In Fall of 2003 the seminar will meet Fridays 12:30-1:30 pm in Room 5417 at the CUNY Graduate Center. The graduate center building is 365 5th Avenue (34th St) in Manhattan. This seminar is organized by Yiannis Petridis, who can be contacted at petridis@comet.lehman.cuny.edu.

Fall 2003 speakers include: Nalini Anantharaman (Ecole Normale Superieure de Lyon), J. Colliander (U. of Toronto), P. Daskalopoulos (Columbia University), Zheng-Chao Han (Rutgers University), Jay Jorgenson (City College, CUNY), Denis Kosygin (Princeton University), S. D. Miller (Rutgers University), T. Tokieda (College of Staten Island-CUNY). The seminar will meet on the following dates:

**September 5, Denis Kosygin (Princeton University)**Title: Statistical properties of spectra of Laplace-Beltrami operators for Liouville surfaces

Liouville surfaces are the most general class of surfaces, on which geodesic flow is still integrable (a surface of revolution is a Liouville surface). The talk discusses the link of Weyl's law for the Laplace-Beltrami operator on Liouville surface to the classical number-theoretical problem of counting integer lattice points inside a region. In particular we consider the relation of eigenvalues to closed geodesics and why the remainder term in Weyl's law behaves as a random variable. The exposition is elementary, no knowledge of probability theory is required.

**September 12, Jay Jorgenson (City College, CUNY)**Title: Asymptotic behavior of canonical and Arakelov metrics and spectral determinants. (joint with Jurg Kramer)

**September 19, Edwin Beggs (University of Wales) at 12:30**Title: Deforming differential structures on manifolds (joint with S. Majid)

**Stephen D. Miller (Rutgers University, New Brunswick) NOTE TIME: 3:30 PM****October 3, Jean Steiner (New York University) postponed for Dec. 5****October 10, No talk, Columbus day weekend****October 17, Panagiota Daskalopoulos (Columbia University)**Title: Rolling Stones with Flat sides: all time regularity of the interface.

Abstract: We consider the free-boundary problem associated with the Gauss Curvature Flow with Flat sides. We study the regularity and geometry of the interface.

**October 24****October 31, John Stalker (Princeton)**Title: Schrodinger Equations with Complex Eigenvalues Recent Improvements on Some Estimates of Kato

Abstract: Schrodinger equations with complex eigenvalues arise naturally in applying resolvent methods to the full Schrodinger equation or the wave equation. Kato has shown that, for reasonable potentials, nice, e.g. L^2, solutions of this equation automatically have exponential decay at spatial infinity. Unfortunately Kato does not get the sharp decay rate. I will present the sharp estimates. They improve Kato's estimates in three directions: 1) The decay rate is provably sharp, 2) the estimates are pointwise rather than L^2, and 3) there is a derivative estimate, analogous to Sommerfeld's outgoing radiation condition.

**November 7, J. Colliander (University of Toronto)**Title: Recent progress on semilinear nonlinear Schrodinger equations.

Abstract: This talk will describe new work establishing global well-posedness and scattering results for certain defocusing nonlinear Schrodinger evolution problems. The progress is based on the construction of almost conservation laws, new Morawetz-type spacetime estimates and multilinear harmonic analysis. This is joint work with M. Keel, G. Staffilani, H. Takaoka and T. Tao.

**November 14, Zheng-Chao Han (Rutgers University, New Brunswick)**Title: Some Results in Conformal Geometry Involving the Schouten Curvature

In the Weyl decomposition of the full Riemann curvature tensor, if one combines the traceless Ricci part and the scalar curvature part as A = E + (n-2)/(2n(n-1)) Rg, then Rm = W + 1/(n-2) A \bcw g, where W is the Weyl tensor. A is called the Schouten curvature tensor. The Weyl tensor has a pointwise convariance transformation property under a conformal change of metric, so the Schouten tensor A encodes the rest of the information of the conformal change of metric, and should exhibit stronger control on the metric than the scalar curvature. In the late 1990's, J. Viaclovsky began to study the variational problem of the total integral of the k-th elementary symmetric function of the eigenvalues of the Schouten tensor, under a volume normalization condition. The critical metrics of such functionals, when 2k not equal to n, the dimension of the manifold, have the property that sigma_k (A) = a constant. The Lagrange-Euler equations satisfied by such metrics are fully nonlinear PDEs when k >1--- when k=1 they are simply the Yamabe equation.

