Geometric Analysis Seminar

In Fall of 2017, we will meet on Thursdays, at 3pm in a room to be announced. Room 6496. The organizers of this seminar are Zeno Huang, Neil Katz and Bianca Santoro. Please email Bianca at bsantoro(NoSpamPlease)ccny.cuny.edu to schedule a guest speaker. The CUNY Graduate Center is located at 365 Fifth Avenue at 34th Street, diagonally across the street from the Empire State Building, just two blocks from Penn Station (NYC).

Fall 2017:

• Thursday, September 14: Yiming Zhao (NYU/St. Johns University)
  Minkowski-type problems in convex geometry
  Abstract: An overview of Minkowski-type problems will be presented. Minkowski-type problems are related to Monge-Ampere equations. Particular focus will be on the dual Minkowski problem, which is the characterization problem for a family of new geometric measures, called dual curvature measures, introduced in [Huang, Lutwak, Yang, Zhang; Acta 2016]. These are the "duals" of Federer's curvature measure. A solution to the dual Minkowski problem in the symmetric case will be demonstrated. The solution is strongly connected to measure concentration phenomenon.
  
  
• Thursday, September 21: No Seminar
  Holiday
  
  
• Thursday, September 28: Klaus Kroencke (University of Hamburg)
  Stability of ALE Ricci-flat manifolds under Ricci-flow
  Abstract: We prove that if an ALE Ricci-flat manifold (M,g) is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to g. By adapting Tian's approach in the closed case, we show that integrability holds for ALE Calabi-Yau manifolds which implies that they are dynamically stable. This is joint work with Alix Deruelle
  
  
• Thursday, October 5: Xiaochun Rong (Rutgers University)
  Collapsed manifolds with Ricci curvature and local rewinding volume bounded below
  Abstract: I'll report some recent progress in the study of manifolds in the title. Given a number rho>0 and a point x in M, a complete n-manifold, the local rewinding volume of the rho-ball at x, B_rho(x), is the volume of the \rho-ball at \tilde xon the (incomplete) Riemannian universal cover of B_rho(x). Part of this work is joint with Hongzhi Huang, Zuohai Jiang and Shicheng Xu of Capital Normal University.
  
  
• Thursday, October 12: Hengyu Zhou (Sun Yat-Sen University)
  Mean Curvature Flows in Warped Product Manifolds
  Abstract: In this talk we discuss the mean curvature flow of starshaped hypersurface in warped product manifolds admitting a totally geodesic slice Sigma. With some natural conditions on the warping function and Ricci curvature, long time existences and convergences to Sigma are established. This is a joint work with Zhou Zhang (Sydney) and Zheng Huang (CUNY).
  
  
• Thursday, October 19: CUNY symposium on nonlinear problems in geometry
  TBA
  
  
• Thursday, October 26: Valentino Tosatti (Northwestern University)
  TBA
  Abstract: TBA
  
  
• Thursday, November 2: Renato Bettiol (University of Pennsylvania)
  TBA
  Abstract: TBA
  
  
• Thursday, November 9: TBA
  TBA
  Abstract: TBA
  
  
• Thursday, November 16: Xinliang An (University of Toronto)
  On Singularity Formation in General Relativity
  Abstract: In the process of gravitational collapse, singularities may form, which are either covered by trapped surfaces (black holes) or visible to faraway observers (naked singularities). In this talk, with three different approaches coming from hyperbolic PDE, quasilinear elliptic PDE and dynamical system, I will provide answers for four physical questions: i) Can black holes form dynamically in vacuum? ii) To form a black hole, what is the least size of initial data? iii) Can we find the boundary of a black hole region? Can we show that a black hole region is emerging from a point? iv) For Einstein vacuum equations, could singularities other than black hole type form in gravitational collapse?
  
  
• Thursday, November 23: No seminar
  Thanksgiving
  
  
• Thursday, November 30: Nan Li (New York City Tech)
  TBA
  Abstract: TBA
  
  
• Thursday, December 7: TBA
  TBA
  Abstract: TBA
  
  

  Spring 2017:

  • Thursday, February 16: Yusheng Wang (Beijing Normal University and Rutgers University)
    Curvature>=1, diameter>=\pi/2 and rigidity (in Alexandrov geometry)
    Abstract: In this talk, we will review some classical and recent rigidity results in Alexandrov geometry with curvature >=1 (including Riemannian geometry with sectional curvature >=1) brought by distance>=\pi/2 between points or convex subsets.
    
