Differential Geometry Seminar Spring 2008

We met on Tuesdays 3-4 pm (and sometimes 11-12 am Friday) at the CUNY Graduate Center in Room 4419. The organizers were Józef Dodziuk, Adam Koranyi, Laurentiu Maxim (Max) and Christina Sormani. Please feel free to contact Christina Sormani regarding scheduling.

Past Seminar Schedules and Abstracts:

• Differential Geometry Seminar: Fall 2007. Spring-Summer 2007. Fall 2006.
• Differential Geometry and Lie Theory Seminar: Spring-Summer 2006. Fall 2005. and Spring 2005
• Differential Geometry and Analysis Seminar:
  Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2002, Fall 2001 and Spring 2001.
• Lie Group Seminar: Fall 2004 and Spring 2004.

Schedule, Spring 2008

• Tuesday February 5: Matthais Keller, Chemnitz University of Technology
  The Laplacian on rapidly branching graphs.
  Abstract: We discuss the unbounded Laplace Operator on graphs with uniformly increasing vertex degree and give a characterization for absence of essential spectra. In particular we present this characterization for planar tessellations in terms of the combinatorial curvature.
• Friday February 15: Marten Fels, University of Bonn and CUNY Graduate Center student,
  A Large Compactification of Symmetric Spaces
  Abstract: The main topic of this talk is the construction of a new compactification of symmetric spaces, which is larger than the known ones, especially larger as the Karpelevich compactification. I will first give an overview of the compactifications of symmetric spaces that has been known so far, then describe the construction of my new compactification and discuss several of its properties. If time permits, will also explain - using an axiomatic approach -, in which sense this compactification can indeed be considered to be the largest "geometric" compactification of symmetric spaces.
• Sat-Sun Feb 23-24: CUNY Geometric Analysis 2008
  Celebrating Józef Dodziuk's 60th Birthday
  See the webpage to register.
• Tuesday March 4: Neil Katz, CUNY City Tech
  "Length normalization of the total scalar curvature"
  Abstract: The Yamabe constant of a compact Riemannian manifold is the minimal total scalar curvature over a conformal class normalized by volume. In this talk we give necessary and sufficient conditions for a metric to have minimal total scalar curvature in its conformal class normalized with a lower bound on the length of a family of curves. The problem is closely related to a conformal isosytolic inequality where volume is minimized instead of total curvature. Examples of conformal minima with non-constant scalar curvature will be given.
• Tuesday March 11: Youngju Kim, CUNY Graduate Center student
  "Quasiconformal stability in hyperbolic 4-space"
  Abstract: An n-dimensional Mobius group is said to be quasiconformally stable if its sufficiently small deformations in Isom+(Hn) are all quasiconformally conjugate to it. Marden has shown that any geometrically finite Kleinian group in dimension 3 must be quasiconformally stable. Here we discuss quasiconformal stability in hyperbolic 4-space. Applying these methods we provide an example demonstrating that Marden stability does not extend to dimension 4.
• Friday March 14: no meeting
  AMS meeting at Courant this weekend
• Tuesday March 18: no meeting
  "Singularities in Geometry and Topology" at Courant Institute March 17-20 see webpage
• Tuesday March 25 Two seminars:
  3-3:50pm: Marcello Lucia, College of Staten Island, CUNY
  Isoperimetric profile and uniqueness for Neumann problems
  Abstract: Consider on a two-dimensional compact Riemannian manifold $(M,g)$ the nonlinear problem: -&Delta _g u = f(u). Under suitable assumptions on the nonlinearity, we prove an inequality which involves the isoperimetric profile of the surface. We apply it to get lower bounds on the first non-zero Neumann eigenvalue, and to derive uniqueness results for mean field type equations of the type where f(u) is a renormalized $e^u$. The precise formula requires tex. Optimal results are obtained on the sphere and some flat torus.
  4-5:50pm: Omri Sarig, Pennsylvania State University
  Generalized laws of large numbers for horocycle flows
  Probability Seminar in 5417 relevant to differential geometers. He will discuss a generalized law of large numbers for a natural class of dynamical systems arising from hyperbolic geometry. More precisely, if T: X -->X is a map and m is a T inveriant ergodic measure, one has a generalized law of large numbers if there is a procedure which accepts as the input the number n such that T^n(x) is in the set E (but not x or E) and provides as output the measure of E.
