In the 2025 Spring semester, the seminars will be held both online and in-person (CU 116).  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 1/20/25 (in-person)

Speaker: Farhan Abedin (Lafayette College)

Title: An Oscillatory Free Boundary Problem

Abstract: I will discuss recent joint work with Will Feldman (University of Utah) on large-scale regularity properties for a free boundary problem of obstacle type in periodic media.

Thursday 1/23/25 (online)

Speaker: Greg Parker (Stanford)

Title: Families of non-product minimal submanifolds with cylindrical tangent cones

Abstract: The study of singularities of minimal submanifolds has a long history, with isolated singularities being the best understood case. The next simplest case is that of minimal submanifolds with families with singularities locally modeled on the product of an isolated conical singularity and a Euclidean space — such submanifolds are said to have cylindrical tangent cones at these singularities. Despite work in many contexts on minimal submanifolds with such singularities, the only known explicit examples at present are global products or involve extra structure (e.g. Kahler subvarieties). In this talk, I will describe a method for constructing infinite-dimensional families of non-product minimal submanifolds in arbitrary codimension whose singular set is itself an analytic submanifold. The construction uses techniques from the analysis of singular elliptic operators and Nash-Moser theory. This talk is based on joint work with Rafe Mazzeo.

Monday 1/27/25 (in-person)

Speaker: Keaton Naff (Lehigh)

Title: Revisiting the planarity estimate in higher codimension mean curvature flow

Abstract: In this talk, we will discuss the mean curvature flow of $n$-dimensional submanifolds in $\mathbb{R}^N$ satisfying a pinching condition $|A|^2 < c |H|^2$ introduced by Andrews and Baker ('10).  Inspired by the work of Brendle-Huisken-Sinestrari (`11), we will discuss a slight improvement to the planarity estimate and applications to ancient pinched solutions of mean curvature flow in higher codimension. We will end by discussing an outstanding problem in the surgery of certain pinched flows in high codimension and potential applications of the improved estimate in this area.

Thursday 1/30/25 (online)

Speaker: Jingbo Wan (Columbia)

Title: Rigidity of contracting maps between positively curved manifolds

Abstract: We discuss the rigidity of contracting maps between positively curved manifolds using geometric heat flows, focusing on distance and area non-increasing maps between spheres or complex projective spaces. The methods rely on energy monotonicity for harmonic map heat flow, and the Strong Maximum Principle for Mean Curvature Flow or the coupled Mean Curvature Flow-Ricci Flow system. These results demonstrate and extend an interesting application of geometric heat flows to the homotopy classes of maps, pioneered by Prof. Mao-Pei Tsui and Prof. Mu-Tao Wang in 2004. This is based on joint works with Man-Chun Lee and Luen-Fai Tam.

Monday 2/3/25 (in-person)

Speaker: Liam Mazurowski (Lehigh)

Title: Stability of the Yamabe Invariant of S^3

Abstract: The Yamabe problem asks whether every closed Riemannian manifold admits a conformal metric with constant scalar curvature. The Yamabe problem has been fully resolved in the affirmative by work of Yamabe, Trudinger, Aubin, and Schoen. The resolution of the Yamabe problem is closely connected to an inequality for the total scalar curvature: the total scalar curvature of (M^n,g) is at most that of the round n-sphere with the same volume. Moreover, if equality holds then (M^n,g) is conformal to a round n-sphere. It is natural to investigate the stability of this inequality. In this talk, we will show that if the total scalar curvature of (S^3,g) is close to that of the round 3-sphere with the same volume, then some metric in the conformal class of g is close to round in a certain weak sense. This is joint work with Xuan Yao.

Monday 2/17/25 (in-person)

Speaker: Conghan Dong (Duke)

Title: Non-collapsing of Ricci shrinkers with bounded curvature

Abstract: In this talk, I will discuss my recent joint work with Yu Li, where we show that for any simply connected Ricci shrinker with finite second homotopy group, uniformly bounded curvature implies uniformly non-collapsing.

