Math 2000

Plenary Speakers / Conférenciers principaux

JAMES ARTHUR, Toronto
An introduction to the trace formula

The trace formula is a vast generalization of the Poisson summation formula. It relates rational conjugacy classes on the one hand, with automorphic representations on the other. The talk will be a general introduction to the trace formula. We shall also try to give some indication of how it can be used to study fundamental arithmetic properties of automorphic representations in terms of corresponding properties of conjugacy classes.

FRANCIS CLARKE, Institut Desargues, Université Lyon I, 69622  Villeurbanne, France
The feedback problem in control theory

In this nontechnical survey, we begin by defining the principal issue in control theory: to design a feedback law having a certain desired effect on the underlying dynamical system. The possible contexts in which this problem can be given a precise formulation are multiple, depending notably on the type of dynamical system considered, the degree of knowledge of the model that is assumed, and the goal that is desired. The actual and potential applications are also widespread, and very important, encompassing as they do such special cases as linear systems theory and optimal control. Here we shall focus primarily upon the stabilization issue for nonlinear systems of ordinary differential equations, a very active area in the last decade. Besides discussing some earlier important results, we shall present more recent work bearing upon the interest (in fact, the absolute necessity) of using discontinuous feedback laws. In particular, this work settles an old problem in the field by proving that every (open-loop) controllable system admits a stabilizing feedback.

IOANNIS KARATZAS, Columbia
Financial mathematics

DUSA MCDUFF, Department of Mathematics, SUNY, Stony Brook, New York  11794, USA
Some open problems in symplectic topology

A symplectic structure is a weakened form of a Kahler structure. Much of the rigidity of complex geometry is lost in the symplectic world, but intriguing topological features remain. A good deal is now known about them, but there are still plenty of open problems. This talk will discuss some of them, concentrating on questions concerning diffeomorphisms that preserve the symplectic structure and the groups that they form.

EUGENE MYERS, Celera Genomics, Rockville, Maryland  20850, USA
Whole genome assembly of the Drosophila and human genomes

We report on the design of a whole genome shotgun assembler and its application to the sequencing of the Drosophila and Human genomes. Celera's whole genome strategy consists of randomly sampling pairs of sequence reads of length 500-600 that are at approximately known distances from each other-short pairs at a distance of 2K, long pairs at 10K, and BAC-end pairs at 150K. For Drosophila, we collected 1.6 million pairs whereby the sum of the lengths of the reads is roughly 13 times the length of the genome ( ~ 120 million), a so called 13X shotgun data set. The reads were further collected so there are two short read pairs to every long read pair, with a sprinkling of roughly 12,000 BAC-end pairs. The experimental accuracy of the read sequences is roughly 98%. Given this data set, the problem is to determine the sequence of Drosophila's 4 chromosomes that are estimated to be 5-10% repetitive sequence.
By layering the ideas of uncontested interval graph collapsing, confirmed read pairs, and mutually confirming paths, we arrive at a strategy that makes remarkably few errors. The assembler correctly identifies all unique stretches of the genome, correctly building contigs for each and ordering them into scaffolds spanning each of the chromosomes. We will present our final results for the Drosophila assembly and report on our progress towards the sequencing and assembly of the human genome.

RAYMOND PIERREHUMBERT, Department of Geophysical Sciences, University of Chicago, Chicago, Illinois, USA
Statistical mechanics of chaotic advection-diffusion in the atmosphere

``Chaotic Advection-Diffusion'' refers to the class of advection-diffusion problems for which the trajectories induced by the advecting velocity field are chaotic, i.e. have a positive Lyapunov exponent. The theory of chaotic advection diffusion has developed extensively in the past two decades, and for suitably idealized classes of flows many quantitative predictions can be made. The main entities treated by the theory are the probability distributions of concentration fluctuation, of concentration gradient, and of concentration differences over finite displacements. All these quantities are of central interest in atmospheric chemistry, in atmospheric dynamics (when applied to potential vorticity), and in diagnosis of atmospheric observations.
In this talk, I will survey the theoretical developments, mostly as applied to once-differentiable velocity fields, and discuss attempts to push the theory to the point to which it can be applied to realistic large scale atmospheric velocity fields. The ideas arising from simple models carry over remarkably well, but there is a clear need to better understand the effects of transport barriers. I will also discuss new results on the extension to chemically reactive tracers, a subject that plays a role in chemical combustion and supernova ignition, aside from its applications to atmospheric chemistry.

