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Plenary Speakers / Conférenciers principaux

PATRICK DEHORNOY, University de Caen, 14032  Caen, France
From set theory to braids via self-distributive algebra

We shall show how studying large cardinals in set theory (whose existence is, and will remain, an unprovable assumption) has led to the construction of new examples of algebraic systems satisfying the left self-distributivity law, and, from there, quite naturally, to the discovery of a linear ordering on Artin's braid groups. The latter has led to new braid applications, and it has now received several equivalent purely geometrical or topological constructions.

RICHARD DURRETT, Cornell University, Ithaca, New York  14853, USA
Mathematical models of biodiversity

Species area curves and species abundance distributions have long been studied by ecologists. We will describe several approaches to this question based on a variety of mathematical techniques. However, we will find that only our two spatial models are successful at explaining the patterns observed in Nature. Furthermore, we will take comfort from the fact that our two quite different approaches lead to the same qualitative predictions.

ROGER HOWE, Yale University
Challenges in teacher preparation and development

As the demand for mathematical skills has spread to more professions, and the need for mathematical understanding has pervaded more aspects of daily life, the clamor for better mathematical education likewise has grown louder. These demands fall directly on university and college mathematics departments. However, mathematics instruction in college can achieve maximal effectiveness only when supported by strong elementary and secondary education in mathematics. For this, the preparation and continuing development of teachers is a focal issue. This talk will discuss recent findings on the magnitude and nature of the challenges involved in building a strong teaching corps in mathematics, with a focus on the role of mathematics departments in higher education. It will be partly based on the forthcoming report from CBMS on the mathematical preparation of teachers.

IZABELLA LABA, University of British Columbia, Vancouver, British Columbia  V6T 1Z2
Gowers's proof of Szemerédi's theorem, and applications to harmonic analysis

Szemerédi's theorem, often regarded as one of the milestones of combinatorics, states that any subset of Z with positive asymptotic density must contain arbitrarily long arithmetic progressions. Recently, W. T. Gowers found a new elementary proof of the theorem. Gowers's proof, which generalizes an earlier combinatorial argument due to Roth, provides significantly better quantitative bounds than those in Szemerédi's and Furstenberg's proofs. It has also led to a better understanding of certain other combinatorial questions, notably in inverse additive number theory; this in turn has had somewhat unexpected consequences in harmonic analysis. The purpose of the talk will be to survey these and other related recent developments.

STANLEY PLISKA, University of Illinois at Chicago, Finance Department, Chicago, Illinois  60607-7124, USA
Risk sensitive asset management

PAUL ROBERTS, University of Utah, Salt Lake City, Utah  84112, USA
Intersection multiplicities of modules of finite projective dimension

In attempting to prove Serre's vanishing conjecture for intersection multiplicities, more general conjectures were formulated for nonregular local rings. Some of these general conjectures turned out to be false, and in particular a counterexample to the general vanishing conjecture was given about fifteen years ago. Recent investigations have led to a new interest in this subject. In this talk I will describe joint work with V. Srinivas in which we give a complete answer to the question of the existence of this kind of example in the case of the localization of a graded ring such that the associated projective scheme is smooth. The answer is given in terms of the group of cycles modulo numerical equivalence of the associated projective scheme. This result includes all previously known examples and shows the existence of many new ones.

Hilbert's Eleventh Problem

Hilbert's Eleventh Problem asks which integers in a number field (resp. rationals) are represented by a quadratic form with integral (resp. rational) coefficients.For the case of rationals the solution was given by Hasse (the Hasse-Minkowski local to global principle). The case of integers is much more difficult. We will describe the developments and recent solution.

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