January 31: Xiaodong Cao, Columbia University
"The Ricci Flow on Manifolds with Positive Curvature Operator"
Abstract: In this talk, we will first give an introduction of the Ricci
flow. Then we will study the singularities of the Ricci flow
on manifolds with positive curvature operator. We first refine Hamilton's
dimension reduction theorem. Then we prove a convergence result of the
dilation limit. Finally we will talk about the limiting case-- the shrinking
Ricci soliton, we show that under an integral inequality, all the compact
gradient shrinking Ricci solitons with positive curvature operator must be
Einstein, hence of constant curvature.
February 7: Morten S. Risager,
University of Aarhus.
"Counting prime geodesics on Riemann surfaces".
February 14: no meeting
February 21: Classes follow a Monday Schedule
February 28: Feng Luo, Rutgers University
Some applications of the cosine law in low-dimensional
geometry
Abstract: In the discrete approach to smooth metrics on
surfaces, the basic building blocks are sometimes
taken to be triangles in constant curvature spaces. In
this setting edge lengths and inner angles of
triangles correspond to the metrics and its
curvatures. For triangles in hyperbolic, spherical
and Euclidean geometries, edge lengths and inner
angles are related by the cosine law. Thus cosine law
should be considered as the metric-curvature relation.
From this point of view, the derivative of the cosine
law is an analogy of the Bianchi identity in
Riemannian geometry. This talk is focus on many
applications of the derivative cosine law. It provides
a unification of many known approaches of constructing
constant curvature metrics on surfaces. These include
the work of Colin de Vedierer, Greg Leibon, Bragger,
Chow and myself on variational approaches and discrete
curvature flows on triangulated surfaces.
March 7: George Willis, University of Newcastle
Directions in Totally Disconnected Locally Compact Groups
Abstract: Particular groups have an associated structure at infinity that is
used to study the group. Examples include: the set of ends of a discrete
group; the Furstenberg boundary of a semi-simple Lie group; and the
spherical Bruhat-Tits building of a semi-simple group over a local field.
The talk will describe the space of directions of a general totally
disconnected locally compact group $G$. In the case when $G$ is the
automorphism group of a homogeneous tree the space of directions is the set
of ends of the tree, and when $G$ is a semi-simple group over a local field
it is the spherical building.
March 14: Ara Basmajian Univerity of Oklahoma,
Hyperbolic surfaces with discrete length spectra
Abstract: It is well known that the length spectrum of a
geometrically finite hyperbolic manifold is discrete.
In dimension two, geometric finiteness is equivalent to
having a finitely generated fundamental group. Hence
being geometrically infinite is dictated purely by the
topology of the surface. In this generality,
it is possible that the spectrum is not discrete and
the main focus of this talk is to uncover necessary and
sufficient conditions for a hyperbolic structure
to have a discrete spectrum. As a by-product of our
work, we have that all surfaces of infinite topological type
admit a hyperbolic structure with a discrete spectrum.
This is joint work with Youngju Kim.
March 21: Djordje Milicevic (Princeton) L-infinity norms of eigenfunctions.
Friday March 24 at 11:00 am: Johannes Härtel (Universität Göttingen) L^2-torsion
March 28:
Friday April 7 at 11:00 am in 3212: Robert Neel (Columbia) The heat kernel at the cut locus Abstract:
It is well known that, on a compact Riemannian manifold, minus t times the
logarithm of the heat kernel converges uniformly to the energy function as t
goes to zero. This limit commutes with spatial derivatives away from the
cut locus, but one expects more complicated behavior at the cut locus. In
this talk we will give formulas for the small time asymptotics of the
gradient and the Hessian of the logarithm of the heat kernel which are valid
everywhere on the manifold and which admit an appealing probabilistic
interpretation. We will also show how these formulas can be used to study
both the pointwise and the distributional limits of derivatives of the
logarithm of the heat kernel.
April 11: no meeting
April 18: Spring Break
April 25: Leonid Parnovski, University College, London,
Bethe-Sommerfeld conjecture for periodic Schroedinger
operators.
