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: Backwards uniqueness and rates of singularity formation in MCF

Abstract: The question of backwards uniqueness for parabolic PDE asks whether the same final state can be reached via the flow starting from two different sets of initial data. Backwards uniqueness is a far more subtle property compared to standard uniqueness but gives insight into the fine structure of solutions and singularity formation.

I will discuss two recent results concerning the comparative rate of singularity formation at compact singularities, and backwards uniqueness at isolated conical singularities in closed flows.:

• For flows with a compact singularity, we show that two flows cannot approach each other faster than polynomially, extending a result of Martin-Hagemayer--Sesum.

• For isolated conical singularities, we show backwards uniqueness by first proving a backwards uniqueness theorem assuming a rate of convergence and then show that this rate is always saturated by two flows that agree on the singular time slice.

This talk is based on joint work with Or Hershkovits.

Monday 10/6/25 (in-person, CU216, 4 PM)

Speaker: Hongyi Liu (Princeton University) 

Title: Poincaré-Einstein 4-manifolds with conformally Kähler geometry

Abstract: Poincaré–Einstein metrics play an important role in geometric analysis and mathematical physics, yet constructing new examples beyond the perturbative regime is difficult. In this talk, I will describe a class of four-dimensional Poincaré–Einstein manifolds that are conformal to Kähler metrics. These metrics admit a natural symmetry generated by a Killing field, which reduces the Einstein equations to a Toda-type system. This approach leads to existence and uniqueness results in the case of complex line bundles over surfaces of genus at least one. The construction produces large-scale, infinite-dimensional families of new Poincaré–Einstein metrics with conformal infinities of non-positive Yamabe type. This is joint work with Mingyang Li.

Monday 10/20/25 (in-person, CU216, 4 PM)

Speaker: Jared Marx-Kuo (Rice University) 

Title: Widths, Index, Intersection, and Isospectrality

Abstract: In this talk, I will discuss a series of works on Gromov's p-widths, $\{\omega_p\}$, on surfaces. For ambient dimensions larger than $2$, $\omega_p$ morally realizes the area of an embedded minimal surface of index p. This characterization was historically used to prove the existence of infinitely minimal hypersurfaces in closed Riemannian manifolds. In ambient dimension $2$, $\omega_p$ realizes the length of a union of (potentially immersed) geodesics, and heuristically, $p$ is equal to the sum of the indices of the geodesics plus the number of points of self-intersection. Joint with Lorenzo Sarnataro and Douglas Stryker, we prove upper bounds on the index and vertices, making progress towards this heuristic. Along the way, we prove a generic regularity statement for immersed geodesics. If time allows, we will also discuss the isospectral problem for the p-widths and how surfaces provide a convenient setting to investigate this.

Thursday 10/23/25 (Online, Zoom, 4:30 PM) (Postponed)

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: from Cones to cylinders: asymptotic geometry of Ricci flat manifolds with linear volume growth.

Abstract: In 2014, Colding—Minicozzi studied the asymptotic cones of a Ricci flat manifold with Euclidean volume growth and proved the uniqueness of the asymptotic cones if one cross section is smooth. Their result generalizes an earlier uniqueness result by Cheeger—Tian which requires integrability of the cross section among other things. In this talk I will talk about some recent results on the study of the similar uniqueness problem of asymptotic limit spaces for Ricci flat manifold with linear volume growth following the path of Cheeger—Tian and Colding—Mincozzi for asymptotic cones. Here one considers translation limits instead of rescaling limits and the limit spaces are no longer cones but cylinders. This is joint work with Zetian Yan.

Thursday 10/30/25 (Online, Zoom, 4:30 PM)

Speaker: Junsheng Zhang (NYU) 

Title: Kähler-Ricci Shrinkers and Polarized Fano Fibrations

Abstract: We prove that every (non-compact) Kähler-Ricci shrinker naturally admits a polarized Fano fibration structure. The proof relies on Kähler reductions and boundedness result in birational geometry. Moreover, we propose several conjectures for Kähler-Ricci shrinkers, unifying the well-developed theories of Kähler-Einstein metrics and Calabi-Yau cones. This is joint work with Song Sun.

Monday 11/3/25 (in-person, CU216, 4 PM)

Speaker: Ruojing Jiang (MIT) 

Title: Stability of Ricci flow and Minimal Surface Entropy for Finite-volume Hyperbolic 3-manifolds

Abstract: In this talk, I will discuss finite-volume hyperbolic 3-manifolds. If the initial metric is close to the hyperbolic metric h_0, then the normalized Ricci-DeTurck flow exists for all time and converges exponentially fast to h_0 in a weighted Holder norm. 

Furthermore, if time permits, I will also present an application to minimal surface entropy, which measures the growth rate of the number of closed minimal surfaces in terms of genus.

Thursday 12/4/25 (Online, Zoom, 4:30 PM)

Speaker: Sahana Vasudevan (IAS/Princeton) 

Title: TBD

Abstract: TBD