CARL POMERANCE, Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey  07974, USA
Prime numbers: what we still don't know

The study of prime numbers goes back over two millennia, yet there are still extremely basic problems that no one can solve. While some of these problems are exciting just for the intellectual challenge, others are given urgency by their connection to the security of many modern forms of cryptography, including those used to protect commerce on the Internet. In this talk I will highlight a personal top ten of unsolved problems, and discuss what meager progress we have made.

MAURICE QUEYRANNE, Faculty of Commerce and Business Administration, University of British Columbia, Vancouver, British Columbia  V6T 1Z2
Scheduling polyhedra: from cutting planes to approximation algorithms

Scheduling problems concern the allocation of resources, usually called ``machines'' or ``processors'', to tasks over time. They arise in numerous practical applications, in particular in the manufacturing, computing and communication industries. We will discuss an approach whereby scheduling problems are modelled as linear programming problems with a small number of variables (typically, one per job or operation) and numerous linear constraints that represent or approximate the scheduling restrictions. This approach relies on the study of polyhedra that arise as the convex hull of vectors representing feasible schedules, and was initiated by Balas (1985) and Wolsey (1985). Some of the main results include the identification of supermodular polyhedra associated with certain classes of ``easy'' scheduling problems and, for harder problems, of classes of linear inequalities (cutting planes) that admit efficient separation algorithms. We will present some recent results with exact solution methods of the branch and cut type, and approximate algorithms with bounded performance guarantees, for certain single and parallel machine shop scheduling problems with precedence constraints and/or release dates.

LAWRENCE SHAMPINE, Mathematics Department, Southern Methodist University, Dallas, Texas  75275, USA
Solving ODEs in new computing environments

For several decades the speaker investigated the numerical solution of ODEs and developed solvers for general scientific computation (GSC). In the last several years he has worked on solving ODEs in a variety of new computing environments that emphasize convenience. What is there to work on? Why don't you just translate one of the solvers popular in GSC? You can, and people have, but this does not provide a quality product because both the goals and the environments differ in important ways. Case studies are used to illustrate some of the differences and their implications for solving ODEs: ODE Architect illustrates packages for teaching ODEs. The simulation language SIMULINK illustrates packages for specific applications. Two problem solving environments are included: MATLAB, which emphasizes numerical computation, and Maple, which emphasizes algebraic computation.

DEMETRI TERZOPOULOS, Toronto

LOU VAN DEN DRIES, Urbana, Illinois, USA
Logarithmic-exponential series

The field of ordinary power series with real coefficients can be extended to a much larger field of formal series in which operations such as exponentiation, taking logarithms, derivatives, anti-derivatives, composition and compositional inverse can be carried out without the usual restrictions. This field of ``logarithmic-exponential series'' is a natural domain for the kind of asymptotic expansions that occur in Hilbert's 16th problem on limit cycles (work of Ecalle, Il'yashenko). This field of LE-series has also remarkable algebraic and model-theoretic properties, such as o-minimality.

SHING-TUNG YAU, Harvard, USA
PDE-to be confirmed

EFIM I. ZELMANOV, Yale, USA
On some algebraic structures related to conformal field theory

During the last 20 years vertex and conformal algebras found amazing applications in various areas of physics and mathematics. We will discuss examples amd classification of algebras of small Gelfand-Kirillov dimension.