Monday 2/24/25 (in-person)

Speaker: Ben Lowe (UChicago)

Title: Minimal Surfaces in Negative Curvature

Abstract: Kahn-Markovic showed that every closed negatively curved 3-manifold contains essential minimal surfaces in great abundance.  Since then the goal of understanding the geometry of these minimal surfaces has been a focus of activity, both in analogy to the geodesic flow one dimension lower and the more positive-curvature-centric  min-max theory of minimal surfaces.   This talk will survey recent developments in this area, which brings together techniques from dynamical systems, geometric analysis, and hyperbolic geometry.

Monday 3/3/25 (in-person)

Speaker: Doug Stryker (Princeton)

Title: Min-max constructions in noncompact manifolds

Abstract: The existence problem for minimal hypersurfaces (or more generally prescribed mean curvature hypersurfaces) in complete noncompact manifolds is fundamental, but little is known. Min-max provides a powerful framework for existence in closed manifolds, but relies critically on compactness. I will present a new localization technique to adapt min-max theory to noncompact manifolds.

Monday 3/17/25 (in-person)

Speaker: Adrian Chun-Pong Chu (Cornell)

Title: Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

Abstract: In 1989, B. White conjectured that every Riemannian 3-sphere has at least 5 embedded minimal tori. I will present a recent work with Yangyang Li, in which we confirm this conjecture for 3-spheres of positive Ricci curvature. While our proof uses min-max theory, the underlying heuristics are largely inspired by mean curvature flow.

Thursday 3/20/25 (online)

Speaker: Jaume de Dios Pont (ETH Zurich)

Title: Convex sets can have interior hot spots

Abstract: A homogeneous, insulated object with a non-uniform initial temperature will eventually reach thermal equilibrium. The Hot Spots conjecture addresses which point in the object takes the longest to reach this equilibrium: Where is the maximum temperature attained as time progresses? Rauch initially conjectured that points attaining the maximum temperature would approach the boundary. Burdzy and Werner disproved the conjecture for planar domains with holes. Kawohl, and later Bañuelos-Burdzy, conjectured that this should still hold for convex sets of all dimensions. 

This talk will draw inspiration from a recurrent theme in convex analysis: almost every dimension-free result in convex analysis has a natural log-concave extension. We will motivate and construct the log-concave analog of the Hot Spots conjecture, and then disprove it. Using this log-concave construction, we will show that the hot spots conjecture for convex sets is false in high dimensions.

Monday 3/24/25 (in-person; Special time: 5 pm - 6 pm)

Speaker: Zetian Yan (UC Santa Barbara)

Title: Rigidity and nonexistence of CMC hyper surfaces of 5 manifolds.

Abstract: We prove that the nonnegative 3-intermediate Ricci curvature and uniformly positive k-triRic curvature implies rigidity of complete noncompact two-sided weakly stable minimal hypersurfaces in a Riemannian manifold (X^5, g) with bounded geometry. The nonnegativity of 3-intermediate Ricci curvature can be replaced by nonnegative Ricci and biRic curvature. In particular, there is no complete non-compact stable minimal hypersurface in a closed 5-dimensional manifold with positive sectional curvature. It extends the result of Chodosh-Li-Stryker [to appear in J. Eur. Math. Soc (2024)] to 5-dimensions. We also prove some nonexistence results of complete noncompact constant mean curvature hypersurfaces with finite index in hyperbolic space H^5.

Monday 3/31/25 (in-person)

Speaker: Dongyeong Ko (Rutgers)

Title: Cohomogeneity two min-max theory and minimal hypersurfaces with arbitrarily large Betti number on high dimensional spheres

Abstract: In this talk, I will explain a new equivariant min-max construction to construct minimal hypersurfaces. Also, I discuss a min-max construction of sequences of minimal hypersurfaces with arbitrarily large Betti number on high dimensional spheres in dimension 4 to 7 as an application.

Monday 4/7/25 (in-person)

Speaker: Yangyang Li (UChicago)

Title: Non-persistence of Strongly Isolated Singularities and Hsiang's minimal hyperspheres

Abstract: In 1969, Shiing-Shen Chern proposed the spherical Bernstein problem, asking whether the equators in a round (n+1)-dimensional sphere are the only smooth, embedded minimal hyperspheres. In 1983, Hsiang provided a negative answer by constructing an infinite sequence of distinct embedded minimal hyperspheres in the round 4-dimensional sphere. This sequence arises from the desingularization of the Clifford football—the spherical suspension of a Clifford torus inside an equator—which has exactly two strongly isolated singular points. 

