The Fermi-Pasta-Ulam (FPU) model of coupled anharmonic oscillators has long been of interest in nonlinear science. It is only recently (Friesecke and Wattis 1994, Frieseck and Pego 1999-2003, and Mizumachi (submitted)) that the existence and stability of solitary waves in FPU has been completely understood. In light of the fact that the Korteweg-deVries (KdV) equation may recovered as a long wave limit of FPU and that the theory of soliton interaction is both beautiful and completely understood in KdV, it is of interest to describe the interaction of two colliding solitary waves in the FPU model. We show that the FPU model contains an open set of solutions which remain close to the linear sum of two long wave low amplitude solitions as time goes to either positive or negative infinity.

In order to construct a viable model of string theory, one seeks to build Calabi Yau threefolds with prescribed conditions. Borcea and Voisin were able to built a family of Calabi-Yau threefolds using elliptic curves and K3 surfaces admitting non symplectic involutions. We will display how the construction can be generalized by studying higher order non symplectic automorphisms on K3 surfaces

A Zoll metric is a metric whose geodesics are all circles of equal length. In this talk, we will first review the definition of the twistor correspondence of LeBrun and Mason for Zoll metrics on the sphere $S^{2}$. It associates to a Zoll metric on $S^{2}$ a family of holomorphic disks in $CP_{2}$ with boundary in a totally real submanifold $P\subset CP_{2}$. For a fixed $P\subset CP_{2}$, we will indicate how one can show that such a family is unique whenever it exists, implying that the twistor correspondence of LeBrun and Mason is in some sense injective. One of the key ingredients in the proof will be the blow-up and blow-down constructions in the sense of Melrose.

I'm sure it will be great!

We consider variations of the usual voter model, which favor types that are locally less common. Such voter models with selection are dual to systems of branching annihilating random walks that are parity preserving. We consider coexistence of types in the voter models which is related to the survival of particles in the branching annihilating random walk. We find conditions for the uniqueness of a homogeneous coexisting invariant law as well as for convergence to this law from homogeneous and coexisting initial laws. For a particular one dimensional model we also show a complete convergence result for any initial condition. This is based on comparison with oriented percolation of the associated branching annihilating random walk.

A bihermitian structure is a Riemannian metric compatible with two distinct orthogonal complex structures. In the mathematical literature this notion appeared in 90's in the study of the curvature of conformal 4-manifolds. However, bihermitian metrics were already studied in the physics literature in the 80's, as a building bloc of what Gates, Hull and Rocek call `the target space for a (2,2) super-symmetric sigma model'. There has been a great deal of interest in bihermitian geometry more recently, motivated by its link with the notion of generalized Kaehler geometry, introduced by Gualtieri and Hitchin. In this talk I will explain some main features of 4-dimensional bihermitian manifolds, as developed in the 90's, and report on recent classification results that I obtained with M. Gualtieri and G. Dloussky.

For many biological systems, the timescale at which ecological interactions occur is much shorter than the timescale at which evolutionary changes occur. For rapidly evolving pathogens such as influenza, however, this is not the case; influenza researchers therefore need to understand both the ecological interactions between the host and the pathogen and the virus?s evolutionary changes in order to ultimately control the disease in humans. Recently, a study looking at the evolutionary patterns of influenza showed that, while the virus?s genetic evolution occurred gradually, its antigenic evolution occurred in a punctuated manner. (Genetic evolution refers to how the virus?s nucleotides change over time; antigenic evolution refers to how the virus changes over time with respect to how our immune system recognizes it.) Previous research from our group hypothesized that these differences in evolutionary patterns could be explained by the presence of /neutral networks/ in the virus?s genotype space: networks of sequences that differ genetically from one another but fold into the same protein conformation and thereby share antigenic properties. Here, I will present a simple epidemiological model that implicitly incorporates these neutral networks. I show that this model can reproduce (1) the seasonal and interannual outbreak patterns of influenza, (2) the quantitative patterns of influenza?s antigenic evolution, and (3) the patterns of the virus?s genetic evolution, including its characteristic phylogenetic tree. I end with how this model may be useful in understanding patterns of viral diversity in other host species (e.g., avian and equine hosts).

