(September 13) Professor Israel Gohberg: Factorization of Matrix Functions and Applications - In this talk we will consider factorization of matrix functions relative to a curve which is used to solve Wiener-Hopf equations, Toeplitz equations and Riemann-Hilbert boundary value problems. The talk will contain a review with the emphasis on recent developments for rational matrix valued functions. The problem of factorization in decomposing algebras will be also considered. The state space method will be described in detail. Different applications will be presented.

(September 20) Dr. Stephen Wolfram: - Stephen Wolfram will describe ideas and discoveries from his book, A NEW KIND OF SCIENCE, their implications for various fields of science, and their personal and historical context. An extended question and answer period will be included.

(September 27) Professor Larry Shepp: The Poisson random set and its applications to wireless telephony and cosmology - The interference noise at any instant and at any frequency at the base station for mobile telephones is modelled in engineering as $Z = \sum_n \frac{X_n}{ {R_n}^\beta}$ where $R_n$ is the $n$\underline{th} largest distance from the base station and $\beta = 3.9$. Suppose $X_n$ are independent and identically distributed. The radiation on earth at any frequency due to the stars (also the total gravitational force on the earth due to the stars) is modelled by cosmologists by the same sum with $\beta = 2$. In both cases we will use a Poisson random set as the locations of the mobile phones and the stars. We will show (following Chandrasekharan, Samorodnitsky, and Taqqu) that the distribution of $Z$ s always one of a two-parameter family of distributions and is never normal.

(October 4) Professor Richard Schwartz: Complex Hyperbolic Triangle Groups - Informally, a geometric structure on a manifold is a description of how to build the manifold out of some specific geometric material. I will explain my discovery of a closed 3 dimensional manifold which admits both a hyperbolic structure and a spherical CR structure. This example is the first one known. On the one hand it can be built out of pieces of 3-dimensional hyperbolic space and on the other hand it can be built out of pieces of the 3 dimensional sphere, glued together by the restrictions of complex projective transformations. This example arises from considerations of complex deformations of the classical reflection triangle groups. In the talk I will explain these ideas more fully.

(October 11) Professor Guido Weiss: Some personal thoughts about pure and applied mathematics and science and mathematical education. - There are strong feelings about the merits of pure science and mathematics vs applied science and mathematics. I hope to present arguments that the two should go hand-in-hand, and these arguments cannot be disassociated from the question of how mathematics should be taught at all levels.

(October 25) Professor Daniel Rudolph: Applications of Orbit Equivalence to Actions of Discrete Amenable Groups - Since the work of Ornstein and Weiss in 1987 (Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987)) it has been understood that the natural category for classical ergodic theory would be probability measure preserving actions of discrete amenable groups. A conclusion of this work is that all such actions on nonatomic Lebesgue probability spaces were orbit equivalent. From this foundation two broad developements have been built.

First, a full generalization of the various equivalence theories, including Ornstein's isomorphism theorem itself, exists. Fixing the amenable group G and an action of it, one can define a metric-like notion on the full-group of the action, called a size. A size breaks the orbit equivalence class of a single action into subsets, those reachable by a Cauchy sequence (in the size) of full group perturbations. These subsets are the equivalence classes associated with the size. Each size possesses a distinguised ``most random" set of classes, the "Bernoulli" classes of the relation. An Ornstein-type theorem can be obtained. Many naturally occuring equivalence relations can be described in this way. Perhaps most interesting, entropy itself can be so described.

Second, one can use the characterization of discrete amenable actions as those which are orbit equivalent to a action of Z to lift theorems from actions of Z to those of arbitrary amenable groups. The most interesting of these are first, that actions of completely positive entropy (called K-systems for Z actions) are mixing of all orders (proven jointly with B. Weiss) and that such actions have countable Haar spectrum (proven by Golodets and Dooley). As all ergodic actions are orbit equivalent, only ergodicity is preserved by orbit equivalences in general, but by considering orbit equivalences restricted to be measurable with respect to a sub-s-algebra, many properties relative to that algebra are preserved. This provides the tool for this method to succeed.

