Topic: Convex Optimization: Interior-point methods and Applications April 20

Presenter: Robert Vanderbei, Princeton University

ABSTRACT: Fifteen years ago, interior-point methods revolutionized the subject of linear programming (LP) by providing the first class of algorithms for LP that are efficient both in theory and in practice. A currently active area of research is the extension of these methods beyond the domain of linear programming. In this talk, we will start with a brief survey of the extension to smooth convex nonlinear optimization. Many real-world problems, especially within engineering, belong to this general class. We will present several examples such as digital filter design, structural design, and minimal surfaces, to name a few. We will offer a multimedia presentation of the optimal solution to many of these problems (technology permitting).

Topic: Representations of covering groups of SL(n) April 20

Presenter: Jeffrey D. Adams, University of Maryland

Abstract: Two methods of constructing automorphic forms are the Langlands program and the theta-correspondence. While there is substantial overlap between these two approaches, the former deals with linear (matrix) groups, and the latter with possibly non-linear groups such as the two-fold cover of Sp(2n). Thus it would be desirable toextend the Langlands program to non-linear groups.

A beginning of an approach can be seen by studying characters of non-linear groups. In this talk I will discuss the case of covering groups of SL(n). Perhaps surprisingly the notion of L-packet can be defined in some cases, and it is not the obvious generalization of the (conjectural) L-packets in the linear case. We give a correspondence between "stable" genuine virtual characters of covers of SL(n) and characters of PGL(n).

Topic: Mathematical Models of Language Evolution April 21

Presenter: Martin Nowak, Institute for Advanced Study

Abstract: The emergence of language was a defining moment in the evolution of modern humans. It was an innovation that changed radically the character of human society. Here I provide a new approach to language evolution based on evolutionary game theory. I will discuss how a proto-language can emerge in a pre-linguistic society and describe the dynamics of signal-object association, word-formation, and the emergence of simple grammatical rules.

Topic: Butterflies and Rudich's Conjecture April 22

Presenter: Cliff Smyth, Rutgers University

Abstract: Let V = {x_1, x_2, x_3, ..., x_n} be a set of independent True/False random variables, with Pr(x_i = True) = Pr(x_i = False) = 1/2. A term on V is a conjunction of variables and negations of variables in V. For example, t_1 = x_1 && x_2 && not(x_3) and t_2 = x_3 && x_4 are both terms on V. Let T = {t_1, t_2, ..., t_n} be a set of terms on V. The neighborhood of a term t_i is the set N_i of all terms in T that share a variable with t_i. In this context, x_i and its negation are considered to be the same variable. For example, if t_1 and t_2 are as before, N_1 contains both t_1 and t_2. If S is a subset of T, let U(S) = Pr(exactly one term in S is True). The following conjecture of S. Rudich has been circulating unpublished for some years: There exist a, b > 0, such that for all pairs (V,T) as above, U(T) > 1 - a implies there exists an N_i so that U(N_i) > b. A short proof of this conjecture will be presented, utilizing the Butterfly Theorem used by D. Reimer in the proof of the Van Den Berg - Kesten Conjecture. This is joint work with Jeff Kahn and Mike Saks.

Topic: Stable Ergodicity April 22

Preseneter: Andrei Torok, Princeton University

Topic: Mixed Volumes and Valuations: A Tale of Two Topics April 22

Presenter: Maragaret Readdy, IAS

Abstract: The work we will discuss is directly influenced by Klain and Rota's recent book on geometric probability. As a warning, this talk has everything to do with old mathematics and geometry and nothing to do with geometric probability. Part I: Laplace proved the volume of the k-th slice of a unit cube is given by an Eulerian number. We generalize Laplace's result by giving a combinatorial interpretation for the mixed volumes of two adjacent slices from the unit cube in terms of a refinement of the Euleriannumbers. Part II: We will discuss valuations and complex hyperplane arrangements. Namely, we present a new combinatorial method to compute the characteristic polynomial of subspace arrangements using the theory of valuations. This method

applies to any subspace arrangement over an infinite field. Examples include complex subspace arrangements, the Dowling divisor lattice and its interpolations. We also consider the effect of the Dowlingization transformation on the characteristic polynomial of real subspace arrangements. This talk is intended for a general audience.

