Differential Geometry and Analysis Seminar

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 isospectrality.
• 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 DeTurck's trick.
• 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