(November 1) Yair Minsky: Thurston's ending lamination conjecture and the classification of hyperbolic 3-manifolds - The ends of an infinite-volume hyperbolic 3-manifold have a rich and mysterious geometric structure, which has been studied using methods of complex analysis, dynamics, topology and geometry. Thurston conjectured in the 1980's that this structure is completely classified by "end invariants" which describe only its asymptotic properties. Recently in joint work with J. Brock and R. Canary we were able to prove this conjecture (in the incompressible-boundary case), using in an essential way the combinatorial structure of the set of closed curves on a surface. I will give an overview of the structure of this field and of these and related developments.

(November 8) Steve Zelditch: Asymptotic Geometry of Polynomials - This talk is about the asymptotic geometry of algebraic varieties defined by systems of polynomials of a large degree N tending to infinity. We endow the space of (systems) of polynomials of degree N with a Gaussian probabillity measure and study the statistics of zeros and critical points. For instance, suppose we have a system of m polynomials in m variables, so that the joint zeros forms a discrete set. Do the zeros repel each other? or behave like particles of a neutral gas? Or clump together like gravitating particles? The answer will be given in my talk. Another issue is the effect of the Newton polytope on the distribution of zeros. It turns out that the Newton polytope gives rise to a `tunneling effect' in zeros: it creates classically allowed regions where zeros concentrate and classically forbidden regions which zeros avoid. The results will be illustrated by pictures and computer graphics. These results are joint work with P. Bleher and B. Shiffman.

(November 15) Andrzej Zuk: On the Problems of Atiyah About L^2 Betti Numbers - I will present recent results about the values of L^2 Betti numbers of closed manifolds. In particular I will present the constructions of manifolds with unexpected L^2 Betti numbers as well as possible candidates for manifolds with irrational L^2 Betti numbers.

(November 22) Professor Eitan Tadmor: High Resolution Methods for Time Dependent Problems with Piecewise Smooth Solutions - A trademark of nonlinear, time-dependent, convection-dominated problems is the spontaneous formation of non-smooth macro-scale features, like shock discontinuities and non-differentiable kinks, which pose a challenge for high-resolution computations. We overview recent developments of modern computational methods for the approximate solution of such problems. In these computations, one seeks piecewise smooth solutions which are realized by finite dimensional projections. Computational methods in this context can be classified into two main categories, of local and global methods. Local methods are expressed in terms of point-values (-- Hamilton- Jacobi equations), cell averages (-- nonlinear conservation laws), or higher localized moments. Global methods are expressed in terms of global basis functions.

High resolution central schemes will be discussed as a prototype example for local methods. The family of central schemes offers high-resolution ``black-box-solvers'' to an impressive range of such nonlinear problems. The main ingredients here are detection of spurious extreme values, non-oscillatory reconstruction in the directions of smoothness, numerical dissipation and quadrature rules. Adaptive spectral viscosity will be discussed as an example for high-resolution global methods. The main ingredients here are detection of edges from spectral data, separation of scales, adaptive reconstruction, and spectral viscosity.

(December 6) Professor John Ball: Aziz Lecture: The Regularity of Minimizers in Elasticity - It is a major open problem of nonlinear elasticity theory to decide whether or not energy minimizers are smooth or can have singularities. Although some singular minimizers related to phase transformations or fracture are known, there remains the possibility that there is a large class of realistic stored-energy functions for which minimizers are smooth. On the other hand there is apparently not a single example known of a stored-energy function for which smoothness can be proved for arbitrary large boundary data. The talk will survey what is known about this problem, and about related questions such as satisfaction of the Euler-Lagrange equation and uniform positivity of the Jacobian.

