We met Tuesdays 12 pm
at the CUNY Graduate Center in Room 4419.
The organizers were Józef
Dodziuk,
Adam Koranyi,
Yiannis Petridis
and Christina Sormani.
Christina Sormani
was in charge of scheduling.
Schedule, Fall 2006

Sept 12 at 1pm: Jyotshana Prajapat (Tata Institute of Fundamental Research, Mumbai),
"Symmetry and classification of solutions of a semilinear equation on the
Heisenberg group".

Sept 19 at 1pm: Maria Calle (NYU),
Ancient Solutions to the Mean Curvature Flow
Abstract:
In the first part of the talk, I'll introduce mean curvature flow. A
family of surfaces in R^{3} (or, in general,
ksubmanifolds in R^{n}) is
said to move by mean curvature flow if its movement satisfies a
particular parabolic PDE. This evolution follows the steepest descent
direction for the area, that is, the surfaces decrease their area at
the fastest possible rate. I present some basic facts about mean
curvature flow solutions, such as a mean value inequality and the
definition of density at a point.
After that, I'll present a result about ancient solutions. An ancient
solution for mean curvature flow is a solution defined for all times
less than 0. I give a bound on the dimension of the ambient space of an
ancient solution, depending on a bound on the density of the evolving
submanifold.

Sept 26 at 1pm: Frederic Rochon (SUNY Stony Brook)
On the topology of the space of invertible pseudodifferential
operators of order zero.

Oct 3 at 1pm: no meeting (Monday schedule)

Oct 10 at 1pm: Michael Munn (CUNY),
Perelman's Contractibility Theorem
Abstract: In Grigori Perelman's 1994 paper, he proves that
Riemannian manifolds with nonnegative Ricci curvature and almost
maximal volume growth is homeomorphic to R^{n}. This
result was later improved upon by CheegerColding.
Here we present Perelman's constructive approach which provides
the contraction maps for S^{k} of all dimensions k via
an application of BishopGromov's Volume Comparison Theorem
and AbreschGromoll's Excess Theorem.

Oct 17 at 1pm: Jesse Ratzkin (U Conn)
Eigenvalues on spherical domains and an application to probability
Abstract:
I will describe joint work with Andrejs Treibergs, where we
estimate the first Dirichlet eigenvalue of certain spherical domains. Among our
tools are a coning construction to reduce the dimension of the problem and a
domain perturbation based on a Fourier decomposition of eigenfunctions.
One application of our estimates is the proof of a conjecture of Bramson and
Griffeath in Brownian pursuit.

Oct 24 at 1pm: Stefan Wenger (NYU)
Gromov hyperbolic spaces and sharp constants for isoperimetric
and filling radius inequalities.
Abstract: Hyperbolic metric spaces in the sense of Gromov have played an
important role both in Geometric Group Theory and in Geometry and have
been subject to intense research over the past 20 years. They were
introduced and first studied by M. Gromov in his seminal paper on
Hyperbolic Groups and can be thought of as spaces with negative curvature
in a coarse sense.
In this talk we will discuss the following optimal characterization of
Gromov hyperbolicity via an isoperimetric inequality. Let $X$ be a
geodesic metric space and suppose there exist $\varepsilon, s_0>0$ such
that the filling area of each rectifiable loop of length $s\geq s_0$ is
bounded above by $(1\varepsilon)(4\pi)^{1}s^2$. Then $X$ is Gromov
hyperbolic. (As is wellknown, $X$ then even admits a linear isoperimetric
inequality, i.e. the filling area is bounded by $Cs$ for some constant
$C$.) Here, the filling area of a loop $\gamma$ is by definition the least
area of a $2$chain with boundary $\gamma$. Our theorem is optimal as
shows the case of Euclidean space and is new even for Riemannian
manifolds. It strengthens results in Gromov's paper in which the constant
$(16\pi)^{1}$ was obtained for a large class of Riemannian manifolds and
$(4000)^{1}$ for geodesic metric spaces.
In the end, we will give a similarly optimal characterization involving
filling radius inequalities.

Oct 31 at 1pm: no meeting (Halloween)

Nov 7 at 1pm: no meeting (doctoral faculty meeting in 9204)

Nov 14 at 1pm: Natasa Sesum (Columbia)
"On the extinction profile of solutions to the fast diffusion equation."
We consider the initial value problem
u_{t} = &Delta log u with u(x,0) = u_{0}(x)\ge 0
in the plane.
This equation describes the conformal evolution of the metric
u g_{0} by Ricci flow. It is well known that the maximal
(complete) solution u vanishes identically after time
T= 1/4&pi times the average of u_{0} at time 0.
Assuming that u_{0} is compactly supported (so the manifold
has a cusp), we describe precisely
the Type II vanishing behavior of $u$ at time $T$: we show the existence of an inner
region with exponentially fast vanishing profile, which is, up to proper scaling, a
soliton cigar solution, and the existence of an outer region of persistence of a
logarithmic cusp. This is the only Type II singularity which has been shown to
exist, so far, in the Ricci Flow in any dimension. If the time allows we will
discuss the asymptotics for the Yamabe flow on R^{N} as well.

Nov 21 at 1pm: Satyaki Dutta (SUNY Stony Brook)
"Rigidity of Conformally Compact Manifolds"
Abstract: Rigidity of complete noncompact manifolds with certain
curvature assumptions have been well studied in the past. A number of Gap
of gap theorems have been proved by Greene and Wu and also by Kasue and
Sugahara that show that such manifolds with proper conditions on sectional
curvatures are isometric to the Hyperbolic space. Several other theorems
have been proved for conformally compact manifolds that are either
Einstein or Spin. In this talk we would present a new rigidity result for
manifolds that are not Einstein.

Nov 28 at 1pm: Eugene Kritchevski (McGill University, Montreal)
Spectral localization in the hierarchical Anderson model
Abstract: The hierarchical Anderson model is a discrete random
selfadjoint operator H=L+cV acting on l2(X), where X is a countable set,
L is a hierarchical Laplacian, V is a random potential given by
(Vf)(x)=v_{x} f(x) with v_{x}
i.i.d. random variables, and c>0 is a coupling
constant. S. Molchanov has proven that the spectrum of H is pure point
with probability one, when the random variables v_{x}
have a Cauchy
distribution. In this talk, I will review the basic properties of the
model and I will present two localization theorems extending Molchanov's
result. Theorem 1: if the spectral dimension of the model is less or equal
than 4, then, for any continuous distribution of v_{x},
the spectrum of H is
pure point with probability one. Theorem 2: for a dense set of
distributions of v_{x} , H has pure point spectrum with probability one, in
any spectral dimension.

Dec 5 at 1pm: Roe Goodman (Rutgers)
"Harmonic Analysis on Compact Symmetric Spaces: the Legacy of Cartan and
Weyl"
Weyl and Cartan established harmonic analysis on compact homogeneous
spaces for Lie groups in the period 19261934. In this talk we approach
the CartanWeyl results by way of complex analysis and algebraic groups.
We describe the algebraic group version of the PeterWeyl decomposition
and the geometric criterion for simple spectrum of a homogeneous space
due to Vinberg and Kimelfeld. For a compact symmetric space the spherical
representations were determined by Helgason. We describe Gindikin's recent
construction of the horospherical CauchyRadon transform based on the
results of Helgason and Clerc. This transform shows that compact symmetric
spaces have canonical dual objects that are complex manifolds.