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
and Christina Sormani.
was in charge of scheduling. For the current schedule see
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
March 6 at 1pm: Anatoly Vershik (Steklov Institute)
"Isometric imbeddings and linear rigidity of metric
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.
March 13 at 1pm: Youngju Kim (CUNY Grad Center student)
Geometrically infinite surfaces with discrete length spectra
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
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
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
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
May 1 at 1pm: Chika Mese (JHU)
May 1 at 4pm (4214-03): Kate Okikiolu (UC San Diego)
The logarithmic Hardy Littlewood Sobolev inequality on the torus and
spectral zeta functions
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
Abstract: I will explain some ideas behind the proof
of the irreducibility of boundary representations for
fundamental groups of compact negatively curved
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
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
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.