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Plenary Speakers / Conférenciers principaux

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D.W. HENDERSON, Cornell University, Ithaca, New York  14853-4201, USA
How can we encourage students to think like a mathematician?

Much in current classrooms and texts (including reform texts) does not encourage students to think mathematically. There are examples of what I label as lying to students for example, saying something is a fact when it is only an assumption and, then when the students express an un-ease with this, calling this un-ease an ``obstacle'' or ``misconception'' to learning the ``fact''. Many pupils loose their own powerful mathematical ideas (ideas which they can use meaningfully and which they can defend) and replace them with memorized ``facts'' and ``formal definitions'' with little meaning.

Effective K-16 mathematics teachers and students in the 21st Century should possess:

a.  a view of mathematics as an active process of thinking about and understanding of our mental and physical experiences.

b.  a view that central to mathematics are proofs: ``convincing communications that answer-Why?''

c.  a joy and confidence in exploring THEIR OWN mathematical ideas.

d.  the practiced ability to give convincing communications that answer-Why?

e.  the practiced ability to listen to the communication of others and to respond appropriately.

The focus will be on undergraduate teaching and what I am now starting to call Educational Mathematics-What do ideas and results in mathematics mean? And how can we understand (and experience) them on the basis of what they mean? How does this relate to what and how we teach?. Because I am a geometer, I will bring geometry examples and geometric meanings that are behind much mathematics.

N. NIKOLSKI, Steklov Institute of Mathematics, University of Bordeaux (France), St. Petersburg, Russia
Resolvents bounds and their use in analysis

We give an account of the use of various types of estimates of the resolvents (lI-T)^-1 of linear operators. In particular, we pay an attention to the following problems.

(1) Exponential decay for semigroups of operators and operator powers.
(2) Efficient resolvent estimates and the phenomenon of the ``invisible spectrum''.
(3) Pseudospectra and spectral hulls in functional calculi.
(4) Resolvent growth, singular integrals and the similarity problem.

C. REUTENAUER, Université du Québec à Montréal
Tableaux bialgebras and noncommutative character theory of symmetric groups

One defines a bialgebra structure on the Z-module based on permutations; by restriction to plactic classes, one obtains a bialgebra based on tableaux. The latter is mapped onto central functions of symmetric groups, and allows the construction of the irreducible characters. The mapping is an isometry that may be constructed using idempotents linked to free Lie algebras. Among many applications, one has the Littlewood-Richardson rule for the product of characters. The notions are elementary and will be explained during the talk. The presentation we give is specially due to Joellenbeck, with previous contributions of Knuth, Lascoux, Schutzenberger, Poirier, Solomon, Schocker, and many others, including the speaker.

PAUL D. SEYMOUR, Princeton University
Strong perfect graph theorem

Claude Berge proposed the conjecture in 1960 that, in every graph with no odd hole or odd antihole, the number of colours needed to properly colour the graph equal the size of the largest complete subgraph. (A ``hole'' means an induced subgraph which is a cycle of length ³ 4, and an ``antihole'' is same in the complement graph.) This has become one of the most well-known and popular open problems in graph theory. Most attempts on it have been based on linear programming methods, studying the properties of a minimal counterexample; they go a long way, but appear eventually to get stuck. Recently, however, a new approach was initiated by Conforti and Cornuejols, an attempt to actually find explicit constructions for all the graphs not containing odd holes or antiholes, and checking directly that they satisfy Berge's conjecture. I am happy to report that this works. In joint work with Maria Chudnovsky, Neil Robertson and Robin Thomas, we have been able to carry out the Conforti-Cornuejols program, and thereby prove Berge's conjecture. (We hope!-we only finished the proof in May, and there are two years worth of details to check, but so far it stands up.)

ISADORE M. SINGER, MIT
Forty years of index theory

Definition of the index of a Fredholm operator. A simple index theorem (for Toeplitz operators). Statement of the index theorem. A review of three proofs of the index theorem. Refinements of the theorem. Current issues.