(December 13) Professor Ron DeVore: The Mathematics of Analogue to Digital Encoding - Digital Signal Processing (DSP) has revolutionized the storage and transmission of audio and video signals in consumer electronics and also in scientific settings. The main advantage of DSP is its robustness: although all of the operations have to be implemented (by necessity) in not quite ideal hardware, the a priori knowledge that all correct outcomes must lie in a very restricted set of well separated numbers makes it possible to recover them by round off appropriately.

However, many signals (audio signals e.g.) are not digital but are rather analog in nature. For this reason the first step in any digital processing of such signals is a conversion of the analog signal to the digital world. The question is then what is the most efficient method to do such a conversion. A first mathematical look would conclude the problem to be trivial: sample at Nyquist rate and encode these samples in binary. However, this is generally not done in practice. Rather engineers use a quite unexpected encoding consisting of high oversampling of the signal followed by very coarse (e.g. one bit) quantization. Such methods of encoding lead to an array of interesting mathematical questions.

This talk will discuss one bit quantization methods with an eye to explaining why engineers prefer this method. The talk requires no background in signal processing and little mathematical sophistication.

(February 7) Dr. Katepalli Sreenivasan : Nonlinear Dynamics in the Wake of Solid Objects - We show that a close connection exists between various flow properties, measured several years ago now, and an elementary dynamical system. A posteriori, it will be shown how the dynamical system can be derived from the equations of fluid motion.

(February 14) Prof. Roman A. Polyak: Nonlinear Rescaling in Constrained Optimization. (Primal, Dual and Primal-Dual aspects). - Transforming a constrained optimization problem into a sequence of constrained optimization problems goes back to R.Courant's (1943) penalty method for equality constrained optimization. The Sequential Unconstrained Minimization Technique (SUMT) became a considerable and important part of modern optimization theory after A.Fiacco and G.McCormick published their classical book in the late 60's. .It became the foundation for the recent advances in Interior Point Methods (IPM). We will discuss an alternative to the SUMT and IPM approach that is based on the Nonlinear Rescaling (NR) principle. It consists of transforming the objective function and/or the constraints of a given constrained optimization problem into an equivalent problem and using the Classical Lagrangian for the equivalent problem for both theoretical analysis and numerical methods. This nonlinear transformation is parameterized by a positive scaling parameter or by a vector of scaling parameters one for each constraint. The NR methods alternate the unconstrained minimization of the Lagrangian for the equivalent problem with Lagrange multipliers update. We will emphasize the primal and dual aspects of SUMT and NR and show their fundamental differences. In particular, we will show that the PDNR method with scaling parameter update converges with 1.5 Q-superlinear rate under the standard second order optimality conditions. Numerical results which strongly corroborate the theory will be discussed.

(February 21) Prof. Edward Swartz: Representations of matroids - What is the nature of linear independence over fields of different characteristics? For a specific vector space, what are the possible geometric point configurations? Matroids, introduced by Whitney in 1935, are a framework for answering these and other questions involving notions of independence such as algebraic independence. In the 70's researchers of real hyperplane arrangements, the simplex algorithm and directed graphs were independently and simultaneously led to oriented matroids. This combinatorial abstraction of linear independence in an ordered field can always be realized by an arrangement of pseudospheres. We now know that if we allow homotopy spheres, then all matroids have such a representation.

(February 28) Professor Alex Eskin: Billiards, Riemann Surfaces and Number Theory - Studying billiards in a polygon is one of the original problems in the theory of dymamical systems, and it still seems very difficult. However if one assumes that the polygon is rational, (i.e. all its angles are rational multiples of pi), then this problem becomes connected to many different areas of modern mathematics. I will attempt to explain some of the connections.

(March 7) Professor Alfio Quarteroni: Mathematical and Numerical Modeling of the Cardiovascular System - The use of numerical simulation in the study of the cardiovascular system (with its inherent pathologies) has greatly increased in the past few years. Blood flow interacts mechanically and chemically with vessel walls producing a complex fluid-structure interaction problem, which is practically impossible to simulate in its entirety.