In many problems in fluids and electromagnetics, we may formulate solutions to the Dirichlet and Neumann problems in terms of double and single layer potentials. Such boundary integral representations can result in computational difficulties at points on and near the boundary due to singularities and near singularities. The case of a smooth closed boundary has been well-studied, but I will focus on computational issues that arise from a boundary that is only piecewise smooth, consisting of connected curve segments. I will give an overview of my research in approximating singular and nearly singular integrals, as well as discuss an approach for the computation of double layer potentials at points on and near a curve segment.

THIS JUST IN - An Abstract: I plan to speak about the good old Calabi Conjecture, and its beautiful solution by Yau, that gave gim the Fields Medal. I will start with some background material, and see how far we can get into the proof!

The talk will contain a review of several recent results on the analytic continuation of the random analytic functions. We will start from the classical theorem on the random Taylor series (going to Borel’ s school), but the main subject will be the random zeta – function (which was introduced implicitly by Cramer) and its generalizations. We will show that “true primes are not truly random “, since zeta – functions for the random “pseudo-primes” (in the spirit of Cramer) have no analytic continuation through the critical line Re (z) = 1/2.

The simplest model of vortex filament motion in an ideal fluid leads to an integrable nonlinear evolution equation, known as the localized induction approximation or the vortex filament flow, closely related to the cubic focusing nonlinear Schroedinger equation. For closed finite-gap solutions of this flow, certain geometric and topological features of the evolving curves appear to be correlated with the algebro-geometric data used to construct them. In this talk, I will briefly discuss this construction, and some low-genus examples (in particular, Kirchhoff elastic rod centerlines) where this correlation is well understood. I will mainly discuss recent joint work with Annalisa Calini, describing how to generate a family of closed finite-gap solutions of increasingly higher genus via a sequence of deformations of the multiply covered circle. We prove that every step in this sequence corresponds to constructing a cable on previous filament; moreover, the cable's knot type (which is invariant under the evolution) can be read off from the deformation sequence.

There are many different ways to make a compact 2-manifold of genus g into a Riemann surface. In fact, there is an entire space of dimension 3g-3 (when g>1) of possible holomorphic structures. This space is called the moduli space of Riemann surfaces of genus g. We will give a definition of moduli spaces and briefly talk about their construction, starting with the "easy" examples of g=0 and g=1. We will also talk about mapping class groups, which play an important part in the construction of moduli spaces.

In this talk we present a recent result on the smoothness of the density for the solution of a semilinear heat equation with multiplicative space-time Gaussian white noise. We assume that the coefficients are smooth and the diffusion coefficient is not identically zero at the initial time. The proof of this result is based on the techniques of the Malliavin calculus, and the existence of negative moments for the solution of a linear heat equation with multiplicative space-time white noise.

We argue that there exists a derived equivalence between Calabi-Yau threefolds obtained by taking hyperplane sections (of the appropriate codimension) of the Grassmannian G(2,7) and the Pfaffian Pf(7). The existence of such an equivalence has been conjectured in physics for almost ten years, as the two families of Calabi-Yau threefolds are believed to have the same mirror. It is the first example of a derived equivalence between Calabi-Yau threefolds which are provably non-birational.

Another Job Audition.

The Jacobian of any compact Riemann surface carries a natural theta divisor, which can be defined as the zero locus of an explicit function, the Riemann theta function. I will describe a generalization of this idea, which starts by replacing the Jacobian with the moduli space of higher rank bundles. These moduli spaces also carry theta divisors, described via "generalized" theta functions. In this talk, I will describe recent progress in the study of generalized theta functions.

The first step in finding a spacetime solution to the Einstein gravitational field equations via the inital value formulation is to construct initial data which satisfy the Einstein constraint equations. There are three ways of carrying out this construction which have been found to be useful: the conformal and conformal thin sandwich methods, the gluing techniques, and the quasi-spherical approaches. We describe each of these, we discuss their advantages and disadvantages, we outline some of their recent successful applications, and we present some of the outstanding questions remaining to be solved from each of these perspectives.

