(October 1) Mike Reed: Probability and Neurobiology - Neurons are inherently variable devices and therefore an important general question in the central nervous system is how we can make accurate calculations with (large numbers of) imperfect devices. This question is particularly pertinent in the auditory system where behavioral studies show that mammals can detect extremely small time differences. After a brief introduction to the auditory system, experimental data will be presented that show that the question is serious indeed. A natural abstraction of the question leads to a (not so simple) problem in probability theory for which analytical and computational results (with Colleen Mitchell) will be presented. Recent theoretical results suggest that the behavior of some neurons is similar to the behavior of the abstract neurons.

(October 8) Richard Schwartz: Reflections, Polygons, and GUIs - In this lecture I will demonstrate some graphical user interfaces I've made. These interfaces are designed to explore and/or illustrate geometric phenomena related to the basic operation of reflecting objects in the sides of a polygon. I will concentrate on two examples:

-Lucy and Lily

-McBilliards

Lucy and Lily is an internet computer game which gives a geometric illustration of some basic ideas in Galois theory. McBilliards is a program which searches for periodic billiards paths in triangles. Using McBilliards I can make substantial progress on the 200 year old triangular billiards conjecture. If times permits, I will also demonstrate some other examples.

(October 22) Sinai Robins : Traditional Dedekind sums and new polynomial Dedekind sums for weighted lattice point enumeration in polytopes - We first give an overview of the history of Dedekind sums, their role in number theory, and their more recent utility in the lattice point enumeration of polytopes. Intuitively, we may think of Dedekind sums as the building blocks of the Discrete volume of a polytope. We next define new types of Dedekind sums which play an important role in evaluating sums of polynomial weights at all lattice points inside rational polytopes. Such sums appear, for example, in the numerical analytic-theory of Euler-MacLaurin summation, and in the combinatorial-geometric theory of lattice polytopes. We find reciprocity laws for the polynomial Dedekind sums in certain cases, which allow us to compute them in linear time. This is joint work with Helaman Ferguson.

(October 29) Larry Shepp: Applications of higher dimensional brownian motion to finance and biology - At each time, t, the value of each of two companies under your control varies as an independent standard Brownian motion, but if the value of a company ever becomes zero, the company dies (you cannot transfer funds). If you devote effort to one or the other company you can add a unit upward drift to its value and can move your effort instantaneously from one to the other company. David Aldous conjectured that the optimal strategy for maximizing the mean number of companies that survive forever is to always push the bottom company. Henry McKean showed that Aldous's strategy does maximize the probability that both companies last forever. Socialism is optimal for this criterion! But this strategy seems to be non-optimal for Aldous's criterion, so maybe capitalism is better after all???

Proteins are supposed to diffuse as three dimensional standard Brownian motions starting from fixed points (Nizar Batada and David Siegmund, PNAS, to appear). Two different proteins are born at fixed locations, distant r apart, at Poisson times and die at independent exponential times. We found the chance that the two Brownian paths ever come within epsilon of each other before they die. This is believed to be a good model for protein-protein bond exchange in interior cell dynamics.

Receptor sites on the surface of a spherical cell diffuse as Brownian motions on the surface of the sphere and also exchange chemical bonds when they meet within epsilon. We compute the rate of meetings in a similar way for this geometry in a paper in progress.

(November 5) John D. Weeks: Screening, Structure and Simulation of Ionic Fluids: The Long and Short of It - Charged ions and dipoles in dielectric solvents play an important role in many chemical and biological processes. We describe a new theoretical approach to such systems by exactly separating the point charge Coulomb interaction into a long-ranged and slowly varying perturbation part u1(r), which we view as arising from a rigid Gaussian charge distribution with width sigma, and the remainder u0(r), which is short-ranged and can be added to the other short-ranged core interactions. When sigma is chosen sufficiently large, the perturbation potential u1 is very slowly varying on the scale of short ranged intermolecular correlations, and its averaged effects can be taken into account in a controlled way by using spatially varying effective fields. The resulting local molecular field theory is exceptionally accurate both for uniform and nonuniform fluids. Characteristic phenomena in strongly coupled uniform ionic fluids such as ion pairing at low density and charge ordering at higher density can be seen in a simpler mimic system with only short-ranged interactions. At very low densities and weak coupling the theory reduces to the exact Debye-Hckel theory. Ions near charged walls can be similarly accurately described. Relations to other treatments of long-ranged interactions such as Ewald sum and fast mutipole methods are briefly discussed.

This work is supported by the National Science Foundation under grant NSF CHE0111104.

