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kyky¿ªÔª1383¹ÙÍø: Schedule

• Wednesday, February 21, 9:00-10:00, Zoom link (ID: 828 0478 1311, Code: 635902)
  Gang Liu (East China Normal University) - Complete Kahler manifolds with nonnegative Ricci curvature - Abstract

  We discuss some recent results on complete Kahler manifolds with nonnegative Ricci curvature:

  1. the invariance of average of scalar curvature at infinity

  2. boundedness of integral of higher power of Ric

  3. A rigidity result for Kahler Ricci flat metric

  
  

• Wednesday, February 28, 9:00-10:00, Zoom link (ID: 811 2485 6853, Code: 301398)
  Yuan Yuan (Syracuse University) - Function theory on quotient domains - Abstract

  Let $f: D \rightarrow \Omega$ be a proper holomorphic map between two bounded domains. $\Omega$ is a quotient domain of $D$ if there exists a finite group $G$ such that $\Omega = X / G$. The function theory on $\Omega$ can be studied by transforming to $D$. In this way, we may study the Bergman projection, the Szeg\H{o} projection and the $\bar\partial$ problem on $\Omega$. In this talk, we will mainly discuss the recent work on the Szeg\H{o} projection.

  We will introduce a boundary value problem for holomorphic functions on $D$ which enables us to define the Hardy space on $\Omega$ and derive a Bell type transformation formula for the Szego projection on $\Omega$. This definition of the Hardy space is different from the existing one in the literature and is a natural generalization of that on the planar domain considered by Lanzani-Stein. When $D$ is the unit ball or the polydisc, we provide a sufficient condition for the solution to the boundary value problem. We further obtain the sharp $L^p$ estimates for Szego projections on some quotient domains in $\mathbb{C}^2$.

  
  

• Wednesday, March 6, 9:00-10:00, Zoom link (ID: 834 8910 4525, Code: 783607)
  Jingbo Wan (Columbia University) - Rigidity of Area Non-Increasing Maps - Abstract

  In this talk, we discuss the approach of Mean Curvature Flow to demonstrate that area non-increasing maps between certain positively curved closed manifolds are rigid. Specifically, this implies that an area non-increasing self-map of $CP^n, n \geq 2$, is either homotopically trivial or is an isometry, answering a question by Tsai-Tsui-Wang. Moreover, by coupling the Mean Curvature Flow for the graph of a map with Ricci Flows for the domain and the target, we can also study the rigidity of area non-increasing maps from closed manifolds with positive 1-isotropic curvature (PIC1) to closed Einstein manifolds, where Prof. Brendle's PIC1 Sphere Theorem is applied. The key to studying the rigidity of area non-increasing maps under various curvature conditions lies in the application of the Strong Maximum Principle along the MCF/MCF-RF. We will focus our attention on one particular case to illustrate the SMP argument. This is a joint work with Professor Man-Chun Lee and Professor Luen-Fai Tam from CUHK.

  
  

• Thursday (Special day), March 14, 9:00-10:00, Zoom link (ID: 842 9937 0734, Code: 296730)
  Artem Pulemotov (The University of Queensland) - Palais--Smale sequences for the prescribed Ricci curvature functional - Abstract

  On homogeneous spaces, solutions to the prescribed Ricci curvature equation coincide with the critical points of the scalar curvature functional subject to a constraint. We provide a complete description of Palais--Smale sequences for this functional. As an application, we obtain new existence results for the prescribed Ricci curvature equation, which enables us to observe previously unseen phenomena. Joint work with Wolfgang Ziller (University of Pennsylvania).

  
  

• Wednesday, March 20, 9:00-10:00, Zoom link (ID: 867 7258 6499, Code: 188353)
  Bo Zhu (Texas A&M University) - Metric invariants of manifolds and Llarull rigidity theorem on four-manifolds - Abstract

  This talk will focus on some metric invariants of Riemannian manifolds, which were introduced by Gromov in the 1980s. I will first introduce several metric invariants of Riemannian manifolds and then explore its connection with curvature on manifolds. After that, I will talk about our recent progress on Llarull rigidity theorem on four-dimensional manifolds.

