Speaker: Vladimir Retakah, Rutgers University, Host: Landsberg
Title: Noncommutative algebra and geometry: a down-to-Earth approach

Abstract: We will consider several basic problems in noncommutative mathematics based on a notion of quasideterminant introduced by I. Gelfand and the speaker twelve years ago. The problems include: roots of noncommutative polynomials and noncommutative symmetric functions, noncommutative Plücker coordinates, relations to graph theory and an introduction to noncommutative integrable systems.

Speaker: Andrea Bertozzi, Duke University, Host: Mucha
Title: New challenges for hydrodynamics: microfluidics, imaging science, and mobile sensors

Abstract: This talk will showcase three new research areas involving mathematical fluid dynamics.

Microfluidics is a rapidly growing field being driven by new technological applications in the medical, materials, and chemical sciences. Surface tension effects (Marangoni stresses) are important on these scales. We consider the basic physics of surface tension gradients (used to move liquids) in conjunction with body forces on fluids and show that the ensuing dynamics can yield multiple shock structures involving undercompressive waves.

In the field of imaging science, Image inpainting involves filling in part of an image or video using information from the surrounding area. We introduce a class of automated methods for digital inpainting using ideas from classical fluid dynamics. The main idea is to think of the image intensity as a 'stream function' for a two-dimensional incompressible flow. The method is directly based on the Navier-Stokes equations for fluid dynamics, which has the immediate advantage of well-developed theoretical and numerical results.

An emerging area of mobile sensor control is the design of algorithms for multiple unmanned vehicles. Taking ideas from mathematical biology, we consider swarming algorithms for fluid-like motion based on simple rules for self-propulsion and local interaction. Applications range from mine detection algorithms to perimeter patrol and gradient searching.

Speaker: Professor Sasha Volberg, Michigan State University, East Lansing, Host: Lubinsky
Title: Nonhomogeneous Calderon-Zygmund operators, singular capacities, and Vitushkin's problems

Abstract: In 1888 Painleve asked for the geometric description of removable sets of bounded analytic functions. In 1960's Vitushkin introduced the quantitative measurement of removability---so-called analytic capacity. It was a capacity only by name. For example, it was not known whether this quantity is subadditive. In the last decade there was a lot of activity around this problem, which turned out to be related closely to geometric measure theory (P. Jones' traveling salesman problem, for example), and to theory of Singular Integral Operators with degeneracies. We will show how the solution of Vitushkin's problem obtained recently by Tolsa is based on Calderon-Zygmund theory with degeneracies. We will also show that one can get rid of the specifics of the complex plane and Cauchy kernel to get a general statement about some singular capacities. It turns out that capacities with Calderon-Zygmund kernels behave essentially as classical capacities with positive kernels.

Speaker: Percy Deift, Host: Geronimo
Title: Toeplitz and Hankel determinants, their prevalence, and their asymptotic evaluation

Abstract: An extraordinary variety of quantities of physical and/or mathematical interest can be expressed in terms of a Toeplitz or Hankel determinant. Such representations arise in the moment problem, in quantum and classical physics, as well as various branches of chemistry. Most often the issue at hand is the asymptotic behavior of the quantity of interest as some parameter in the problem becomes large. In this talk the speaker will describe some recent developments in the asymptotic evaluation of Toeplitz and Hankel determinants using Riemann-Hilbert techniques. Universal aspects of the asymptotic behavior will also be discussed.

Speaker: Percy Deift, Host: Geronimo
Title: Universality for mathematical and physical systems

Abstract: All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. The speaker will recount some of the history of these ideas starting with Wigner's model for the scattering of neutrons and how they have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting. The talk is aimed for the non-specialist.

Speaker: Leonid Bunimovich, Georgia Tech, School of Mathematics
Title: Kinematics, equilibrium and shape: lab effect

Abstract: Consider a gas of N identical particles inside a container. Suppose that all particles were initially located in the left half of the container. Then we would expect that after a sufficiently long time the particles will occupy entire container and will be distributed there uniformly. It occured that both these expectations are generically not true. This seems to contradict to the basic principles of statistical mechanics but, in fact, it does not. This effect of the container's shape is likely to be observable in a lab.

Speaker: Dr. Julie Mitchell (Job Candidate Seminar), San Diego Supercomputer Center, University of California at San Diego, Host: Shonkwiler

Title: Using Mathematics to Predict Protein Interactions

Abstract: The human genome contains instructions for encoding thousands of different proteins, and these proteins perform many important biological functions. A protein is a linear sequence of amino acids that folds into a distinctive three-dimensional structure. Once the protein has folded, it can perform its intended function(s) through interaction with proteins, DNA, small molecules or ions. For two molecules to interact, they must bind to form a single unit, called a "bound complex." The lecture will discuss mathematical approaches to predicting the bound complex between a pair of proteins or other macromolecules. Two important factors determining how molecules can interact are their electrostatic and shape properties, and the Docking Mesh Evaluator and Fast Atomic Density Evaluation programs were developed in order to study these properties.

The Docking Mesh Evaluator (DoME) program uses adaptive mesh solutions to the Poisson-Boltzmann equation to model electrostatic interactions. The Poisson-Boltzmann equation is a nonlinear partial differential equation that treats water and solvent as a continuum. This time-averaged "implicit solvent" approach is more computationally efficient than direct electrostatic calculation with solvent ions and water molecules. DoME is presently in development, and initial results for its parallel scanning and optimization capabilities will be given.

The Fast Atomic Density Evaluation (FADE) program calculates molecular shape features of an individual protein or shape complementarity for bound complexes. FADE is based on atomic density methodology, and it uses Fast Fourier Transforms and convolution integrals for rapid calculation. FADE has been successfully used in the prediction of bound protein complexes, and the complementarity markers found by FADE correlate well with experimental mutation data for several studied examples.

Speaker: Gui-Qiang Chen, Northwestern University, Host: Su

Speaker: Blain Lawson, Host: Landsberg
Title: Singularities and Chern-Weil Theory
Abstract

Fall 2002

Applied Mathematics | CDSNS Colloquium | CDSNS/ACELab | Colloquium | Combinatorics | Geometry | Geometry/Topology/Algebra | Nonlinear Science | Research Horizons | Stochastics | Topology |