(November 12) A. Tzavaras: Entropy methods in hyperbolic systems - The notion of entropy is motivated by the second law of thermodynamics and has played an important role in the theory of hyperbolic systems of conservation laws. The term has various uses In the mathematical literature usually connected with additional conservation laws that the system satisfies. In this talk i will review various uses of entropies: (i) The notion of relative entropy and how it is used to assess the stability properties of approximating theories such as viscosity or relaxation theories. (ii) the role that nonlinear transport equations play in structural properties of polyconvex elastodynamics. (iii) The representation of entropy pairs through the kinetic formulation and their role in the study of oscillations for systems of two conservation laws, and the existence of weak solutions in the functional framework of the energy norm.

(November 19) Micheal Vogelius: Electromagnetic Imaging for small inhomogeneities - Electromagnetic Imaging in this context refers to the identification of internal characteristics of a medium based on boundary (or near-field) measurements of the electric and/or magnetic fields. After a brief review of some of the main mathematical results in Electromagnetic- and Impedance Imaging (from the last 20 years, or so) I shall proceed to discuss some very recent, extremely efficient representation formulas that lead to a surprisingly accurate identification of the size, and the location of relatively small inhomogeneities.

These representation formulas take into account polarization effects, and they may be derived by variational techniques related to $H-$ (or $\Gamma-$) convergence. The magnitude of the polarization effects may be estimated in ways that are very reminiscent of effective media bounds (of the Hashin-Shtrikman type). A precise assessment of the polarization effects is very important for highly accurate size estimates.

Finally, these representation formulas lend themselves very naturally to the application of reconstruction methods of a linear sampling- or MUSIC (MUltiple SIgnal Chararcterization) character. On this matter I shall discuss some general ideas, and implementation issues, as well as provide examples of computational reconstructions.

(February 4) Teresa Przytycka : Graph Theoretical Insights into Protein Evolution - Proteins are polymers responsible for performing majority of functions in a living cell. Most domains can be divided into two or more evolutionary units called domains. Each domain is subject to local evolutionary changes (mutations) while larger scale evolution of multidomain proteins involves gaining/loosing and exchanging whole domains. In this work we focus on understanding of such large scale protein evolution.

In order to focus on the properties of multidomain proteins and the relationships between them, we introduce and study graph theoretical representation of multidomain proteins called domain overlap graph. In the domain overlap graph, the vertices are protein domains and two domains are connected by an edge if there is a protein that contains both domains. We demonstrate how properties of this graph such as chordality and the Helly property can indicate various evolutionary mechanisms. We apply our graph theoretical results to address several interrelated questions: do proteins acquire new domains infrequently, or are common enough that the same combinations of domains will be created repeatedly through independent events? Once domain architectures are created, do they persist? In other words, is the existence of ancestral proteins with domain compositions not observed in contemporary proteins unlikely?

(February 11) Michal Misiurewicz: Branched derivatives - The notion of a derivative is central to the whole of Mathematical Analysis. In particular, if one iterates a smooth map in a neighborhood of its fixed point, the derivative at this point determines the dynamics (at least in generic cases). For a branched map of a plane to itself, in order for a derivative to exist at a branching point, this derivative has to be zero. In a joint paper with Alexander Blokh, we define a derivative-like notion in order to remove this restriction. The results are especially interesting if the map locally preserves area.

(February 25) Bard Ermentrout: Learning Synchrony Oscillations and spike-time dependent plasticity - Abstract: In this talk, I discuss ways that the nervous system adjusts interactions in a such a way as to foster synchrony. I first discuss methods for changing intrinsic parameters (frequency) and apply it to the synchronization of the the SE Asian firefly P. mallacae. I use perturbation methods to show that the mechanism leads to synchronous (zero phase lag) locking. Next I show some consequences of spike-time dependent plasticity (STDP) in networks of spiking neurons. I show that the empirical (experimentally determined) rules along with some resonable assumptions about the effects of excitation will result in synchrony. I use the fact that there will be a large dimensional center manifold which enables an infinite number of solutions to the synchrony problem

(March 4) Alexander Kechris: Logic, Ramsey theory and topological dynamics - In this talk, I will discuss some recently discovered interactions between topological dynamics, concerning the computation of universal minimal flows and extreme amenability, the Frass theory of amalgamation classes and homogeneous structures, and finite Ramsey theory.

(March 11) Ivo Babuska: What I Know and Would Like to Know About The Meshless and Generalized FEM - Abstract: Recently the Meshless and Generalized FEM became to be in the center of interest, especially in the engineering community. I have identified more than 200 engineering papers and four books ([1],[2],[3],[4]). I found very few papers of mathematical character (assumptions, theorems, proofs). Hence there is a large experience and heuristics but not too much mathematical understanding of the methods and their range of the applicability and the performance.

