the area functional in the space of embedded tori and spheres in S^3. Applying this, we show the existence of at least nine embedded minimal tori in bumpy positively Ricci curved S^3. This talk is based on joint work with Zhichao Wang.
Monday 9/29/25 (in-person, CU216, 4 PM)
Speaker: Eric Chen (University of Illinois Urbana-Champaign)
Title: Expanding Ricci solitons asymptotic to cones with nonnegative scalar curvature
Abstract: In dimensions four and higher, the Ricci flow may encounter singularities modelled on cones with nonnegative scalar curvature. It may be possible to resolve such singularities and continue the flow using expanding Ricci solitons asymptotic to these cones, if they exist. I will discuss joint work with Richard Bamler in which we develop a degree theory for four-dimensional asymptotically conical expanding Ricci solitons, which in particular implies the existence of expanders asymptotic to a large class of cones.
Thursday 10/2/25 (Online, Zoom, 4:30 PM)
Speaker: Joshua Daniels-Holgate (Queen Mary University of London)
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
Thursday 10/23/25 (Online, Zoom, 4:30 PM)
Speaker: Zhenhua Liu (Princeton)
Title: The Hasse Principle for Geometric Variational Problems
Abstract: The Hasse principle in number theory states that information about integral solutions to Diophantine equations can be pieced together from real solutions and solutions modulo prime powers. We show that the Hasse principle holds for area-minimizing submanifolds: information about area-minimizing submanifolds in integral homology can be fully recovered from those in real homology and mod n homology for all integers n at least 2. As a consequence we derive several surprising conclusions, including: area-minimizing submanifolds in mod homology are asymptotically much smoother than expected and area-minimizing submanifolds are not generically calibrated. We conjecture that the Hasse principle holds for all geometric variational problems that can be formulated on chain space over different coefficients, e.g., Almgren-Pitts min-max, mean curvature flow, Song's spherical Plateau problem, minimizers of elliptic and other general functionals, etc.
Monday 10/27/25 (in-person, CU216, 4 PM)
Speaker: Xingyu Zhu (Michigan State University)
Title: TBD
Abstract: TBD
Thursday 12/4/25 (Online, Zoom, 4:30 PM)
Speaker: Sahana Vasudevan (IAS/Princeton)
Title: TBD
Abstract: TBD
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