In the 2024 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 4:00 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 09/02/24 (in-person, 4:00 - 5:00, CU 215)

Speaker: Bin Guo (Rutgers at Newark)

Title: Uniform estimates for complex Monge-Ampere and fully nonlinear equations

Abstract: Uniform estimates for complex Monge-Ampere equations have been extensively studied, ever since Yau’s resolution of the Calabi conjecture. Subsequent developments have led to many geometric applications to many other fields, but all relied on the pluripotential theory from complex analysis. We will discuss a new PDE-based method of obtaining sharp uniform $L^\infty$ estimates for complex Monge-Ampere (MA) and fully nonlinear PDEs, without the pluripotential theory. This new method extends more generally to other interesting geometric estimates for MA and Hessian equations. This is based on joint works with D.H. Phong and F. Tong.

Monday 09/09/24 (in-person, 2:00 - 3:00, Maginnes Hall 104) Please be aware of the special location and time!

Speaker: Pei-Ken Hung (UIUC)

Title: On the Horowitz-Myers conjecture

Abstract: In 1998 Horowitz and Myers constructed a static manifold with a toroidal infinity, later referred as the Horowitz-Myers geon. They further conjectured that the Horowitz-Myers geon minimizes the mass among manifolds with same asymptotics and the scalar curvature lower bounds. In this talk, I will discuss a joint work with Simon Brendle, in which we confirm this conjecture.

Monday 09/16/24 (in-person, 3:00 - 4:00, CU 239) Please be aware of the special location and time!

Speaker: Tamás Darvas (University of Maryland)

Title: The trace operator of quasi-plurisubharmonic functions on compact K\"ahler manifolds

Abstract: We introduce the trace operator for quasi-plurisubharmonic functions on compact Kahler manifolds, allowing us to study the singularities of such functions along submanifolds where their generic Lelong numbers vanish. Using this construction we obtain novel Ohsawa-Takegoshi extension theorems and give applications to restricted volumes of big line bundles (joint work with Mingchen Xia).

Thursday 09/19/24 (online, 4:30 - 5:30, Zoom)

Speaker: Daniela Di Donato (University of Pavia)

Title: Rectifiability in Carnot groups

Abstract: Intrinsic regular surfaces in Carnot groups play the same role as C^1 surfaces in Euclidean spaces. As in Euclidean spaces, intrinsic regular surfaces can be locally defined in different ways: e.g. as non critical level sets or as continuously intrinsic differentiable graphs. The equivalence of these natural definitions is the problem that we are studying. Precisely our aim is to generalize some results proved by Ambrosio, Serra Cassano, Vittone valid in Heisenberg groups to the more general setting of Carnot groups. This is joint work with Antonelli, Don and Le Donne

Monday 09/23/24 (in-person, 4:00 - 5:00, CU 215)

Speaker: Xinrui Zhao (MIT)

Title: Closed mean curvature flow with asymptotically conical singularities

Abstract: In this talk, we will talk about the proof of that for any asymptotically conical self-shrinker, there exists an embedded closed hypersurface such that the mean curvature flow starting from it develops a singularity modeled on the given shrinker. As a corollary, it implies the existence of fattening level set flows starting from smooth embedded closed hypersurfaces. This addresses a question posed by Evans-Spruck and De Giorgi. The talk is based on the joint work with Tang-Kai Lee.

Monday 09/30/24 (in-person, 4:00 - 5:00, CU 215)

Speaker: Theodora Bourni (University of Tennessee Knoxville)

Title: Constructing solutions to curve shortening and related flows

Abstract: We will discuss the construction of certain interesting solutions to  curve shortening and related flows. Some of these lead to classification results for ancient solutions.

Monday 10/7/24 (in-person, 4:00 - 5:00, CU 215)

Speaker: Guangbo Xu (Rutgers University)

Title: Transversality on Orbifolds and Counting Holomorphic Curves

Abstract: (Joint with Shaoyun Bai) The notion of transversality is of fundamental importance in many areas of geometry and topology. However, it usually conflicts with symmetry: symmetric objects are not in "general position". As a result, one may not be able to achieve transversality on orbifolds. This feature is often accompanied with another feature of orbifolds: Poincar\'e duality only holds over rational numbers but not integers. 

Inspired by an old proposal of Fukaya-Ono in symplectic topology, we define a new notion of transversality for sections of complex orbifold vector bundles. For sections satisfying this transversality condition, its zero locus is a space stratified by smooth manifolds and each stratum defines a Euler cycle in integer coefficients. We also obtained several applications in symplectic topology, including the definition of integer-valued Gromov-Witten invariants and proved the Arnold conjecture over integers.

