(Sep. 10) Humpherys Viscous-Dispersive traveling waves---
We will discuss the general stability problem for traveling waves and examine some new results for class of viscous-dispersive and higher order conservation laws. In particular we can generalize to higher order the important notions of symmetrizability, dissipativity, and genuine coupling, as is well-understood in the context of viscous conservation laws. We will show that genuine coupling and dissipativity are equivalent for general higher-order symmetrizable systems. We will also discuss open problems and outline the speaker's current research program.

(Sep. 17) F. Finster The Cauchy Problem for the Wave Equation in the Kerr Geometry---
After a brief general introduction to curved space-time and black holes, linear hyperbolic equations in a black hole geometry are considered. Such equations describe the behavior of quantum mechanical particles (Dirac equation) and are also relevant to the stability problem for black holes (wave equation, Teukolsky equation). We briefly compare the different equations and discuss the methods used for their analysis. After that, I will report on recent work on the Cauchy problem for the scalar wave equation. The main result is an integral representation which expresses the solution of the Cauchy problem as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables. This is joint work with Joel Smoller, Niky Kamran, and Shing-Tung Yau.

(Sep. 24) L. Capogna Heisenberg wave maps---
We define wave maps with range in the heisenberg group endowed with its Carnot-Caratheodory metric. Such space can be obtained as a degenerate limit of sequences of Riemannian manifolds, and the wave map problem is in some sense a limit problem corresponding to such approximation. We compare heisenberg wave maps with riemannian wave maps into lagrangian manifold and use such comparison to prove some form of well-posedness for the corresponding Cauchy problem. Among the new features of this problem there is a lack of global well-posedness, even for smooth data with arbitrarily small norm. This is a joint work with Jalal Shatah.

(Oct. 1) L. Barannyk Fully nonlinear gravity-capillary interfacial waves in a two-fluid system of finite depth---
We study the nonlinear flow which results when two immiscible inviscid incompressible fluids of different densities and separated by an interface which is free to move and which supports surface tension, are caused to flow in a straight infinite channel. Gravity is taken into consideration and the velocities of each phase can be different, thus giving rise to the Kelvin-Helmholtz instability. Our objective is to study the competing effects of the Kelvin-Helmholtz instability coupled with a stably or unstably stratified fluid system (Rayleigh-Taylor instability) when surface tension is present to regularize the dynamics. Our approach involves the derivation of two-dimensional model evolution equations using long-wave asymptotics and the ensuing analysis and computation of these models.

(Oct. 15) J. Rauch Intuitive and Counterintuitive Energy Flux---
This theme of this talk is the computation and estimation of the flux and absolute flux of energy across hypersurfaces for evolution equations. The main surprise is that for smooth rapidly decreasing solutions of the wave equation or Shrodinger equation in dimensions greater than one, the absolute flux of energy across a hyperplane can be arbitrarily large compared to the total energy. For the heat equation the absolute flux never exceeds half the energy.

(Oct. 22) E. Kashdan Complex approach to the solution of the time-dependent Maxwell's equations using high order accuracy numerical methods---
The classical finite-difference time-domain (FDTD) approach to the numerical solution of the time-dependent Maxwell's equations is based on the second order, in space and time, Yee algorithm. However, for an increasing number of applications this algorithm has insufficient accuracy. We replace it by a compact implicit 4th order accuracy scheme that uses the same stencil, but doesn't have drawbacks of the Yee algorithm. We implement this scheme in 3D for the solution of Maxwell's equations in various coordinate systems with a Perfectly Matched Layer (PML) for the truncation of the unbounded domains. A major difficulty with high order methods is the treatment of the dielectric coefficient which is discontinuous across the interface. So we also study the asymptotic and numerical behavior of the solution of the wave equation with discontinuous coefficients in one dimension in both time and frequency space. We present a method for the treatment of the discontinuity that preserves a high order of accuracy for the numerical scheme. Finally, we introduce new parallel strategy for solution of the 3D time-dependent Maxwell's equations using compact implicit scheme. This strategy is based on the use of virtual topologies implemented in the MPI standard. This is joint work with Professor Eli Turkel.

