Program with Abstracts

The webpage has links to participant preprints and short schedule.**Friday: **

**10-10:30: Registration (show ID on 1st floor, then come to 4214)****10:30-10:45: Opening remarks (4102)****10:45-11:45: Iosif Polterovich, U. Montreal (4102)***"The spectral function and the remainder in local Weyl's law: View from below"*Abstract: The talk focuses on asymptotic lower bounds for the spectral function of the Laplacian and for the pointwise error term in Weyl's law on manifolds. I will first discuss some results valid on any manifold, and then explain how the techniques of thermodynamic formalism for hyperbolic flows yield stronger estimates in the negatively curved case. This is a joint work with Dmitry Jakobson (McGill).

**11:45-1:00: Lunch at the cafeteria on the 8th floor is quick****1:00-2:00 pm: Peter Buser, EPF Lausanne (4102)***"A chat on simple closed geodesics"*

transparencies**2:20-3:20 pm: Regina Rotman, Penn State and U. Toronto (4102)***"The length of a shortest periodic geodesic and a shortest geodesic loop at a point"*Abstract: I would like to present several results, (some are joint with A. Nabutovsky) providing upper bounds for the length of a shortest periodic geodesic on a closed Riemannian manifold and for the length of a shortest geodesic loop at a prescribed point on a closed Riemannian manifold. My work was motivated by the following question asked by M. Gromov: Is there a constant c(n), such that the length of a shortest closed geodesic on a closed Riemannian manifold, M

^{n}, is bounded from above by c(n) vol(M^{n})^{1/n}, where n denotes the dimension of a manifold and vol(M^{n}) denotes its volume.In particular, I will prove that the length of a shortest geodesic loop based at any prescribed point of a closed n-dimensional Riemannian manifold does not exceed 2nd, where d denotes the diameter of the manifold. I will also speak about the connection between the length of a shortest geodesic and the existence of "many" short geodesic loops at each point of a manifold.

Lastly, I will speak about the curvature-free upper bounds for the length of a shortest stationary 1-cycle. Stationary 1-cycles can be regarded as homological analogs of closed geodesics. We proved that the length of a shortest stationary 1-cycle can be majorized by c(n)vol(M

^{n})^{1/n}thereby obtaining a positive solution for the analog of the question posed by M. Gromov.**3:20-3:50: Tea (4214)****3:50-4:50: Ruth Gornet, University of Texas at Arlington (4102)***"Laplace and Length Spectra and the Wave Invariants on Riemannian Two-Step Nilmanifolds"*Abstract: We compare the behavior of the length spectrum and the wave invariants on certain families of isospectral nilmanifolds. The clean intersection hypothesis is discussed in detail. transparencies

**5:00-6:00: Christopher Croke, University of Pennsylvania (4102)***"Some rigidity and stability results of sharp isoperimetric inequalities"*

**9:30-10:00: Registration (show ID on 1st Floor, come to 4214)****10:00-11:00: Carolyn Gordon, Dartmouth College (4102)***"The torus action method for isospectrality in the compact and noncompact settings"*Abstract: Over the past dozen years, a technnique involving torus actions and Riemannian submersions has produced many examples of isospectral manifolds with different local geometry, including many simply connected examples. More recently, the technique has been extended to the scattering setting to construct isoscattering metrics with different local geometry. Modifications of the method give isoscattering potentials and obstacles, although for the examples constructed thus far, the background metrics have nonconstant curvature. We will discuss the technique and survey the examples.

**11:15-12:15: Toshikazu Sunada, Meiji University (4102)***``Perturbation techniques in geometry"*Abstract: I will explain some ideas in counting closed geodesics together with lattice vibrations in my talk.

**12:15-1:30: Lunch (cafeteria is closed)**The Japanese/Korean buffet, Minado, at 6 East 32nd Street is quick.

There are many other fast food places in all directions including chains.

