Differential Geometry and Analysis Seminar
In the Fall of 2004,, the seminar will meet
Fridays 12:30-1:30 at the CUNY Graduate Center Room 5417.
This seminar was organized by Christina Sormani
Schedule, Fall 2004
September 3: Amy Novick-Cohen, Technion-IIT, (hosted by Chavel)
Coupled Surface Diffusion and Motion by Mean Curvature
October 1 [Special Time 12-1pm]: Iosif Polterovich, U. Montreal,
Extremal problems for the first eigenvalue
Estimating the first eigenvalue of the Laplacian under different
geometric assumptions is a classical problem in spectral geometry.
We survey some recent results motivated by the following questions:
(1) How large can the first eigenvalue be on a surface of a given area?
(2) How small can the supremum of the first eigenvalues be for metrics in a
fixed conformal class on a manifold of a given volume?
We explore the links between these two questions and other topics
in geometric analysis, such as minimal surfaces, Sobolev inequalities and
October 8: Juan Carlos Alvarez, Polytechnic University
Expect the unexpected: volumes on normed and Finsler spaces.
The theory of volumes and areas on normed and Finsler spaces
holds a number of surprises: there are a number of natural, inequivalent
definitions, it is not clear that flats are area-minimizing, and giving
sharp estimates for the area of unit spheres of normed spaces is a
48-year old open problem of Busemann and Petty. Neverthless, the theory
comprises and unifies large domains of convexity, geometric tomography,
and integral geometry. It opens many classical unsolved problems in these
fields to powerful techniques in global differential geometry, and
suggests new challenging problems that are delightfully geometric and
simple to state.
The talk will be a non-technical introduction to the subject concentrating
mostly on two and three-dimensional phenomena.
October 15: Mao-Pei Tsui, Columbia University
Mean Curvature Flows and Isotopy of Maps Between Spheres
Let f be a smooth map between unit spheres of possibly
different dimensions. We prove the global existence and convergence of
the mean curvature flow of the graph of f under various geometric
conditions. A corollary is that any area-decreasing map between
unit spheres (of possibly different dimensions) is homotopic to a
constant map. This is joint work with Mu-Tao Wang.
October 22: no meeting
October 29: (12:30-1:30) Morten Risager, IAS, Arhus
Distribution of closed geodesics and spectral perturbations.
I will discuss how character perturbations of the Laplacian
can give results on the distribution of closed geodesics on a Riemann
surface of genus greater than two. The main tool is the trace formula
which relates the length spectrum to the spectrum of the Laplacian of
the surface. This is joint work with Yiannis Petridis.
November 5: Sean Paul (posponed until Spring)
November 12: (11:30-12:30)
Penny Smith, Lehigh University
Perron's Method for Symmetric Hyperbolic Systems,
Comparison principle and Application to the Einstein Cauchy Problem
A vector comparison principle, and a definition of viscosity solution
for symmetric hyperbolic systems allows existence proofs via Perron's
method. Application is given to the Einstein Cauchy Problem using
November 19: (11:30-12:30) Nancy Hingston, College of New Jersey
Subharmonic solutions of Hamiltonian equations on tori
Let the torus T^2n be equipped with the standard symplectic
structure and a Hamiltonian H that is periodic in the time variable.
A subharmonic solution is a periodic orbit of the Hamiltonian flow with
minimal period an integral multiple m of the period of H, with m>1.
We prove: If the Hamiltonian flow has only finitely many orbits
with the same period as H, then there are subharmonic solutions with
arbitrarily high minimal period. Thus there are always infinitely
many distinct periodic orbits. This was proved in the nondegenerate
case by Conley and Zehnder, and in the case n=2 by Le Calvez.
November 26: holiday
December 3: Craig Sutton (posponed until Spring)
March visitor: Miatello Rosetti