(Oct. 29) Fengbo Hang On the approximation problem for Sobolev maps between manifolds---
In this talk I will describe the solution for the problem of whether smooth maps are dense in the space of Sobolev maps between two compact Riemannian manifolds. Earlier developments in this direction by Schoen, Uhlenbeck, White, Bethuel, Zheng and Hajlasz etc play important roles in our arguments. I will also discuss the problem (proposed by Brezis and Li) on classifying path connected components of Sobolev mapping spaces between manifolds and its relation to the density problems. In particular, I will present the solutions to two conjectures of Brezis and Li, which motivated the present work. Finally, I will discuss the open problem on classifying weak sequential limits of smooth maps and some recent progress. These are joint works with Fanghua Lin. Detailed papers may be found at http://www.math.princeton.edu/~fhang/papers.html

(Nov. 5) Mark Ashbaugh On the Buckling Eigenvalues of a Clamped Plate---
This talk will survey some of the known results and open problems for the eigenvalues of the buckling problem for a clamped plate. In its classical formulation, this problem consists of an eigenvalue equation involving the biharmonic operator on one side and the Laplacian on the side with the eigenvalue, the equation holding on a given bounded domain (the ``plate''). The clamped boundary condition just means that the eigenfunction and its derivative must vanish on the domain's boundary. The first eigenvalue of this problem is of particular interest due to its physical importance as the critical buckling load, the maximal amount of lateral compressive loading that a clamped plate of the given shape can sustain on its edges before failure by buckling. Lower bounds for the first eigenvalue will be discussed, including Payne's theorem (originally Weinstein's conjecture), the Polya-Szego conjecture, and related partial results. Time permitting, corresponding results and open problems for domains in spheres will also be discussed.

(Nov. 10) Tai-Ping Liu Green functions for Boltzmann equation---
The Boltzmann equation has the dual particle-fluid property. With Shih-Hsien Yu we study this property through the Green function. Our result also reveals the generalized Huygens' principle.

(Nov. 26) Henryk Kudela The Vortex-in-Cell Method for the Study of 3D Vortex Ring Interactions---
The interest in computational vortex methods stems from the fact that the vorticity plays a fundamental role in all real fluid dynamics phenomena. The vortex particles introduced in the computation permit direct tracking of the vorticity and the analysis of the flow phenomena in terms of the vorticity. One can distinguish two different types of vortex methods, the direct method based on the Biot-Savart law where the velocity of each vortex particle is calculated by summing the contribution of all particles in the domain, and the vortex-in-cell (VIC) method where the velocity is obtained on grid nodes and interpolated to the position of the vortex particles. Despite the development of fast summation algorithms, VIC methods are still several orders faster than direct methods. In the literature one can find VIC calculations related mainly to 2D flow and the extension to 3D flow still requires further investigation. In this work we validate a 3D VIC method using examples of vortex ring dynamics: the motion of a single vortex ring, the leap-frogging of two vortex rings, and the reconnection of two colliding vortex rings. The numerical results are in good agreement with experimental data.

(Dec. 3) Bruce Kleiner Mean curvature flow of mean convex surfaces in 3-space---
The level set flow defines, for any compact set K in R^n, a family of compact sets K_t; when the initial compact set has smooth boundary then the boundary of K_t evolves by classical mean curvature flow for short time. I will discuss the structure of singularities of the flow when n = 3 and the boundary of the initial domain is mean convex, i.e. has positive mean curvature. Roughly speaking, the results justify the naive picture of mean curvature flow where the only singularities are neck pinches and components which collapse to asymptotically round balls. This is joint work with Toby Colding.

(Dec. 10) Ping Zhang Semiclassical limit of the Gross-Pitaevskii equation in the an exterior domain---
In this talk, I will present a result on the semi-classical limit of G-P equation (a cubic nonlinear Schrodinger equation) with Neumann boundary condition in the exterior domain. We prove that before the formation of singularity in the limit system, the quantum density and quantum momentum converge to the unique solution of the compressible Euler equation with the slip boundary condition as the scaling parameter approach 0.