Several reduced models have been developed which may give a reasonable approximation of averaged quantities, such as mean flow rate and pressure, in different sections of the cardiovascular system. They are, however, unable to provide the details often needed for understanding a local behavior, such as the effect on the shear stress distribution due a modification in the blood flow following to a partial stenosis.

The derivation of these heterogeneous models, and their coupling, will be presented together with schemes for their numerical solution. These techniques may be extended by including models for chemical transport. Several numerical results on cases of real life interest will be presented.

(March 14) Professor Sijue Wu: Recent Progress in the Mathematical Analysis of Vortex Sheets - In this talk, I will give an overview of the research for the motion of the vortex sheets, and I will speak about recent results concerning the existence and regularity of the vortex sheet solutions.

(March 21) Professor Michael Keane : On Spontaneous Emergence of Opinions - One of the distinguishing properties of the present scientific method is reproducibility.

In one of its guises, probability theory is based on statistical reproduction, near certainty being obtained of truth of statements by averaging over long term to remove randomness occurring in individual experiments.

When one assumes, as is often the case, that events farther and farther in the past have less and less influence on the present, the probabilistic paradigm is currently well understood and is successful in many scientific and technological applications.

Recently, however, we have come to realize that precisely in these applications important stocahstic processes occur whose present outcomes are significantly influenced by events in the remote past.

This behaviour is not at all well understood and some of the simplest questions remain today irritatingly beyond reach.

A salient example occurs in the theory of random walks, where there is a dichotomy between recurrent and transient behaviour.

After explaining this classical dichotomy, we present a very simple example with infinite memory which is neither known to be transient nor recurrent.

Then, using a reinforcement mechanism due to POLYA, we explain the nature of a particular infinite memory process in terms of spontaneous emergence of opinions.

Finally we would like to discuss briefly some of our recent results towards understanding the recurrence-transience dichotomy for reinforced random walks, and indicate an application to universal coding used in optical CD technology.

(April 4) Professor Roger E. Howe: Invariant Theory and Quantum Computation - Quantum computation has attracted considerable interest in recent years, since it holds out the possibility of rapid factorization, with implications for public key cryptography. Just as ordinary computation manipulates collections of binary choices, quantum computation manipulated collections of 2-dimensional quantum systems, known as "q-bits". This talk will discuss how invariant theory can shed some light on the complications of systems composed of several q-bits.

(April 18) Professor Robert Glassey: The Vlasov-Maxwell System - The Vlasov-Maxwell equations are the equations of motion for a collisionless plasma: a high temperature, low density ionized gas in which electromagnetic forces dominate collisional effects. In this lecture we will probe the major open question: does the initial-value problem have a smooth global solution for smooth data of unrestricted size? That is, are there shocks in collisionless plasmas? Partial answers and known results will be surveyed, including weak solutions, solutions with small data, the resolution of the large-data problem in two space dimensions and classical vs. relativistic formulations.

(April 25) Dr. Simon Levin: The Evolution of Biocomplexity and Biocomplexity Research - Biological systems are complex adaptive systems, in which patterns and processes on macroscopic scales emerge from the collective dynamics of individual units, and in which consequently mathematical models must relate phenomena across multiple scales. In this talk, I will discuss a range of problems from animal grouping, to the evolution of cooperation, to the emergence and resiliency of ecosystems, and illustrate some of the mathematical challenges.

I will also discuss some of the trends in research in mathematical biology over the last century, and reminisce about University of Maryland Mathematics and mathematical biology 40 years ago.

(May 09) Professor Katrin Becker: The Ground State of String Theory - In this lecture I will be giving an introduction to string theory. This is our most promising candidate to be a theory of quantum gravity which unifies all the forces in nature. String Theory has borrowed from and revolutionized many aspects of mathematics. From the physics point of view it is the only known theory that could predict quantities like the pattern of masses of the elementary particles. We will see how predictions like this could be made and how they are related to known results in mathematics.