There are four basic directions:

1. The construction of the meshless and generalized FEM spaces and their approximation properties.

2. The discretization of the elliptic, parabolic and hyperbolic equations (solids, fluids etc) and its convergence.

3. The Implementation.

4. Applications.

The talk will address various problems of these categories with the goal to prioritize the major mathematical questions.

1a. Translation invariant single shape functions and their approximation properties were analyzed in 1970. The case of multiple shape functions is still open. These functions are used and there is numerical experience and conjectured statements.

1b. Construction of the single and multiple shape functions is based on various assumptions, which are not exactly specified or are very hard to verify. The approximability properties and their robustness in various norms for the classes of the approximated functions, for example classes of functions, which satisfy some differential equations, are still open problem.

1c. Construction of the spaces when various constrains conditions have to be satisfied.

1d. Problem of the adaptive construction of the spaces of the meshless functions.

2a. Various discretization procedures based on the Petrov Galerkin approach, when the trial and the test functions are different, (it contains for example various collocation methods) are suggested in the engineering literature and tested on numerical examples Nevertheless there is very little mathematical understanding on the convergence etc.

2b. Mixed problems where the variational form is of the saddle point type.

2c. Superconvergence and interior estimates theory.

2d. The locking problems or in general solving the problem which are nearly degenerated.

2e. Problems of large displacements as in the crash problems.

2f. Problems with changing domains or problems when the boundary has to be found as in the freezing, seepage problems etc.

2g. Problems of a-posteriori error estimates in different norms or for the quantities of interest.

2h. Problems of fluid mechanics, mathematical theory of the Smoothed Particle Hydrodynamics.

3a. Problem of numerical integration.

3b. Problem of matrix conditioning.

3c. Problem of solving large system, multigrid and other iterative methods.

3d. Problem of nonlinear solvers, turning and bifurcation problems.

4a. Generalized FEM for highly heterogeneous materials and lattices. Adaptive approaches for quantities of interest.

4b. Methods when very smooth test functions are needed as when the boundary conditions are very rough.

4c. Problems on very complicated domains and the multisite problems.

4d. Characterization of the problem when the meshless and generalized FEM are superior to the classical approaches.

Some typical problems in the above mentioned classes will be presented.

References.

[1] Atluri, S.N., The Meshless Method for domains and BIE discretization. Tech Sci Press 2004, 680 pages.

[2] Atluri, S.N. and Shen, S., The Meshless Method. Tech Sci Press 2002, 430 pages.

[3] Liu, G.R., Mesh Free Method, CRC Press 2003, 680 pages.

[4] Li, S. and Liu, W.K. Meshfree Particle Method, Springer 2004, 500 pages.

(March 18) Elon Lindenstrauss: On measures invariant under tori and other actions - An important open problem in the theory of flows on locally homogeneous spaces is the classification of measures invariant under actions of multidimensional diagonalizable groups. A similar classification due to Ratner for measures invariant under actions of unipotent groups has had numerous applications in many areas, and already the partial results we have towards measure classification for diagonalizable actions have found some applications in number theory and arithmetic quantum chaos.

I will survey some recent results regarding these invariant measures and their applications.

(April 1) Burkhard Wilking: Fundamental groups of manifolds with lower Ricci curvature bound - In general the class of compact manifolds with given lower sectional > curvature > bound is much more rigid than the class of compact manifolds with a > lower Ricci curvature bound. However there is a general belief that > the same structure theorems for fundamental groups should hold. > In this talk (joint work with V. Kapovitch) I will present > several results which confirm this philosophy.

(April 15) Cathleen Morawetz: Conservation Laws and Some Consequences - In this talk we shall begin with simple linear and nonlinear examples which under a stretching of the independent variables produce another solution.One can then obtain time decay from the usual energy estimate. But more general examples of time decay based on invariance are found by Noether's theorem. From there one can move to some 2D results for equations that are not necessarily hyperbolic such as the Tricomi equation, the hyperbolic wave equation and also Born Infeld equations.

(May 6) Yann Brenier: String integration of some MHD equations - We first review the link between strings and some Magnetohydrodynamics equations. Typical examples are the Born-Infeld system, the Chaplygin gas equations and the shallow water MHD model. They arise in Physics at very different (from subatomic to cosmologic) scales. These models can be exactly integrated in one space dimension by solving the 1D wave equation and using the d'Alembert formula. We show how an elementary "string integrator" can be used to solve these MHD equations through dimensional splitting. A good control of the energy conservation is needed due to the repeted use of Lagrangian to Eulerian grid projections. Numerical simulations in 1 and 2 dimensions will be shown.