2023 Fall
In the 2023 Fall semester, the seminars will be held both online and in-person (CU 215). Online talks are usually on Thursdays starting at 4:30 PM, and in-person talks will be on Mondays at 4:10 PM.
Organizers: Huai-Dong Cao, Ao Sun, Xiaofeng Sun.
If you are interested in participating in the seminar, please email Ao (aos223 at lehigh dot edu).
Thursday 09/07/23 (Zoom)
Speaker: Christos Mantoulidis (Rice University)
Title: Improved generic regularity for minimizing hypersurfaces
Abstract: I will discuss recent joint work with O. Chodosh and F. Schulze in which we show that minimizing hypersurfaces are generically smooth in ambient dimensions up to 10. In higher ambient dimensions n+1 >= 11, we also showed that minimizers are generically smooth outside an (n-9-eps_n)-dimensional set.
Thursday 09/14/23 (Zoom)
Speaker: Lvzhou Chen (Purdue University)
Title: The Kervaire conjecture and the minimal complexity of surfaces
Abstract: We use topological methods to solve special cases of a fundamental problem in group theory, the Kervaire conjecture, which has connection to various problems in topology. The conjecture asserts that, for any nontrivial group G and any element w in the free product G*Z, the quotient (G*Z)/<<w>> is still nontrivial. We interpret this as a problem of estimating the minimal complexity (in terms of Euler characteristic) of surface maps to certain spaces. This gives a conceptually simple proof of Klyachko's theorem that confirms the Kervaire conjecture for any G torsion-free. We also obtain injectivity of the map G->(G*Z)/<<w>> when w is a proper power for arbitrary G. Both results generalize to certain HNN extensions.
Monday 09/18/23 (in-person)
Speaker: Zilu Ma (Rutgers)
Title: On Four-dimensional Noncollapsed Steady Ricci Solitons
Abstract: Steady Ricci Solitons may arise as singularity models of the Ricci flow and they are essential to the singularity analysis in order to achieve any topological applications using surgeries, particularly in dimension 4. In this talk, we shall present some results on four-dimensional noncollapsed steady gradient Ricci solitons in some recent joint works. We first classify the tangent flows at infinity in dimension four. With the classification at infinity, we give a classification of Kaehler steady solitons. Under the assumption of positive curvature, we present some recent results on rough geometric asymptotics and precise analytic asymptotics.
Monday 9/25/23 (in-person)
Speaker: Liam Mazurowski (Cornell University)
Title: Recent Developments in Constant Mean Curvature Hypersurfaces
Abstract: A constant mean curvature surface is a critical point of the area functional subject to a volume constraint. Min-max theory is a powerful method for finding saddle type critical points of functionals. Recently, Xin Zhou and Jonathan Zhu developed a min-max theory for finding constant mean curvature surfaces in closed manifolds. In this talk, I will discuss some recent results in the min-max theory of constant mean curvature hypersurfaces. In particular, I will discuss an extension of the CMC min-max theory to certain non-compact manifolds. I will also discuss joint work with Xin Zhou on min-max theory with a volume constraint.
Monday 10/9/23 (in-person)
Speaker: Paul Minter (Princeton)
Title: Area minimising currents: singular set rectifiability and tangent cone uniqueness
Abstract: One of the great achievements of geometric measure theory is Almgren’s work (1983) on area minimising n-currents, showing that such objects are smoothly embedded away from a (generally unavoidable) singular set of dimension at most n-2.
In this talk, I will discuss recent work with Camillo De Lellis (IAS) and Anna Skorobogatova (Princeton) showing that the singular set is countably (n-2)-rectifiable, with unique tangent cones at almost every point.
Thursday 10/19/23 (Zoom)
Speaker: Daniel Stern (Cornell University)
Title: New minimal surfaces in S^3 and B^3 via eigenvalue optimization
Abstract: I'll describe joint work with Karpukhin, Kusner, and McGrath, in which we produce many new families of closed minimal surfaces in S^3 and free boundary minimal surfaces in B^3 via constrained optimization problems for Laplace eigenvalues on surfaces. Along the way, I'll highlight some new techniques for establishing existence of extremal metrics in more general situations, and point to some key open problems in this area.
Thursday 10/26/23 (Zoom)
Speaker: Shuli Chen (Stanford)
Title: A Generalization of the Geroch Conjecture with Arbitrary Ends
Abstract: The Geroch conjecture (proven by Schoen-Yau and Gromov-Lawson) says that the torus T^n does not admit a metric of positive scalar curvature. In this talk, I will explain how to generalize it to some non-compact settings using μ-bubbles. In particular, I will talk about why the connected sum of a Schoen-Yau-Schick n-manifold with an arbitrary n-manifold does not admit a complete metric of positive scalar curvature for n <=7; this generalizes work of Chodosh and Li. I will also discuss about how to generalize Brendle-Hirsch-Johne’s non-existence result for metrics of positive m-intermediate curvature on N^n = M^{n-m} x T^m to to manifolds with arbitrary ends for n <= 7 and certain m. Here, m-intermediate curvature is a new notion of curvature interpolating between Ricci and scalar curvature.
Monday 11/6/23 (in-person)
Speaker: Xiaochun Rong (Rutgers)
Title: Quantitative Maximal Diameter Rigidity of Positive Ricci Curvature
Abstract: In Riemannian geometry, the Cheng's maximal diameter rigidity theorem says that if a complete n-manifold M of Ricci curvature is at least (n-1), has the maximal diameter pi, then M is isometric to the unit n-sphere.