About a decade ago, André Neves asked whether such a phenomenon persists under a small perturbation of the round metric. In this talk, I will discuss how to show the non-persistence of these strongly isolated singular points under a generic perturbation by analyzing the Fredholm index of the Jacobi operator for a certain class of varifolds. As a geometric application, we provide a negative answer to Neves’ question. This is based on joint work with Alessandro Carlotto and Zhihan Wang.

Monday 4/14/25 (in-person)

Speaker: Kai Xu (Duke)

Title: Drawstrings and the geometry of scalar curvature

Abstract: We will discuss a new phenomenon called drawstrings and its effects on the geometry of scalar curvature. Roughly speaking, this phenomenon allows one to collapse any codimension 2 submanifold while only decreasing the scalar curvature by ε. Some applications to scalar curvature stability problems will be addressed. This talk is based on recent joint works with Demetre Kazaras.

Monday 4/21/25 (in-person)

Speaker: Boyu Zhang (University of Maryland)

Title: On the topology of stable minimal surfaces in PSC 4-manifolds

Abstract: It is known that a closed 2-sided stable minimal hypersurface in a 4-manifold with positive scalar curvature (PSC) must be Yamabe positive, and hence it is diffeomorphic to a connected sum of S^1 x S^2’s and spherical space forms.  We show that using a new compactness result for minimal surfaces in covering spaces and techniques from 4-manifold topology, one can obtain further control of the topology of stable minimal hypersurfaces. As an application, we show that the outermost apparent horizons of a smooth, asymptotically flat manifold with nonnegative scalar curvature must be diffeomorphic to S^3 or connected sums of S^1 x S^2’s.  This is an extension of Hawking’s black hole topology theorem to dimension 4. The talk is based on joint work with Chao Li.

Tuesday 4/22/25 (in-person) Special Analysis Seminar at CU115

Speaker: Chengyang Shao (University of Chicago)

Title: Paradifferential Approach to “KAM for PDEs”

Abstract: In this talk, I will explain how paradifferential calculus can be applied to construct quasiperiodic-in-time solution for PDEs (“KAM for PDEs”). Due to the loss of regularity caused by “small divisors”, these problems are traditionally resolved using Nash-Moser/KAM type iterative schemes. One step of the Nash-Moser scheme is to reduce a non-autonomous linear operator that involves “small divisors” into constant coefficient form, for which a Nash-Moser/KAM reducibility argument is necessary, yielding a complicated “Nash-Moser within Nash-Moser” formalism. However, it is discovered that paradifferential calculus can be used to avoid such formalism, yielding “fixed point” style proof. In particular, it is possible to reduce the nonlinear equation itself into constant coefficient form (modulo smoothing remainder), not just its linearization. This is because paradifferential operators share all the algebraic structures of (pseudo)differential operators while gain back regularities due to J.-M. Bony’s paralinearization process. I will use the existence problem for quasiperiodic-in-time solution of quasilinear hyperbolic systems as illustrative example. This talk is based on joint works with Thomas Alazard.

Monday 4/28/25 (in-person) 

Speaker: Junming Xie (Rutgers)

Title: Convexity of 2-convex translating and expanding solitons to the mean curvature flow in $R^{n+1}$

Abstract: In this talk, we will discuss the convexity of 2-convex self-similar solutions to the mean curvature flow in $R^{n+1}$. Specifically, inspired by the work of Spruck and Xiao, we show that complete 2-convex translators are convex, and complete 2-convex self-expanders asymptotic to mean convex cones are convex. This talk is based on joint work with Jiangtao Yu.

Monday 5/5/25 (in-person) 

Speaker: Max Hallgren (Rutgers)

Title: Finite-Time Singularities of the Ricci Flow on Kähler Surfaces

Abstract: By work of Song-Weinkove, it is understood that the Ricci flow on any Kähler surface can canonically be continued through singularities in a continuous way until its volume collapses. This talk will discuss recent progress in understanding a more detailed picture of the singularity formation in this context.