Differential Geometry and Lie Theory Seminar

We meet Tuesdays 4-5pm at the CUNY Graduate Center in Room 4419. When we have a double header, the first speaker speaks 2:30-3:30pm in 8405. Tea is at 3:30pm in the mathematics lounge (4214). The CUNY Graduate Center building is 365 5th Avenue at 34th Street. The organizers are Józef Dodziuk, Adam Koranyi, Martin Moskowitz, and Christina Sormani. Christina Sormani is in charge of scheduling.

In the past there were two different seminars and we just merged to promote more interaction between the two groups of researchers. The our first merged schedule is Spring 2005. Past schedules for the Differential Geometry and Analysis Seminar are Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2002, Fall 2001 and Spring 2001. Past schedules for the Lie Group Seminar are Fall 2004 and Spring 2004. This seminar has a 30 year history of meeting at the CUNY Graduate Center.

Schedule, Fall 2005

• September 20: Double Header
  • 2:30 pm: (in 8405) Roe Goodman (Rutgers University)
    "The Lie Algebra of K-invariant vector fields on a symmetric space G/K"
    Abstract: Let G be a reductive complex algebraic group, and let K be the fixed-point subgroup of an involution of G. We study the (infinite-dimensional) Lie algebra of K-invariant algebraic vector fields on G/K using the geometry of G/K and the (finite-dimensional) K-spherical representations of G. (joint work with Ilka Agricola, Humboldt-University at Berlin)
  • 4:00 pm: (in 4419) Toshio Oshima (Graduate School of Mathematical Sciences, University of Tokyo)
    "Differential Equations attached to generalized flag manifolds"
    Abstract: I will explicitly construct generators of the invariant system of differential equations which kill functions over generalized flag manifolds whose typical examples are Grassmannians. I will explain their applications to integral geometry, such as, Poisson transforms, Penrose transforms, Radon transforms, hypergeomegtric systems, Whittaker models etc.
• September 27: A joint seminar with the Probability Seminar.
  • 2:45 pm: (in 8405) Max von Renesse (CIMS)
    "Probability Techniques used in Metric Measure Geometry"
    Abstract: This is an introductory lecture for geometers who will be attending the 4pm talk.
  • 4:00 pm: (in 4419) Max von Renesse (CIMS)
    "Mass Transportation and Synthetic Ricci Curvature Bounds".
    Abstract: The problem of optimal mass transportation has appeared first in the 18th century in economic models. In recent years the theory has undergone a remarkable development with applications in PDE, probability and geometry. The talk will give a short review of some basic concepts involved. The focus of the second part will be geometric. Mass transportation is used for the defintion and analysis of generalized lower Ricci curvature bounds for metric measure spaces with no or almost no regularity.
• October 4: no meeting (holiday)
• October 11: no meeting (monday schedule!)
• October 18:
  • 4:00 pm: (in 4419) Vladimir Gol'dshtein (Department of Mathematics, Ben-Gurion University of the Negev)
    "Sobolev Inequalities for Differential Forms."
    Abstract: We study relations between Sobolev inequalities for differential forms on Riemannian manifolds and the L_q,p cohomology of these manifolds. The L_q,p cohomology is defined to be the quotient of the space of closed p-integrable differential forms modulo the exact forms which are exterior differentials of q-integrable forms. We show also that L_q,p cohomology can be relevant in the study of some non-linear PDE.
• October 25:
  • 2:00 pm: (in 6417) Jean-Michel Bismut will speak at the Einstein Chair Seminar
  • 4:00pm: (in 4419) Peter Zograf (Steklov Institute of Mathematics, St.Petersburg)
    "Volumes of moduli spaces of Riemann surfaces"
    Volumes of moduli spaces of Riemann surfaces (complex algebraic curves) with respect to the celebrated Weil-Petersson metric are interesting invariants that play a role in both mathematics and theoretical physics. We will explain how these volumes appear in 2-dimensional gravity, how they are related to each other for different genera and how they can be effectively computed. We will also discuss their asymptotical behavior when the genus or the number of marked points (punctures) become large.
• November 1:
  • 4:00 pm: (in 5417) Gady Kozma, IAS will speak at the Probability Seminar
    "Isoperimetric inequalities in probability"
    Abstract: This talk will survey connections between isoperimetric inequalities on infinite graphs and random walk and percolation on these graphs.
• November 8: GC Doctoral Faculty Meeting at 4pm
• November 15:
  • 2:30 pm: (in 8405) Mu-Tao Wang (Columbia)
    "On quasi-local mass and its positivity"