Abstract:
We consider Schroedinger operator with smooth periodic potential. It is
well-known that in the one-dimensional case the number of gaps in it's
spectrum is generically infinite. It was conjectured in the 1930's that
if the dimension of the problem is at least two, then the number of gaps in
the spectrum is always finite. This problem has rather interesting
interactions with other areas of mathematics, including number theory
(distribution of lattice points) and geometrical combinatorics. I will
review the recent progress made in this area.
May 9: Dariush Ehsani (Penn State) The del-bar-Neumann problem on product domains
in C^n. Abstract:
We discuss the solution to the del-bar-Neumann problem on
a product of domains in Cn. We set up the problem for
(0,1)-forms on a product of upper half planes, with a diagonal
metric tensor. We analyze singularities of the solution, and we
also consider the canonical solution to the del-bar
problem.
Summer 2006: we will meet either at 2 or 4 pm.
May 30 (4419 4pm): Christina Sormani (Lehman College and CUNY GC) The Topology of complete open Manifolds with nonnegative
Ricci Curvature
This is the one hour talk I presented at the Midwest Geometry Conference. It is understandable to
graduate students.
Abstract:
While there have been many advances in the understanding of the topology of complete noncompact manifolds of nonnegative Ricci curvature, this area is wide open for further study. In particular, Milnor's famous 1969 conjecture that such a manifold has a finitely generated fundamental group is still open. The speaker will survey a number of theorems and examples, including her work with Zhongmin Shen classifying the codimension one integer homology of these manifolds and her proof of the Milnor Conjecture when the manifolds are assumed to have small linear diameter growth. Unlike the algebraic approach in Burkhard Wilking's reduction of the Milnor conjecture to manifolds with abelian fundamental groups and the analytic proof by Shing-Tung Yau that a manifold with positive Ricci curvature has trivial codimension one real homology, the proofs of these results are purely geometric and can be described with a few key diagrams and lemmas.
June 6 (4419 4pm): Mikhail Khovanov, Columbia University
"Some well-known 3 manifolds with interesting topology" Abstract: The following talk will be accessible to graduate students
with some background in Differential Geometry. The speaker will present
a 3 manifold whose fundamental group is the rationals. He will also present
a contractible 3 manifold which is not diffeomorphic to Euclidean space.
June 13: no meeting
June 20: Daniel Garbin (CUNY GC student)
"Convergence of the hyperbolic and parabolic Eisenstein
series through degeneration." This is a joint work with Mike Munn and
Jay Jorgensen.
July 11: (4419 4pm) Jozef Dodziuk (CUNY GC and Queens College)
"Harmonic forms representing cuspidal cohomology classes on Riemann
surfaces"
aimed at advanced graduate students and faculty
July 18 (4419 4pm): Isaac Chavel (CUNY GC and City College)
"Introduction to Isoperimetric Inequalities" aimed at graduate students, faculty welcome
July 25 (4419 4pm) Neil Katz (City Tech, CUNY)
"A modulus of curves from distance" Abstract:
A modulus is an outer measure on the set of curves
in a manifold. In this talk a modulus related to a
distance function such as from a Riemannian metric
is constructed.
Such a modulus is not a conformal invariant and yet
produces an equivalent definition of quasi-conformal
maps. It also allows one to determine the modulus
of a very general class of curve families. The talk
is aimed at both faculty and graduate students.
August 1: (4419 1pm) Krishnan Shankar (University of Oklahoma)
"Spherical rank rigidity and Blaschke manifolds" Abstract:
Let M be a complete Riemannian manifold whose sectional curvature is
bounded above by 1. We say that M has positive spherical rank if along
every geodesic one hits a conjugate point at t=pi. The following
theorem is then proved: If M is a complete, simply connected Riemannian
manifold with upper curvature bound 1 and positive spherical rank, then
M is isometric to a compact, rank one symmetric space (CROSS) i.e.,
isometric to a sphere, complex projective space, quaternionic
projective space or to the Cayley plane.
The notion of spherical rank is analogous to the notions of Euclidean
rank and hyperbolic rank studied by several people (see references).
The main theorem is proved in two steps: first we show that M is a
Blaschke manifold with extremal injectivity radius (equal to diameter).
Then we prove that such M is isometric to a CROSS.