    
  • Thursday, February 23: Jorge Basilio (Graduate Center, CUNY)
    Sequences of three-dimensional manifolds with positive scalar curvature
    Abstract: Here we explore to what extent one may hope to preserve geometric properties of three dimensional manifolds with lower scalar curvature bounds under Gromov-Hausdorff and Intrinsic Flat limits. We introduce a new construction of three dimensional manifolds with positive scalar curvature called sewing. We produce sequences of such manifolds which converge to spaces demonstrating that almost rigidity theorems for manifolds with positive or nonnegative scalar curvature fail to hold for these limits including the Scalar Torus Rigidity Theorem and the rigidity part of the Positive Mass Theorem. Since the notion of nonnegative scalar curvature is not strong enough to persist alone, we propose that one pair a lower scalar curvature bound with a lower bound on the area of a closed minimal surface when taking sequences as this will prevent the sewing of manifolds and possibly also the existence of counter examples. This is joint work with Christina Sormani.
    
    
  • Thursday, March 16: Dan Ketover (Princeton University)
    Variational constructions of minimal surfaces
    Abstract: I will describe how variational methods can be applied to produce many new examples of minimal surfaces with various symmetries. IÕll focus on the construction of free boundary minimal surfaces resembling a desingularization of the critical catenoid and flat disk, which were first conjectured by Fraser-Schoen.
    
    
  • Thursday, March 23: Jacob Bernstein (Johns Hopkins University)
    Surfaces of Low Entropy
    Abstract: Following Colding and Minicozzi, we consider the entropy of (hyper)-surfaces in Euclidean space. This is a numerical measure of the geometric complexity of the surface. In addition, this quantity is intimately tied to to the singularity formation of the mean curvature flow which is a natural geometric heat flow of submanifolds. In the talk, I will discuss several results that show that closed surfaces for which the entropy is small are simple in various senses. This is all joint work with L. Wang.
    
    
  • Thursday, March 30: Florentin Munch (Potsdam University)
    Rigidity properties of the hypercube via discrete Ricci curvature
    Abstract: We give rigidity results for discrete Bonnet-Myers diameter bound and Lichnerowicz eigenvalue estimate. Both inequalities are sharp if and only if the underlying graph is a hypercube. The proofs use well-known semigoup methods as well as new direct methods which translate curvature to combinatorial properties. The results can be seen as first known discrete analogues of Cheng's and Obata's rigidity theorems.
    
    
  • Thursday, April 6: Ronan Conlon (Florida International University)
    New examples of gradient expanding Kahler-Ricci solitons
    Abstract: A complete Kahler metric g on a Kahler manifold M is a gradient expanding Kahler-Ricci soliton if there exists a smooth real-valued function f:M -> R with $\nabla^{g}f$ holomorphic such that Ric(g)-Hess(f)+g=0. I will present new examples of such metrics on the total space of certain holomorphic vector bundles. This is joint work with Alix Deruelle (Universite Paris-Sud).
    
    
  • Thursday, April 27: Liming Sun (Rutgers University)
    Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant
    Abstract: We study one of Yamabe problems on compact manifolds with boundary. For negative Yamabe type manifolds, we are able to prescribe scalar curvature and boundary mean curvature. Precisely, given any negative smooth functions f in M and h on the boundary $\partial M$, there exists a unique conformal metric of g_0 such that its scalar curvature equals f and mean curvature curvature equals h. One family of flow with dynamic boundary mean curvature is constructed to prove this result. At the end, I will mention some ongoing results on positive type manifold using flow approach.
    
    
  • Thursday, May 4: TBA
    TBA
    Abstract: TBA
    
    
  • Thursday, May 11: Jiayin Pan (Rutgers University)
    A proof of Milnor conjecture in dimension 3
    Abstract: In this talk, we present a proof of Milnor conjecture in dimension 3 based on the Cheeger-Colding theory on limit spaces of manifolds with Ricci curvature bounded below.
    
    

  Fall 2016:

  • Tuesday, September 6: Irina Markina (University of Bergen, Norway)
    Pseudo H-type Lie algebras: research in progress
    Abstract: The talk is devoted to new results in the study of pseudo H-type Lie algebras. Pseudo H-type Lie algebras is a natural generalisation of nowadays classical Heisenberg-type algebras, introduced by A. Kaplan in 1980 in order to study hypoelliptic operators. The pseudo H-type Lie algebras admit rational structural constants, that lead to the existence of lattices on the corresponding Lie groups according to the Malcev theorem. The pseudo H-type Lie groups give rise to a chain of examples of nilmanifolds that are isospectral but non-diffeomorphic. We also will discuss the groups of automorphisms of pseudo H-type algebras and the classification. Some questions of rigidity and a relation to complex simple graded Lie algebras will be explained.
    