    Abstract: The quasi-local mass is a quantity associated with a compact space-like region $\Omega$ in the space-time. It is expected that this information can be derived from the boundary, $\partial \Omega$, which is a two-dimensional space-like surface. By Throne's hoop conjecture, the quasi-local mass is supposed to be closely related to the formation of black holes in the enclosed region. We shall discuss some recent developments in the construction and the positivity of the quasi-local mass. The construction relies on the solutions of some canonically defined elliptic and parabolic equations associated to the geometry of the surface, and the application of the positive mass theorem.
  • 4:00 pm: (in 4419) Jean Steiner (CIMS)
    "Hide and Seek and Spectral Invariants (on Surfaces and Markov Chains)"
    Abstract: In this talk we prove the expected duration of a game of Hide-and-Seek played on a Riemannian Manifold under the laws of Brownian Motion is a spectral invariant: it is a zeta-regularized version of the `trace' of the Laplacian. Our proof relies on the fact that the trace and its density may be approached {\it via } the Green's function for the Laplacian, and the fact that Green's functions and expected first hitting times are closely connected. We will also consider analogous Hide-and-seek games played on a Markov chain, where a classical spectral invariant known as Kemeny's constant emerges. Finally, as an application on surfaces, we describe conjectured extremal behavior for the regularized trace. This is joint work with Peter Doyle.
• November 22:
  • 4:00 pm: (in 4419) Stefan Wenger (CIMS)
    "Area-minimizers and isoperimetric inequalities in CAT(0)-spaces"
    Abstract: In this talk we discuss an isoperimetric inequality for generalized k-dimensional surfaces in complete CAT(0)-spaces. We show how it can be used to prove existence of area-minimizers in locally non-compact spaces. We furthermore use it to study some filling invariants at infinity.
• November 29: no meeting
• December 6:
  • 2:30 pm: (in 8405) Kenneth Clarkson (Lucent Bell Labs)
    "Approximating Surfaces with Meshes"
    Abstract: How hard is it to approximate a smooth surface with a piecewise-linear mesh? When the smooth surface is the boundary of a convex body, remarkably tight bounds are known for the smallest Hausdorff distance possible using a mesh with n simplices. In the case of more general surfaces, less is understood. I'll show that the best possible distance for a d-manifold M is about O(S/n)^2/d, where S is the integral over M of the square root of the Gaussian curvature. (The constant factor here depends only on the dimension.) Also, under some reasonably "natural" conditions on the surface and the mesh, this expression is also a lower bound, up to a constant factor.
  • 4:00 pm: (in 4419) Tim Steger (IAS, Universita di Sassari)
    "Unitary Free Group Representations: Duplicity on the Boundary"
    Abstract: Let $\Gamma$ be a finitely generated free group. Consider an irreducible unitary representation $\pi$ of $\Gamma$ on a Hilbert space $H$. The Cayley graph of $\Gamma$ with respect to its standard generators is a tree. Let $\Omega$ be the ideal boundary of that tree. One asks when it is possible to identify $H$ with an $L^2$-space defined on $\Omega$. More precisely, one would like to identify $H$ with the space of $L^2$-sections of some $\Gamma$-equivariant Hilbert-space bundle on $\Omega$. In operator algebra language, one would like to extend the representation $\pi$ of $\Gamma$ to a representation $\pi'$ of the crossed product $C^*$-algebra $\Gamma\ltimes C(\Omega)$. In fact, one can identify, or at least inject $H$ in such an $L^2$-space if and only if $\pi$ is \emph{tempered}, that is weakly contained in the regular representation. In many cases, one can identify $H$ with an $L^2$-space on $\Omega$ in two different ways. This fact is to be compared with the possibility of obtaining the same principal series representation of $SL(2,\RR)$ by induction in two different ways. Conjecture: (roughly) in no case can $H$ be identified with an $L^2$-space on $\Omega$ in \emph{more} than two ways. The speaker and WALDEMAR HEBISCH have partial results which (1) prove this conjecture for many examples and (2) when applicable, provide a powerful method to prove that $\pi$ is irreducible. The speaker and GABRIELLA KUHN have constructed a big family of examples, which cover most of the irreducible, tempered representations of $\Gamma$ which have been explicitly constructed in the past.
• December 13:
  • 4:00 pm: Gopal Prasad (U Michigan and IAS)
    "Fake projective planes"
    Abstract: A fake projective plane is a smooth compact complex surface which is not the complex projective plane but has the same Betti numbers as the projective plane. Such a surface is projective algebraic and is a quotient of the complex 2-ball by a torsion-free cocompact arithmetic subgroup of PU(2,1). The first example of such a surface was constructed by David Mumford and two more examples were found later.

    Using my volume formula and some interesting number theoretic estimates, in a joint work with Sai Kee Yeung we have been able to get a complete list of all fake projective planes and give their construction in a direct and natural way. The list contains several new examples. It also helps us to discover interesting geometric properties of these surfaces.