We present experiments on pattern formation and front propagation in the Belousov-Zhabotinsky (BZ) chemical reaction in flowing systems with chaotic advection. The flow is a chain of alternating vortices that oscillate and/or drift in the lateral direction. Mixing between the vortices is chaotic in this flow with either (enhanced) diffusive or superdiffusive transport. Experiments with the excitable BZ reaction are used to study the motion of reaction fronts in this system. If the vortices oscillates laterally, reaction fronts typically mode-lock to the external forcing. If the vortices drift with constant velocity, fronts typically pin to the leading vortex, remaining motionless in a reference frame that drifts with the vortices. Experiments with the oscillatory BZ reaction are used to study synchronization of a network of oscillators by chaotic mixing. We find that the system is globally-synchronized only if the long-range transport is superdiffusive, characterized by Levy flight trajectories. Time-permitting, we will also present results of experiments on chemical fronts and patterns in a two-dimensional array of vortices.

Coarsening is the phenomenon where many objects (water drops, molecular islands, particles in a freezing liquid) becoming a smaller number of objects in an orderly way. This talk will examine modeling one such system, tiny liquid drops, through three models: a PDE for the fluid, a coarsening dynamical system for the drops, and an LSW-type ensemble model for the group of drops. We will study self-similarity in the dynamics and extensions of the model to examine very long times when drops grow large enough that gravity distorts their shape.

Reaction-diffusion equations are used in mathematical models of many physical and biological phenomena involving front propagation and pulse propagation. How do variations in the environment effect these phenomena? In this seminar, I will describe recent progress in understanding how fronts propagate in heterogeneous media. In particular, I will describe properties of generalized traveling waves for one-dimensional reaction-diffusion equations with variable excitation. I also will discuss multi-dimensional fronts in stationary random media, a model relevant to premixed-turbulent combustion. Along the way, I plan to describe interesting topics for future research.

Social networks are often represented by directed graphs where the nodes are individuals and the edges indicate a form of social relationship. A simple way to model the diffusion of ideas, innovative behavior, or word-of-mouth effects on such a graph is to consider a stochastic process of ``infection'': each node becomes infected once an activation function of the set of its infected neighbors crosses a random threshold value. I will prove a conjecture of Kempe, Kleinberg, and Tardos which roughly states that if such a process is ``locally'' submodular then it must be ``globally'' submodular on average. The significance of this result is that it leads to a good algorithmic solution to the problem of maximizing the spread of influence in the network--a problem known in data mining as "viral marketing"'. This is joint work with Elchanan Mossel.

Yau's theorem on Kahler manifolds states that there exists a unique Kahler metric in every Kahler class with prescribed volume form. This has many applications in complex geometry. I will discuss symplectic manifolds. In a different direction, I will talk about the problem of existence of constant scalar curvature Kahler metrics, which can also be considered a generalization of Yau's theorem.

This work is motivated by a fundamental problem in sensor networks -- the need to aggregate redundant sensor data across a network. We focus on a simple problem of enumerating targets with a network of sensors that can detect nearby targets, but cannot identify or localize them. We show a clear, clean relationship between this problem and the topology of constructable sheaves. In particular, an integration theory from sheaf theory that uses Euler characteristic as a measure provides a computable, robust, and powerful tool for data aggregation.

The probabilistic method has redefined functional analysis in high dimensions. Random spaces and operators are to analysis what random graphs are to combinatorics. They provide a wealth of examples that are otherwise hard to construct, suggest what situations we should view as typical, and they have far-reaching applications, most notably in convex geometry and computer science. With the increase of our knowledge about random structures we begin to wonder about their universality. Is there a limiting picture as the dimension increases to infinity? Is this picture unique and independent of the distribution? What are deterministic implications of probabilistic methods? This talk will survey progress on some of these problems, in particular a proof of the conjecture of Von Neumann and Goldstine on random operators and connections to the Littlewood-Offord problem in additive combinatorics.

My research focuses in three areas of evolutionary biology: the structure of viral populations, the evolution of drug resistance, and phylogenetics. Knowledge of the diversity of viral populations is important for understanding disease progression, vaccine design, and drug resistance, yet it is poorly understood. New technologies (pyrosequencing) allow us to read short, error-prone DNA sequences from an entire population at once. I will show how to assemble the reads into genomes using graph theory, allowing us to determine the population structure. Next, I will describe a new class of graphical models inspired by poset theory that describe the accumulation of (genetic) events with constraints on the order of occurrence. Applications of these models include calculating the risk of drug resistance in HIV and understanding cancer progression. Finally, I'll describe a polyhedral method for determining the sensitivity of phylogenetic algorithms to changes in the parameters. We will analyze several datasets where small changes in parameters lead to completely different trees and see how discrete geometry can be used to average out the uncertainty in parameter choice.