(Sept 3) R. Pego, UMCP: Gronwall inequalities, Wasserstein metrics, measure-valued solutions, and domain coarsening --  The classical theory of domain coarsening during phase transitions in materials science involves a conservation law for the size distribution of a family of particles.  I'll describe a theory of well-posedness for the initial value problem that: (a) admits arbitrary probability measures for initial data; (b) obtains existence, uniqueness, and continuous dependence on initial data for measure-valued weak solutions; and (c) uses a physically natural topology on the space of size distributions, given mathematically by aWasserstein distance between probability measures.  Wasserstein distances are related to the classic Monge problem of efficient transport; they measure the size changes required to rearrange one distribution into another.  Some charming variations of Gronwall's inequality are needed to handle some special difficulties. (This is joint work with Barbara Niethammer, University of Bonn. A preprint is available from the Los Alamos e-print server.)

(Oct 1) Prof. Tom Seidman, UMBC:  A convection-reaction problem with discontinuous dynamics ---
 We consider a model of bioremediation whose nature is characterized by discontinuous transitions of the bacteria between`dormant' and `active' states, determined hysteretically by the concentration of a soluble critical nutrient. Our object is to show, for a nondispersive spatial coupling (pure convection), well-posedness in some sense appropriate to the consideration of this as a distributed parameter optimal control problem.

(Oct 8) O. Lakkis, UMCP:  Existence of solutions for a class of semilinear polyharmonic equations with critical exponential growth --- We consider the semilinear equation (-\Delta)^m u = g(x,u), subject to Dirichlet-type boundary conditions on a bounded domain in R^2m. The notion of nonlinearity of critical growth is for this problem is introduced. It turns out that the critical growth rate is of exponential type and the problem is closely related to the Trudinger embedding and Moser-type inequalities. The main result is the existence of non-trivial weak solutions to the problem.

(Oct 15) A. Lifschitz-Lipton, Banker's Trust NY& U Illinois:  Stability of fluids and plasmas and spectral theory ---In this talk we discuss several important stability problems arising in hydrodynamics and magneto-hydrodynamics and show how they can be solved via methods of modern spectral theory. In particular, we analyze the spectra and stability of elliptic and hyperbolic plane-parallel fluid and plasma flows, and Riemann ellipsoids.

(Oct 16) A. Lifschitz-Lipton, Banker's Trust NY& U Illinois:  Linear and nonlinear pricing problems in mathematical finance --- In the present talk we discuss several interesting and important problems of mathematical finance and show how they can be solved in the framework of stochastic calculus and partial differential equation theory.

(Oct 22) A. L. Krylov, Earth Physics Institute, Russian Academy of Science: Stochastic soliton lattice ---
We introduce a new concept, Stochastic Soliton Lattice, as a random process generated by a finite-gap potential of the Schroedinger(Sturm-Liouville) operator. We study its basic properties and consider the Korteweg-de Vries evolution of this stochastic process.
 
(Oct. 29) C. Elmer, NIST: Analysis and computation of traveling wave solutions for bistable nonlinear differential-difference equations --- In this talk we consider traveling wave solutions to nonlinear bistable differential-difference equations. We study propagation failure, lattice anisotropy, bifurcation points in the wave speed and detuning parameter relation, and step-like solution profiles.    Traveling wave solutions satisfy a two-point boundary value problem but standard BVP codes do not handle the discrete difference operator. We present two relaxation techniques to deal with the delay terms.  The first is fixed point iteration which allows us to solve equations where we represent the bistable nonlinearity with a piecewise linear function. The second is a variation of Newton's method which allows us to solve equations containing smooth bistable nonlinearities.  We also present a series of numerical examples along with current and future research directions.

(Nov. 5) P. Milewski, U Wisconsin, Madison: Effects of a Varying Bottom on Nonlinear Surface Waves ---
Waves on the surface of a fluid can be greatly affected by variations in the bottom topography. Particularly interesting are cases where the topography couples with nonlinear effects either by enhancing the strength of these or by serving as a catalyst for new effects. Three examples shallow water will be presented: 1) The resonant forcing of gravity and gravity--capillary waves by flow over a bump. 2) The interaction of periodic wavetrains on a gently sloping beach. 3) The resonant interaction of waves over periodic topography. Some aspects of the application of these effects to geophysical flows (shallow water in the presence of rotation) will be discussed.

(Nov. 19) J.-R. Quintero, UMCP: Three-dimensional long water waves with small amplitude --- In this talk, first we will discuss briefly how to describe the evolution of long water waves with small amplitude as an approximation of the full water wave problem. We will see that for three-dimensional water waves with surface tension this reduces to studying the solution of an isotropic Benney-Luke equation. Second, we will discuss the relationship between the Benney-Luke equation and the Korteweg-de Vries (KdV) equation and the Kadomtsev-Petviashvili (KP) equation when the amplitude and the long-wave parameters are small. Third, we will discuss which solutions of the Benney-Luke equation have relevance from the physical viewpoint as the parameters tend to zero. And finally, we will describe how to obtain 2-D traveling waves for the Benney-Luke equation by using the concentration-compactness method and the center manifold method, depending upon the traveling wave speed and the Bond number.

(Dec. 3) G. Domokos, Cornell: Symbolic dynamics of infinite depth: finding global invariants for BVPs --- Continuing earlier research by Domokos & Holmes, we explore the global bifurcation diagrams of elastic chains, serving both as mechanical and  mathematical discretization of Euler's classical beam buckling boundary value problem. Moreover, the mapping describing the chain's physical shape turns out to be the standard map. Using properties of the latter, coupled with ideas from mechanics and global bifurcation theory, we give an explicit construction for unique integer labels that can be attached to each branch of the extremely complicated bifurcation diagram. These labels (which are actually suitably chosen symbolic dynamics of the standard map with finite length but infinite depth) turn out to be useful not only as global invariants for understanding the structure of the bifurcation diagrams, but also for identifying self-similar patterns for stable solutions and for numerically "safe" path continuation.

(Dec. 10?) W. Gangbo, Georgia Tech:  Wasserstein distance and its applications ---
The Kantorovich-Rubinstein-Wasserstein metric defines the distance between two probability measures $\mu$ and $\nu$ on ${\bb R}^{d+1}$ by computing the cheapest way to transport the mass of $\mu$ onto $\nu$,  where the cost per unit mass transported is a given function $c({\bf x},{\bf y})$ on ${\bb R}^{2d+2}$.  Motivated by applications to shape recognition, we analyze this transportation problem with the cost $c({\bf x},{\bf y}) = |{\bf x}-{\bf y}|^2$ and measures supported on two curves in the plane,  or more generally on the boundaries of two domains $\Omega, \Lambda \subset {\bb R}^{d+1}$. Unlike the theory for measures which are absolutely continuous with respect to Lebesgue,  it turns out not to be the case that $\mu$-a.e.\ ${\bf x} \in \partial \Omega$ is transported to a single image ${\bf y} \in \partial \Lambda$; however, we show the images of ${\bf x}$ are almost surely collinear and parallel the normal to $\partial \Omega$ at ${\bf x}$.  If either domain is strictly convex, we deduce that the solution to the optimization problem is unique.  When both domains are uniformly convex,  we prove a regularity result showing the images of ${\bf x} \in \partial \Omega$ are always collinear,  and both images depend on ${\bf x}$ in a continuous and (continuously) invertible way. This produces some unusual extremal doubly stochastic measures.