Fall 2005
We meet Tuesdays 45pm
at the CUNY Graduate Center in Room 4419. When we have a
double header, the first speaker speaks 2:303:30pm 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.
Our upcoming CUNY Geometric Analysis Conference will be held February 35, 2006.
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 Kinvariant vector fields on
a symmetric space G/K"
Abstract:
Let G be a reductive complex algebraic group, and let K be the
fixedpoint subgroup of an involution of G. We study the
(infinitedimensional) Lie algebra of Kinvariant algebraic vector
fields on G/K using the geometry of G/K and the (finitedimensional)
Kspherical representations of G.
(joint work with Ilka Agricola, HumboldtUniversity 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, BenGurion 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
pintegrable differential forms modulo the exact forms which are exterior
differentials of qintegrable forms.
We show also that L_{q,p} cohomology can be relevant in the study of some
nonlinear PDE.
 October 25:
 2:00 pm: (in 6417) JeanMichel 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 WeilPetersson metric are interesting
invariants that play a role in both mathematics and
theoretical physics. We will explain how these volumes
appear in 2dimensional 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) MuTao Wang (Columbia)
"On quasilocal mass and its positivity"
Abstract: The quasilocal mass is a quantity associated with a
compact spacelike region $\Omega$ in the spacetime. It is
expected that this information can be derived from the boundary,
$\partial \Omega$, which is a twodimensional spacelike surface.
By Throne's hoop conjecture, the quasilocal 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 quasilocal 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 HideandSeek
played on a Riemannian Manifold under the laws of Brownian Motion is a
spectral invariant: it is a zetaregularized 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
Hideandseek 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)
"Areaminimizers and isoperimetric inequalities in CAT(0)spaces"
Abstract: In this talk we discuss an isoperimetric inequality for
generalized kdimensional surfaces in complete CAT(0)spaces. We show how it
can be used to prove existence of areaminimizers in locally noncompact
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
piecewiselinear 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 dmanifold 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
Hilbertspace 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 2ball by a torsionfree
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.