    
  • Tuesday, September 13: Hung Tran (University of California Irvine)
    Index characterization for free boundary minimal surfaces.
    Abstract: In this talk, we derive a computation of the Morse index for a free boundary minimal sub-manifold by data from the fixed boundary problem and Steklov eigenvalues associated with Jacobi fields. As applications, we characterize the critical catenoid as the only free boundary minimal annulus in a ball with index equal to 4.
    
    
  • Tuesday, September 20: Eleonora Di Nezza (Imperial College, London)
    The space of Kaehler metrics on singular varieties
    Abstract: The geometry and topology of the space of Kaehler metrics on a compact Kaehler manifold is a classical subject, first systematically studied by Calabi in relation with the existence of extremal Kaehler metrics. Then, Mabuchi proposed a Riemannian structure on the space of Kaehler metrics under which it (formally) becomes a non-positive curved infinite dimensional space. Chen later proved that this is a metric space of non-positive curvature in the sense of Alexandrov and its metric completion was characterized only recently by Darvas. In this talk we will talk about the extension of such a theory to the setting where the compact Kaehler manifold is replaced by a compact singular normal Kaehler space. As one application we give an analytical criterion for the existence of Kaehler-Einstein metrics on certain mildly singular Fano varieties, an analogous to a criterion in the smooth case due to Darvas and Rubinstein. This is based on a joint work with Vincent Guedj.
    
    
  • Tuesday, September 27: Jose Espinar (IMPA, Brazil)
    Escobar's Type Theorems for elliptic fully nonlinear equations
    Abstract: In this talk we prove nonexistence and classification results for elliptic fully nonlinear elliptic degenerate and nondegenerate conformal equations on certain subdomains of the sphere with prescribed constant mean curvature along its boundary. Such subdomains are the hemisphere (or a geodesic ball of S^m), punctured balls and annular domains. Our results extend those of Escobar when m is greater or equal to 3, and Hang-Wang and Jimenez when m=2.
    
    
  • Tuesday, October 4: no seminar
    Holiday
    
    
  • Tuesday, October 11: no seminar
    Holiday
    
    
  • Tuesday, October 18: Otis Chodosh (Princeton University)
    A splitting theorem for scalar curvature
    Abstract: I'll discuss a minimal surface rigidity result for three-manifolds with non-negative scalar curvature which is a natural analogue of the splitting theorem. This is joint work with M. Eichmair and V. Moraru.
    
    
  • Tuesday, October 25: Anusha Krishnan (University of Pennsylvania)
    Ricci Flow and non-negative curvature on four-dimensional cohomogeneity one manifolds
    Abstract: We show that $S^4$, $S^2\times S^2$, $\C P^2$ and $\C P^2 \# \overline{\C P^2}$ admit metrics of non-negative (sectional) curvature which under the Ricci flow immediately acquire some negatively curved two-planes. Although this was previously known for compact manifolds of dimension greater than five and for non-compact manifolds, these are the first compact four-dimensional examples showing such behaviour. Our proof makes use of the fact that these four-manifolds $M$ admit cohomogeneity one actions, i.e. an isometric action by a Lie group $G$ such that the orbit space $M/G$ is one-dimensional. This talk is based on joint work with Renato G. Bettiol.
    
    
  • Tuesday, November 1: Peter Smillie (Harvard University)
    Degeneration of constant mean curvature foliations in Minkowski space
    Abstract: A natural question in general relativity is how foliations of spacetime by surfaces of constant mean curvature degenerate near the singularities of the spacetime. WeÕll look at a very special case of spacetimes covered by a domain U in Minkowski space, whose singularities therefore admit bordifications by the boundary of U. In this special case, the question is simply whether the leaves of the constant mean curvature foliation converge in the appropriate sense to the metric space at the singularity. In dimension 2+1, Belraouti, using the work of Bonsante, has proved this in the Gromov-Hausdorff sense. I will give another approach to this question inspired by work of Moncrief, with partial results.
    
