




Plenary Speakers / Conférenciers principaux
 D.W. HENDERSON, Cornell University, Ithaca, New York 148534201, 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 unease with this, calling this unease 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 K16 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 answerWhy?''
c. a joy and confidence in exploring THEIR OWN mathematical ideas.
d. the practiced ability to give convincing communications
that answerWhy?
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 MathematicsWhat 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 (lIT)^{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 Zmodule 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 LittlewoodRichardson 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 wellknown 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 ConfortiCornuejols 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.

