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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
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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
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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
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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
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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
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- PAUL ROBERTS, University of Utah, Salt Lake City, Utah 84112, USA
Intersection multiplicities of modules of finite projective
dimension
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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.
- PETER SARNAK, Princeton
Hilbert's Eleventh Problem
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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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