  
  

• Wednesday, March 27, 9:00-10:00, Zoom link (ID: 836 0795 5992, Code: 960544)
  Antoine Song (California Institute of Technology) - Minimal surfaces in spheres from random permutations - Abstract

  The main result I will discuss states that there exists a sequence of closed minimal surfaces in high-dimensional Euclidean spheres which converge (around most points) to the hyperbolic plane. The proof is based on a surprising connection between minimal surfaces in spheres, random permutations and convergence of unitary representations.

  
  

• Wednesday, April 3, 9:00-10:00, Zoom link (ID: 861 1625 0516, Code: 824116)
  Tran-Trung Nghiem (University of Montpellier) - Calabi-Yau metrics on complex symmetric spaces - Abstract

  On complex symmetric spaces of rank one, Stenzel constructed explicit examples of Calabi-Yau metrics with smooth cross-section asymptotic cone. A new feature in higher rank is that the possible candidates for asymptotic cones generally have singular cross-section.

  After an introduction and survey of known results, I will present an existence theorem of Calabi-Yau metrics on symmetric spaces of rank two with asymptotic cone having singular cross-section. This provides new examples of Calabi-Yau manifolds with irregular asymptotic cone besides the only known example of Conlon-Hein, and covers the rank two symmetric spaces left by Biquard-Delcroix. The metrics on the decomposable cases turn out to be asymptotically a product of two Stenzel cones. If time allows, I will also try to explain why some special symmetric spaces of rank two don't have any invariant Calabi-Yau metrics with a given asymptotic cone, using an obstruction on the valuation induced by such metric if exists.

  
  

• Wednesday, April 10, 9:00-10:00, Zoom link (ID: 859 0910 7027, Code: 438807)
  Erik Hupp (Northwestern University) - Lower Ricci Bounds and Nonexistence of Manifold Structure - Abstract

  By Gromov compactness, any sequence of complete Riemannian n-manifolds with uniform lower Ricci bounds has a subsequence (pointed Gromov-Hausdorff) converging to a limit metric space. How close is this limit to being a manifold itself? A cornerstone result of Cheeger-Colding gives an answer if one also assumes that the limit is volume non-collapsed: it is a topological manifold on an open dense set whose complement has dimension at most n - 2. This talk will describe a family of counterexamples to the corresponding statement in the collapsed setting. These limit spaces can be constructed to approximate any given complete (smooth) Riemannian 4-manifold with lower Ricci bounds, but have the property that no open set is homeomorphic to R^k, for any k. Everything discussed is joint work with Aaron Naber and Kai-Hsiang Wang.

  
  

• Wednesday, April 17, 9:00-10:00, Zoom link (ID: 840 9110 8991, Code: 004748)
  Jian Wang (Stony Brook University) - Positive mass theorem for asymptotically locally flat 4-manifolds with $\mathbb{S}^1$-symmetries - Abstract

  The ADM mass, a crucial global geometric invariant, intricately relates with scalar curvature in the different setting. I will explain the relation between asymptotically locally flat (ALF) 4-manifolds and their ADM mass. Specifically, I will talk about how the topology at infinity influences the ADM mass within the ALF setting.

  
  

• Wednesday, April 24, 9:00-10:00, Zoom link (ID: 811 6179 4210, Code: 985997)
  Kai Xu (Duke University) - Inverse mean curvature flow with outer obstacle - Abstract

  The weak inverse mean curvature flow, initially introduced by Huisken and Ilmanen, has been a powerful tool in approaching scalar curvature problems. In recent years, on the other hand, the analytic and measure-theoretic structure of the inverse mean curvature flow itself has drawn growing attention. In this talk, I will introduce a new theory for the (weak) inverse mean curvature flow inside bounded domains. In our setting, the boundary of the domain plays the role of an outer obstacle, and the hypersurfaces in the flow stick tangentially to the boundary upon contact. We will discuss the relevant motivations for considering such a problem, as well as the analytic/geometric behaviors of the solutions. Then we will explain an existence and C^{1,\alpha} regularity theorem for smooth boundary, and the ideas involved.