Thursday 10/17/24 (online, 4:30 - 5:30)

Speaker: Jingrui Cheng (Stony Brook University)

Title: Interior W^{2,p} estimates for complex Monge-Ampere equations

Abstract: The classical estimate by Caffarelli shows that a strictly convex solution to the real Monge-Ampere equations has W^{2,p} regularity if the right hand side is close to a constant. We partially generalize this result to the complex version, when the underlying solution is close to a smooth strictly plurisubharmonic function. The additional assumption we impose is related to the lack of Pogorelov type estimate in the complex case. The talk is based on joint work with Yulun Xu

Monday 10/28/24 (in-person, 4:00 - 5:00, CU 215)

Speaker: Tang-Kai Lee (MIT)

Title: Non-uniqueness of mean curvature flow

Abstract: The smooth mean curvature flow often develops singularities, making weak solutions essential for extending the flow beyond singular times, with applications for geometry and topology. Among various weak formulations, the level set flow method is notable for ensuring long-time existence and uniqueness. However, this comes at the cost of potential fattening, which reflects genuine non-uniqueness of the flow after singular times. In the talk of Xinrui Zhao, it was presented that even for flows starting from smooth, embedded, closed initial data, such non-uniqueness can occur. The examples we constructed extend to higher dimensions, complementing the surface examples obtained by Ilmanen and White. Thus, we can't expect genuine uniqueness in general. Addressing this non-uniqueness issue is a difficult problem. With Alec Payne, we establish a generalized avoidance principle. We prove that level set flows satisfy this principle in the absence of non-uniqueness.

Monday 11/4/24 (in-person, 4:00 - 5:00, CU 215)

Speaker: Xuan Yao (Cornell)

Title: Scalar curvature comparison of compact manifolds

Abstract: We prove a rigidity result on comparison of scalar curvature and scaled mean curvature on the boundary for compact domain in Euclidean space, which is a generalization of Gromov's conjecture on polyhedra and a parallel of Shi-Tam's results.

Thursday 11/7/24 (online, 4:30 - 5:30)

Speaker: Valentino Tosatti (NYU Courant Institute)

Title: Immortal solutions of the Kähler-Ricci flow

Abstract: I will discuss the problem of understanding the long-time behavior of Ricci flow on a compact Kähler manifold, assuming that a solution exists for all positive time. Inspired by an analogy with the minimal model program in algebraic geometry, Song and Tian posed several conjectures which describe this behavior. I will report on recent work (joint with Hein and Lee) which confirms these conjectures.

Monday 11/11/24 (in-person, 4:00 - 5:00, CU 215)

Speaker: Antonio De Rosa (Bocconi University)

Title: Min-max construction of anisotropic minimal hypersurfaces

Abstract: We prove the existence of closed optimally regular hypersurfaces with constant anisotropic mean curvature  with respect to elliptic integrands in closed n-dimensional Riemannian manifolds. This proves a conjecture posed by Allard in 1983. The talk is based on joint work with Guido De Philippis and Yangyang Li.

Monday 11/18/24 (in-person, 4:00 - 5:00, CU 215)

Speaker: Michael Schultz (Virginia Tech)

Title: Holomorphic Conformal Geometry is Special

Abstract: In algebraic or complex geometry one often encounters families of varieties or complex manifolds depending on many parameters. The set of parameters for which the associated object is smooth fits together to form its own space, known as a parameter space or moduli space, which is often an analytic space in its own right. Elements of the geometry and topology of the generic smooth member of the family in turn reflects the local geometry of the parameter or moduli space. It is a fundamental problem to understand the geometry of such spaces. In this talk, we will examine how this story plays out for certain families of algebraic K3 surfaces. It will be shown that moduli spaces of K3 surfaces carry a rich differential geometric structure known as a holomorphic conformal structure. Moreover, crucial aspects of the geometry of the associated K3 surfaces necessitate this structure is integrable, or locally conformally flat. This geometric structure parallels a rich differential geometric structure on moduli spaces of Calabi-Yau threefolds known as "special geometry". In this way, holomorphic conformal geometry is special. Based on joint work with Andreas Malmendier, https://arxiv.org/abs/2401.10950.

Monday 11/25/24 (in-person, 4:00 - 5:00, CU 215)

Speaker: Hunter Stufflebeam (UPenn)

Title: Stability Theorems for Spheres

Abstract: In this talk, I will present several stability theorems for the widths of spheres. Some of this is joint with Davi Maximo, and some is joint with Paul Sweeney.

Monday 12/2/24 (in-person, 4:00 - 5:00, CU 215)

Speaker: Jingze Zhu (MIT)

Title: Arnold-Thom conjecture for the arrival time of surfaces

Abstract: Following Łojasiewicz's uniqueness theorem and Thom's gradient conjecture, Arnold proposed a stronger version about the existence of limit tangents of gradient flow lines for analytic functions. In this talk, I will explain the proof of Łojasiewicz's theorem and Arnold's conjecture in the context of arrival time functions of mean convex mean curvature flows of surfaces. This is joint work with Tang-Kai Lee.

Thursday 12/5/24 (online, 4:30 - 5:30)

Speaker: Lucas Ambrozio (Instituto Nacional de Matemática Pura e Aplicada - IMPA)

Title: Why Zoll metrics?

Abstract: In the beginning of the nineteen hundreds, Otto Zoll discovered smooth, rotationally symmetric spheres in the Euclidean space, not the round ones, which have the remarkable property that all of their nontrivial geodesics are closed, embedded and have the same length. 

Besides the obvious interest in such curious metrics from a dynamical and geometric point of view, it has been observed more recently that Zoll metrics are also central objects from variational points of view. This leads to a curious possibility: "Zoll like" objects may be found in other variational theories, and may play a central role. 

I will review these developments and discuss some contributions towards a theory of "Zoll-like" metrics in the context of minimal submanifolds theory, which include classification and rigidity results. The talk will be based on joint works with F. Marques, A. Neves and R. Montezuma.