    
  • Tuesday, November 8: Sylvester Eriksson-Bique (NYU)
    Establishing Poincare inequalities and notions of differentiation for Metric Spaces
    Abstract: The talk will start by discussing some background of Poincare inequalities on metric measure spaces. We will then introduce a new condition that is equivalent to admitting a Poincare inequality. This metric condition is flexible enough to have a number of applications, and we immediately obtain new classes of spaces admitting Poincare inequalities. We also observe a connection to classical analysis on Euclidean space through Orlicz-Poincare inequalities and Muckenhoupt weights. In the second half of the talk we introduce an asymptotic version of our condition, and explain how it guarantees a new type of rectifiability result in terms of spaces admitting Poincare inequalities. This result is applied to resolve, in a particular context, a conjecture of Cheeger and Kleiner on so called differentiability spaces. The talk should be understandable without particular knowledge of the applications.
    
    
  • Tuesday, November 15: Ian Adelstein (Trinity College)
    Inaudible properties of the G-invariant spectrum
    Abstract: I'll start with an introduction to the inverse spectral problem by discussing Mark Kac's famous question: Can you hear the shape of a drum? We'll see how to produce isospectral non-isometric orbit spaces using a generalization of the Sunada technique. By examining isospectral quotients of the sphere we'll see that properties like constant sectional curvature and the presence of non-orbifold singularities are inaudible to the G-invariant spectrum. This talk should be accessible to graduate students. This work is joint with Mary Sandoval.
    
    
  • Tuesday, November 22: Double Header - Ioana Suvaina (Vanderbilt University) . Usual time and place: room 3212, 2:45pm to 3:45pm.
    ALE Kahler manifolds
    Abstract: The study of asymptotically locally Euclidean Kahler manifolds had a tremendous development in the last few years. This talk presents a survey of the main results and the open problems in this area. When the manifolds support an ALE Ricci flat Kahler metric the complex surfaces and their metric structures are well understood. The remaining case to be studied is that of ALE scalar flat Kahler manifolds. In this direction, the underlying complex manifold is described. It is exhibited as a resolution of a deformation of an isolated quotient singularity. As a consequence, there exists only finitely many diffeomorphism types of minimal ALE Kahler surfaces.
    
    
  • Tuesday, November 22: Double Header - Rares Radeasconu (Vanderbilt University) . Please note the different time and location: Math Thesis Room, 4-5pm.
    Complex Manifolds and Special Hermitian Metrics
    Abstract: In this talk, we discuss several classes of hermitian metrics on closed complex manifolds and the relations between them. The equality between the cones of balanced and Gauduchon metrics will be addressed in several situations. We will see that the equality of such cones does not hold for arbitrary balanced closed complex manifolds, but it holds on Moishezon manifolds. Moreover, we prove that a SKT manifold of dimension three on which the balanced cone equals the Gauduchon cone is in fact Kahler. (Joint work with I. Chiose and I. Suvaina)
    
    
  • Tuesday, November 29: Brian Allen (USMA)
    Non-Compact Solutions of Inverse Mean Curvature Flow in Hyperbolic Space
    Abstract: Unlike Mean Curvature Flow (MCF), the investigation of non-compact solutions of Inverse Mean Curvature Flow (IMCF) has been very minimal until recently. In this talk we will survey previously known results on compact solutions of IMCF as well as non-compact solutions of MCF before moving on to my results on non-compact solutions of IMCF. Specifically, my results deal with bounded graphs over horospheres in the upper half space model of hyperbolic space and we will investigate long time existence of the flow as well as asymptotic convergence to horospheres. Along the way a new ODE maximum principle at infinity will be introduced and explored in relation to non-compact solutions of IMCF.
    
    
  • Tuesday, December 6: Mark Stern (Duke University)
    Instantons on ALF spaces
    Abstract: I will discuss progress on establishing Cherkis's Nahm transform for multicenter Taub NUT spaces. This is joint work with Sergey Cherkis and Andres Larrain-Hubach.
    
    
  • Tuesday, December 13: no seminar
    Happy Holidays!
    
    

    Past Seminar Schedules and Abstracts:

    Spring 2016 , Fall 2015 , Spring 2014 , Fall 2013 , Spring 2013 , Fall 2012 , Spring 2012 , Fall 2011 , Spring 2011 , Fall 2010 , Spring 2010, Fall 2009, Spring-Summer 2009, Fall 2008, Spring 2008, Fall 2007, Spring-Summer 2007, Fall 2006, Spring-Summer 2006, Fall 2005, Spring 2005, Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2002, Fall 2001, Spring 2001.