Differential Geometry Seminar

In Spring -Summer 2007 the CUNY Differential Geometry Seminar met on Tuesdays 1-2 pm (and sometimes 4-5pm Tuesday or 11-12 am Friday) at the CUNY Graduate Center in Room 4419. We usually met for lunch at noon gathering at 4214. The organizers were Józef Dodziuk, Adam Koranyi, Yiannis Petridis and Christina Sormani. Christina Sormani was in charge of scheduling.

Schedule, Spring-Summer 2007

• Feb 6 at 1pm: Christina Sormani (Lehman College and CUNY Grad Center)
  The Covering Spectrum of a Riemannian Manifold Part I
  Abstract: The covering spectrum of a Riemannian manifold was first defined by Sormani-Wei in 2004. We will review the definition of this spectrum which roughly measures the sizes of holes in the space using a special sequence of covering spaces called delta covers. On compact spaces, we proved the covering spectrum is a subset of the (1/2) length spectrum and is determined by the marked length spectrum. This is not true on complete noncompact manifolds. In recent work with Wei, we have begun to prove related weaker theorems. To capture more information about complete manifolds we have defined two other spectra: the cut-off covering spectrum and the rescaled covering spectrum.
• Wed Feb 7 2pm: Jean-Michel Bismut
  The hypoelliptic Dirac operator
  More info at the Einstein Chair Seminar Webpage.
• Feb 13 at 1pm: Christina Sormani (Lehman College and CUNY Grad Center)
  The Covering Spectrum of a Riemannian Manifold Part II
  Abstract: We will continue to discuss the covering spectrum with a brief review for those who missed the first talk.
• Feb 20 at 1pm: No Meeting
  Winter break for public schools
  CUNY follows a regular Tuesday schedule.
• Feb 27 at 1pm: Isaac Chavel (City College and CUNY Grad Center)
  The Isoperimetric Inequality for Three Dimensional Cartan-Hadamard Manifolds
  Abstract: This will be an expository talk describing Bruce Kleiner's proof of the Aubin conjecture for three dimensions. This conjecture concerns the isoperimetric inequality for simply connected spaces with nonpositive sectional curvature. It can be proven for dimension 2 using Fourier series on the circular boundary. It was proven in dimension 4 by Croke and now in dimension 3 by Kleiner.
• March 6 at 1pm: Anatoly Vershik (Steklov Institute)
  "Isometric imbeddings and linear rigidity of metric spaces"
  Abstract: The Isometry of metric space to Banach spaces can be considered from point of view Kantoroich metric. There exist remarkable metric spaces which have only one such isometry. We call them linear rigid metric spaces. The first example of such space is Urysohn space. (Holms) Recently Melleray-Petrov and author found all such examples. The crucial idea here is to study the distance matrices of a given metric space. preprint preprint preprint preprint preprint preprint
• March 13 at 1pm: Youngju Kim (CUNY Grad Center student)
  Geometrically infinite surfaces with discrete length spectra
  Abstract: It is well-known that the length spectrum of a geometrically finite hyperbolic manifold is discrete. We show that geometrically infinite surfaces admits both an infinite dimensional family of quasiconformally distinct hyperbolic structures having a discrete length spectrum, and an infinite dimensional family of quasiconformally distinct structures with a nondiscrete spectrum. This is a joint work with Prof. Ara Basmajian.
• March 20 at 4pm: Xiaodong Cao (Cornell University)
  Cross Curvature Flow on Locally Homogenous Three-manifolds.
  Recently, Chow and Hamilton introduced the cross curvature flow on three-manifolds, which is a weakly parabolic partial differential equation system when the sectional curvatures have a definite sign. They also conjectured the long time existence and convergence of cross curvature flow on closed three-manifolds with negative sectional curvature. In this talk, we will study the cross curvature flow on locally homogenous three-manifolds. We will describe the long time behavior of the cross curvature flow for each case. This is a joint work with Yilong Ni and Laurent Saloff-Coste.
• March 27 at 1pm: Regina Rotman (U. Toronto and Penn State)
  Curvature-free upper bound for the length of stationary objects
  Let M be a closed Riemannian manifold of dimension n. I will talk about curvature-free estimates for the length of various minimal 1-dimensional objects, such as geodesic loops, periodic geodesics, stationary geodesic nets and cycles and the "kth" geodesic segment between two points of M.
• April 3 and 10 Spring Break
  No meeting
• April 17 at 1pm: meeting cancelled
  Okikiolu will speak on May 1 at 4pm after Mese's 1pm talk.
• April 24 at 1pm: Iosif Polterovich (Universite de Montreal)
  Dynamical and universal features of spectral asymptotics on manifolds
  Abstract: The talk will focus on the asymptotic behaviors of the spectral function and the eigenvalue distribution for the Laplacian on a Riemannian manifold. It is well-known that some properties of spectral asymptotics are "universal", i.e. valid on any manifold (such as Weyl's law), and some are dependent on the dynamics of the geodesic flow. I will present new results of both "dynamical" and "universal" nature. A special emphasis will be made on negatively curved manifolds. In particular, I will discuss two results of Burton Randol proved for surfaces of constant negative curvature, one of which is "dynamical" and the other is in fact "universal".
• Thursday April 26 at 1pm: R. Parthasarathy (Tata Institute for Fundamental Research, Bombay)
  Unitary representations, Dirac inequality, and cohomology
  Abstract: TBA
• May 1 at 1pm: Chika Mese (JHU)
  Title TBA
  Abstract: TBA
• May 1 at 4pm (4214-03): Kate Okikiolu (UC San Diego)
  The logarithmic Hardy Littlewood Sobolev inequality on the torus and spectral zeta functions
  Abstract: TBA
• May 8 at 1pm: Marcus Khuri (SUNY Stony Brook)
  Compactness Issues Related to the Yamabe Problem
  Abstract: The Yamabe problem consists of finding constant scalar curvature metrics on a compact Riemannian manifold via conformal deformation. Typically there are many high energy solutions with high Morse index for this variational problem. We will describe resent results concerning the compactness of the space of solutions, as well as the close connection these resulta share with the positive energy theorems of General Relativity.
• July 24 at 1pm: Roman Muchnik (Lehman College)
  Irreducibility of boundary representation of hyperbolic group
  Abstract: I will explain some ideas behind the proof of the irreducibility of boundary representations for fundamental groups of compact negatively curved manifold. I will describe some analytical ideas involved in the course of the proof, and in particular our construction of averaging operators. These operators allows us to show that 2 representations are equivalent if and only if the corresponding marked length spectra are proportional. This is a joint work with Uri Bader. If time permits, I will explain how to extend these arguments to general hyperbolic groups.
• August 14 at 1pm: Jeff McGowan (Central Connecticut State University)
  Regular trees in random regular graphs
  Abstract: We investigate the size of the embedded regular tree rooted at a vertex in a d regular random graph. We show that almost always, the size of this tree will be (1/2)log n, where n is the number of vertices in the graph. We use this to give an asymptotic estimate for Gauss' Hypergeometric Function, and consider application to the study of Riemann surfaces. Joint work with Eran Makover.