Topic: TBA April 22

Presenter: Robert Gompf, University of Texas

Topic: On Spectral Gap for Nonarithemtic Subgroups of SL(2, Z)

Presenter: Alex Gamburd, Princeton University

Abstract: A celebrated theorem of Selberg states that for congruence subgroups of SL(2, Z) there are no exceptional eigenvalues below 3/16. Extending the work of Sarnak and Xue for cocompact arithmetic lattices, we give a new proof of Selberg's theorem (with somewhat weaker bound) and present its generalization to infinite index subgroups. For such subgroups with a high enough Hausdorff dimension of the limit set we establish a spectral gap property and consequently solve a problem of Lubotzky pertaining to expander graphs.

Topic: The Topological Structure of Stable Plasma Flows April 23

Presenter: Jason Cantarella, University of Pennsylvania

Topic: Active and Passive Scalars April 23

Presenter: Peter Constantin, University of Chicago

Topic: Certain integrals encountered in random matrix theory April 26

Presenter: M.L. Mehta, Paris

Topic: TBA April 26

Presenter: Michael Weinstein, A T & T

Topic: Codes and Models April 26

Presenter: Bin Yu, Bell Laboratories, Murray Hill and University

of California at Berkeley

Abstract: In this talk, we explore the intricate connections between codes (representations of messages in a communication system) and models (mathematical postulations about data). We draw parallels between theories of information and algorithmic complexity on the one side and concepts and approaches to building statistical models on the other. In particular, statistical models give rise to effective codes -- as in current wavelet image coding schemes; while codes provide insight into models -- through the Principle of Minimum Description Length (MDL) for model selection.

Topic: Monodromy in families of abelian varieties April 27

Presenter: Jeffrey Achter, University of Massachusetts

Topic: Quantum State Reduction as a Gravitational April 27

Phenomenon: Theory and experiment

Presenter: Roger Penrose, Mathematics Institute, Oxford University

Abstract: The problem of quantum state-vector reduction is addressed in many different ways by the various schools of thought in quantum mechanics. To support my own particular viewpoint that reduction is an actual objective physical process, I shall point out a fundamental conflict between the basic principles of general relativity and those of quantum mechanics. This leads to the conclusion that a quantum superposition -- i.e. a "Schrodinger's cat" -- is unstable and will decay into one or the other of the two states in quantum superposition in a period of time that can be calculated from the gravitational energies involved. For a real cat, this time would be a tiny fraction of a second, which is why we do not actually see quantum-superposed Schrodinger's cats. But for a small enough object, like a very small crystal, this predicted effect should be measurable. A specific space-based experiment to detect this effect (now being actively investigated) will be described. Some recent theoretical developments will also be presented.

Topic: A Discrete Time Optimisation Problem for the Lognormal Distribution and April 28

its Relation to Passport Options

Presenter: Professor F. Delbaen, Rutgers University

Place: Rutgers University Operation Research Department

Abstract: Passport options are financial instruments whose price is calculated through an optimisation problem in continuous time. We relate the problem to a discrete time optimisation for the lognormal distribution as well as to an $\Cal H^1$-like equality of martingale norms. This is joint work with Marc Yor.

Topic: On the rank of elliptic curves April 29

Presenter: Joseph Silverman, Brown University

Topic: Stable minimal surfaces and holomorphic curves in April 30

Kahler-Einstein manifolds

Presenter: Claudio Arezzo, MIT

Time: 1:30 p.m.

Topic: Some symmetry and monotonicity results for elliptic PDE on manifolds

Presenter: Luis Almeida, ENS Cachan, France

Time: 2:30 p.m.

Date: Friday, April 30, 1999

Room: Fine 314

Topic: The what and the why of quantum mechanics April 30

Presenter: Sheldon Goldstein, Rutgers University

Topic: Complexity and Generation of Information in Molecular Evolution May 5

Presenter: Manfred Eigen, Scripps Research Institute

Abstract: Combinatorial versus dynamical complexity. Evolving molecular complexity: NP-Complete? An experiment demonstrating evolving ccomplexity. Darwinian algorithms: The generation of information, What is life?

Topic: TBA May 14

Presenter: JingYi Chen, MIT