**1:30-2:30: Peter Sarnak, Princeton University (4102)***"Reciprocal geodesics on the modular surface"**Letter to J. Davis about Reciprocal Geodesics*the letter**2:40-3:40: Zoltan I. Szabo, CUNY Graduate Center and Lehman College (4102)**transparencies*"Laplace Spectra on Open and Compact Zeeman Manifolds"*Abstract: By a recent observation, the Laplacians on the Riemannian manifolds the author used for isospectrality constructions are nothing but the Zeeman-Hamilton operators of free charged particles. These manifolds can be considered as prototypes of the so called Zeeman manifolds. This observation allows to develop a spectral theory both on open Z-manifolds and their compact submanifolds.

The theory on open manifolds leads to a new nonperturbative approach to the infinities of QED. This idea exploits that the quantum Hilbert space decomposes into subspaces (Zeeman zones) which are invariant under the action both of this Zeeman-Laplace operator and the natural Heisenberg group representation. Thus a well defined particle theory and zonal geometry can be developed on each zone separately. The most surprising result is that quantities divergent on the global setting are finite on the zonal setting. Even the zonal Feynman integral is well defined. The results include explicit computations of objects such as the zonal spectra, the waves defining the zonal point-spreads, and the zonal Wiener-Kac resp. Dirac-Feynman flows.

The observation adds new view-point also to the problem of finding intertwining operators by which isospectral pairs of metrics with different local geometries on compact submanifolds can be constructed. Among the examples the author constructed the most surprising are the isospectrality families containing both homogeneous and locally inhomogeneous metrics. The observation provides even quantum physical interpretation to the isospectrality.

**3:40-4:30: Tea (4214)****4:30-5:30: Motoko Kotani, Tohoku University (4102)***"Spectra of the Magnetic Transition Operators on a Crystal Lattice"*Abstract: A crystal lattice is an abelian cover of a finite graph. Typical examples are the integer lattices, the triangular lattice, and the hexagonal lattice. We will consider electrons on a crystal lattice under the magnetic fields, and discuss how spectrum of the magnetic transition operators changes when magnetic fields do.

**5:30-6:00: Open Problems (4102)**

**Please arrange to meet for brunch before conference somewhere.****12:00-12:30: Registration (show ID on 1st Floor and come to 4214)****12:30-1:30: Andras Vasy, Stanford University and MIT (4102)***"Scattering theory on symmetric spaces and N-body scattering".*Abstract: I will explain how techniques from quantum N-body scattering shed light on scattering theory on symmetric spaces. In particular, this connection explains which features of symmetric spaces are `typical' in a larger context, and which are due to the special algebraic features. Work with Rafe Mazzeo. transparencies

**1:40-2:40: Mahta Khosravi, Institute of Advanced Study (4102)***"Spectral Asymptotics of Heisenberg Manifolds"*Abstract: In this talk, we present some new results( obtained in a joint work with Y. Petridis) on the error term in Weyl's law for (2n+1)-dimensional Heisenberg manifolds.

**2:40-3:10: Tea (4214)****3:10-4:10: Nancy Hingston, The College of New Jersey, IAS (4102)**transparencies*"The Loop Product and Closed Geodesics"*Abstract: The critical points of the length function on the free loop space L(M) of a compact Riemannian manifold M are the closed geodesics on M. The geometry of the length function interacts with the algebraic structure provided by the Chas-Sullivan loop product on the chain complex of L(M). Local nilpotence of the loop product has consequences for the length spectrum.

**4:20-5:20: Alexander Nabutovsky, U. Toronto and Penn State (4102)***"Curvature-free estimates for minimal surfaces and geodesics between points".*Abstract: In the first part of this talk we will discuss curvature-free upper bounds for the smallest area of a minimal surface in a closed Riemannian manifold.

Then we will discuss lengths of different geodesics connecting two points in a closed Riemannian manifold. According to a classical theorem of J. P. Serre every two points in a closed Riemannian manifold can be connected by an infinite set of geodesics. The length of the shortest of these geodesics obviously does not exceed the diameter of the manifold. But what can be said about the other geodesics?

We conjecture that there exist k geodesics between any two points of a closed Riemannian manifold such that their lengths do not exceed kd, where d is the diameter of the manifold. We will present several results supporting this conjecture. (Joint work with Regina Rotman.)

**5:30-6:00: Open problems (4102)**