• Friday March 28: Jesenko Vukadinovic, College of Staten Island, CUNY
  Inertial manifolds for a nonlinear Fokker-Planck equation
  Abstract: Although intrinsically infinite-dimensional, many dissipative parabolic systems exhibit long-term dynamics with properties typical of finite-dimensional dynamical systems. The global attractor, often considered the central object in the study of long-term behavior of dynamical systems, appears to be inadequate in capturing this finite-dimensionality. If a very restrictive spectral gap condition is satisfied, an exponentially attracting invariant finite-dimensional manifold - termed inertial manifold - appears much more appropriate. On it, the PDE reduces to an ODE termed inertial form. However, most physically relevant systems fail to satisfy the spectral-gap condition. A transformation is used to prove the existence of inertial manifolds for a class of nonlinear nonlocal Fokker-Planck equations.
• Tuesday April 1: Haydee Herrera, Rutgers at Camden,
  "Rigidity and Vanishing Theorems on Almost Quaternionic Manifolds"
  Abstract: In this talk, I will consider manifolds whose structure group can be reduced to $Sp(n)Sp(1)$, known as almost quaternionic manifolds. On them we can define twisted Dirac operators. I prove the rigidity under circle actions of several these operators and the vanishing of the indices of some of them.
• Friday April 4: Luis Fernandez (BCC- CUNY)
  "The moduli space of minimal 2-spheres in spheres"
  Abstract: I will give a description of the twistor space construction of harmonic (aka minimal) maps from the round 2-sphere to higher-dimensional spheres. Using this approach, one can give a complete algebraic description of the moduli space of harmonic 2-spheres in 2n-dimensional spheres. I will also sketch the proof of the fact that the dimension of the space of harmonic 2-spheres in the 2n-sphere with twistor degree d is 2 d + n^2, as was conjectured in the early 90's.
• Tuesday April 8: no meeting
  conflict
• Friday April 11: Chris Croke (U Penn)
  Zoll surfaces as counterexamples to a geodesic length conjecture.
  Abstract: In this talk we discuss the construction of Zoll surfaces (surfaces all of whose geodesics are closed of length $2\pi$) and how they can be used to find interesting examples. In particular we (discussing a joint work with Balacheff and Katz) construct Riemannian metrics on the 2-sphere with $L>2D$ relating the diameter~$D$ and the least length~$L$ of a nontrivial closed geodesic. This gives a counterexample to a conjecture of Nabutovsky and Rotman. The construction relies on Guillemin's theorem concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd deformation of the round metric. We conclude that the round metric is not even locally optimal for the ratio~$L/D$.
• Tuesday April 15: no meeting
  conflict
• Mon-Fri April 21-5: Spring Recess
  no meetings
• Tuesday April 29: no meeting
  administrative meeting conflict
  Abstract: TBA
• Tuesday May 6: Daniel Garbin, CUNY Graduate Center student
  "Spectral convergence of elliptically degenerating Riemann surfaces"
  Abstract: In this note we study spectral theory of the Laplacian on a sequence of hyperbolic Riemann surfaces which have elliptic elements whose order of ramification is unbounded. By extending results from Jorgenson-Lundelius-Huntley, we establish an asymptotic expansion for the regularized heat trace in terms of the degeneration parameters. We then derive asymptotic expansions for various spectral functions including wave kernels, resolvent kernels, spectral counting functions, spectral zeta functions, and Selberg zeta functions. As a specific example, we consider the sequence of elliptically degenerating surfaces formed from the Hecke triangle groups, thus quantifying an example first considered by Selberg. Our methods extend to sequences of metrics which are hyperbolic near the diverging elliptic points. This is joint work with J. Jorgenson.
• Friday May 9: Fengbo Hang, Courant Institute, NYU
  "Some rigidity results on the half sphere"
  We will discuss some rigidity theorems for compact manifolds with boundary and positive Ricci curvature. In particular we will show if a n dimensional manifold has Ricci curvature at least n-1, the boundary is totally geodesic and isometric to standard S^{n-1}, then the manifold is isometric to standard upper half sphere. This is the Ricci version of a conjecture by Min-Oo on scalar curvature rigidity. Similar statements for nonpositive curvature had been established earlier by many authors (joint work with Xiaodong Wang).
• Tuesday May 27: Prof. Noel Lohoue (Orsay)
  "The Poisson equation for forms on locally symmetric spaces"