  
  

• Wednesday, May 8, 9:00-10:00, Zoom link (ID: 817 5226 6172, Code: 303016)
  Yanir A. Rubinstein (University of Maryland) - Tian's stabilization problem - algebraic meets complex & convex geometry - Abstract

  Coercivity thresholds are a central theme in geometry. They appear classically in the Yamabe problem (constant scalar curvature in a conformal class), in the Nirenberg problem (prescribed curvature on the 2-sphere), and in numerous problems on determining best constants in Sobolev embeddings and related functionals inequalities. In 1980's Aubin and Tian introduced the first such thresholds in the Kahler-Einstein problem and their study has been a central and still very active field. In 1988 Tian observed that these thresholds have quantum versions and he posed the so-called Stabilization Problem: do the equivariant quantum thresholds become constant (and hence equal to the classical thresholds)? Cheltsov conjectured that these invariants coincide with the algbero-geometric log canonical thresholds (lct), and this was verified by Demailly (2008). The best result so far has been Birkar's theorem (2019) that shows that the quantum lcts are constant along a subsequence in the absence of group actions. In joint work with Chenzi Jin (PhD student at UMD) we offer a new approach and solve Tian's problem in the toric case. Surprisingly, the equivariant lcts are constant already from the first quantum level. For more general Grassmannian lcts we offer counterexamples to stabilization and determine when it holds. The key new ideas are understanding the effect of finite group actions on these invariants, and relating these thresholds to support and gauge functions from convex geometry. Time permitting I will discuss extensions and generalizations to other invariants, e.g., the Fujita-Odaka stability thresholds.

  
  

• Wednesday, May 15, 9:00-10:00, Zoom link (ID: 879 2483 3067, Code: 702831)
  Bin Wang (The Chinese University of Hong Kong) - Global curvature estimates for Hessian type equations - Abstract

  We derive global curvature estimates for closed strictly starshaped (n-2)-convex hypersurfaces in warped product manifolds, satisfying the prescribed (n-2)-curvature equation with a general right-hand side. The proof is inspired by the recent breakthrough of [Guan-Ren-Wang CPAM 2015] and it can be readily adapted to establish curvature estimates for semi-convex and (k+1)-convex solutions to the general k-curvature equations; it can also be used to prove the same estimates for prescribed curvature measure type equations. If time permits, we may also discuss the estimates for another class of Hessian type equations.

  
  

• Wednesday, May 22, 9:00-10:00, Zoom link (ID: 895 7783 1648, Code: 781326)
  Yaxiong Liu (University of Maryland) - A Le Potier-type isomorphism twisted with multiplier submodule sheaves - Abstract

  Multiplier ideal sheaves with line bundles have played an important role and been well-studied in complex geometry. For general vector bundles, we consider the L^2 multiplier submodule sheaf associated to a singular Hermitian metric, defined by M. A. de Cataldo. we obtain a Le Potier-type isomorphism theorem twisted with multiplier submodule sheaves, which relates a holomorphic vector bundle endowed with a strongly Nakano semipositive singular Hermitian metric to the tautological line bundle with the induced metric. As applications, we obtain a Kollár-type injectivity theorem, a Nadel-type vanishing theorem, and a singular holomorphic Morse inequality for holomorphic vector bundles and so on.

  
  

• Wednesday, May 29, 10:00-11:00 (Special time), Zoom link (ID: 878 6963 6072, Code: 665065)
  Ramesh Mete (Indian Institute of Science) - Deformed Hermitian Yang Mills equation in the unstable case - Abstract

  The dHYM equation is a special type of complex Hessian equations which has connection to mirror symmetry in string theory. Recently, in the smooth setting it is shown that there exists a smooth solution of the "super-critical" dHYM equation on compact K\"ahler manifolds if and only if certain Nakai-Moishezon type criterion holds. In this talk, we will focus when the NM-type criterion fails - which is the so-called "unstable" case. We will show the existence and uniqueness of solutions of the "weak" dHYM equation, where the wedge product is replaced by the non-pluripolar product. We will also discuss the convergence of the (dHYM) cotangent flow in the unstable case. Based on a joint work with Prof. Ved Datar (IISc, Bengaluru) and Prof. Jian Song (Rutgers University).

  
  

kyky¿ªÔª1383¹ÙÍø: Schedule for the past years:

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