These nonlinear functions \sigma_k (A) of the Schouten tensor have strong interplays with the underlying geometry and topology, especially in low dimensions. This has been demonstrated through a series of recent fundamental work of A. Chang, M. Gursky, J. Viaclovsky, and P. Yang. Much progress in the analytical aspect has been achieved in recent work of the above authors, and those of Y.Y. Li and A. Li, and of P. Guan, C. Lin, and G. Wang. In this talk, I will report on some results in joint work with A. Chang and P. Yang. These include an imbedding criterion for the development map of certain locally conformally flat manifolds with negative Yamabe constant, and some analytical results of the \sigma_k (A) equations concerning local estimates and singular behavior of solutions.-
**November 17, at 11:00 AM Nalini Anantharaman (Ecole normale Superieure de Lyon) NOTE CHANGE OF DATE AND TIME**Title: The vanishing viscosity limit for eigenstates of Schroedinger operators.

Abstract : On a compact manifold, we study the behaviour of the fundamental eigenstate of the operator $$\exp(<\omega, x>/\epsilon).(\epsilon^2\Delta/2 +V(x)).\exp(-<\omega, x>/\epsilon)$$ as $\epsilon$ tends to 0; in other words, we study the problem of "quantum unique ergodicity" for this system, but after performing the Wick rotation which consists in replacing $i\hbar$ by the real number $\epsilon$. Thus, one should speak about "vanishing viscosity limit" rather than "semi-classical limit". We first show that -- up to extraction -- the sequence of invariant states converge to an invariant measure of the classical system, which is action-minimizing in the sense of J. Mather. This is usually not enough to prove uniqueness of the limit. We prove, in addition, that any limit point must minimize the gap in the Pesin-Ruelle inequality, which compares entropy and Lyapunov exponents.

**December 5, Jean Steiner (New York University)**Title: Spectral Invariants and a Positive Mass Theorem on Spheres

Abstract: We describe two mass-like quantities arising from the Green's function for the Laplacian operator on surfaces. The Robin's mass is obtained by regularizing the logarithmic singularity of the Green's function. We show that the Robin's mass is the density for a spectral invariant. We also introduce a related "geometrical mass", which is (a priori) a point-wise varying quanitity. On the sphere, we show that for any metric conformally equivalent to the round metric, the goemetrical mass is independent of the point, and it is a spectral invariant. Moreover, a connection to a Sobolev-type inequality reveals that the geometrical mass is minimized at the standard round metric.

**December 16, 3:00 PM Galia Dafni (Concordia University and IAS) NOTE CHANGE OF DATE AND TIME**Title: Some function spaces related to BMO (joint work with J. Xiao)

Abstract: The space BMO of functions of bounded mean oscillation is well known in connection with PDE, singular integral operators, Carleson measures, quasiconformal mappings and as the dual of the Hardy space H^1. Recently some interesting subspaces of BMO, called Q spaces, were introduced, initially in the context of complex analysis and later on R^n. They play an intermediary role between BMO and Sobolev spaces. We study the properties of the spaces Q_\alpha(R^n) in connection with fractional Carleson measures, Hausdorff capacity, duality and atomic decomposition.

Title: On Weyl's Law for Higher Rank Spaces

Abstract: I'll give an overview of what's known about Weyl's law for eigenvalues of the Laplacian (and other invariant differential operators) for non-compact quotients of SL(n,R). This problem has a rich history; for n=2 it is what motivated Selberg to develop his famous trace formula. The talk will include an introduction to the basic setup of the trace formula, as well as a description of the "partial trace" technique, which sometimes allows one to bypass the full trace formula if only an asymptotic result such as the Weyl Law itself is desired. Applications will be given to the Ramanujan conjectures.