In this talk, after a brief reviewing previous work asserting M a sphere (homeomorphic or diffeomorphic) when diam(M) is almost pi with various additional restrictions, we will discuss our recent work on a quantitative maximal diameter rigidity: if diam}(M) is almost pi and the Riemannian universal cover of every metric ball in M of a definite radius satisfies a Reifenberg condition, then M is diffeomorphic and bi-Holder close to the unit n-sphere. This is a joint work with Tianyin Ren.
Thursday 11/9/23 (Zoom)
Speaker: Antoine Song (Caltech)
Title: Stability of the volume entropy inequality
Abstract: A fundamental result about the dynamics and geometry of hyperbolic manifolds is Besson-Courtois-Gallot's entropy inequality. The volume entropy of a Riemannian metric measures the growth rate of geodesic balls in the universal cover. The result says that given a closed hyperbolic manifold (M,g_0), the metric with same volume as g_0 which minimizes the volume entropy is the hyperbolic one. We will discuss the corresponding stability problem: if a volume normalized metric g has entropy close to that of the hyperbolic metric g_0, is it true that g is "close" to g_0? We will give a positive answer, whose proof involves an area-minimization problem of independent interest. We will also sketch some intriguing examples suggesting that our answer may be optimal.
Monday 11/13/23 (in-person)
Speaker: Rafael Montezuma (Universidade Federal do Ceará)
Title: THE WIDTH OF EMBEDDED CIRCLES
Abstract: The min-max theory for the area functional is a Morse theory on the space of surfaces contained in a three-dimensional Riemannian manifold. The theory experienced remarkable developments and found deep applications in differential geometry. The min-max widths are invariants that naturally emerge from this theory as special critical values of the area. It is very interesting to compare these numbers to other geometric quantities, such as the volume and curvature bounds of the ambient manifold.
In this talk, we briefly discuss some classification results of Riemannian manifolds involving the min-max widths of the area functional. The main topic of this talk is a new notion of width of circles embedded in Riemannian manifolds, generalizing the classical one for plane curves. When this width coincides with the diameter of the curve with respect to the distance function of the ambient space, the curve has properties similar to those of plane curves of constant width. We also found a connection between curves whose width equals half of their length and Riemannian fillings of the circle.
Thursday 11/16/23 (Zoom)
Speaker: Daniele Semola (FIM-ETH Zürich)
Title: Ricci Curvature, Fundamental Groups, and the Milnor Conjecture
Abstract: It was conjectured by John Milnor in 1968 that the fundamental group of a complete Riemannian manifold with nonnegative Ricci Curvature is finitely generated. I will present joint work with Elia Bruè and Aaron Naber where we construct a complete 7-dimensional Riemannian manifold with nonnegative Ricci Curvature and infinitely generated fundamental group, thus providing a counterexample to the Milnor conjecture.
Monday 11/20/23 (in-person)
Speaker: Anna Skorobogatova (Princeton)
Title: Structure of flat singularities for mod(p) area-minimizing surfaces
Abstract: One possible framework in which to study the Plateau problem is by using currents with mod(p) coefficients, for a fixed integer p. This setting allows for minimizing hypersurfaces to exhibit codimension 1 singularities like triple junctions, and has close connections to the known regularity theory for stable minimal hypersurfaces. Early works on the regularity theory in this framework date back to Federer & L. Simon (p=2), J. Taylor (p=3) and White (p=4), while for general p, the recent works of De Lellis-Hirsch-Marchese-Stuvard-Spolaor and Minter-Wickramasekera lead to a complete structural characterization of the codimension 1 part of the singular set for minimizing hypersurfaces.
In this talk, I will discuss the presence of flat singularities in mod(p) area-minimizing hypersurfaces when p is even, and a recent result establishing (m-2)-rectifiability for such singularities in m-dimensional mod(p) area-minimizing hypersurfaces. I will also discuss joint work in progress with Camillo De Lellis and Paul Minter towards establishing a structural result on the interior singular set in higher codimension.
Monday 11/27/23 (in-person)
Speaker: Chao Li (NYU Courant)
Title: Stable minimal hypersurfaces in R^4
Abstract: I will discuss the Bernstein problem for minimal hypersurfaces in R^4: a complete, two-sided, stable minimal immersion in R^4 is flat. The proofs are motivated by several ideas originated in the study of positive scalar curvature.
Thursday 11/30/23 (Zoom)
Speaker: Aaron Kennon (UC Santa Barbara)
Title: Identifying Promising Initial Data for Geometric Flows of G2-Structures
Abstract: A primary goal motivating the study of geometric flows of G2-structures is to better understand which 7-manifolds admit certain types of these metrics. Of particular interest are the cases of G2-holonomy metrics and nearly-parallel G2-structures, both of which are intricately related to broader themes in differential geometry. I will survey what is known for specific promising flows of G2-structures, what would be desirable to prove, and the relevance of some of my work for determining when these flows behave well analytically and flow towards these desired geometric objects.
Monday 12/4/23 (in-person)
Speaker: Daoyuan Han (Lehigh)
Title: Calabi’s Hyperkähler Manifold and Singularity Models of U(3)-Invariant Ricci Flow
Abstract: It was proved by Alexander Appleton in 2022 that the Eguchi-Hanson space can occur as a singularity model of Ricci flow in dimension 4. I will present a generalization of this result of occurrence of singularity modeled on the Calabi's hyperkahler space.