


Plenary Speakers / Conférenciers principaux
 MICHAEL BENNETT, University of British Columbia, Vancouver, BC
Classical Diophantine equations via modern and notsomodern
methods

In this talk, I will attempt to survey some recent developments in the
field of Diophantine equations, with particular emphasis on a number
of classical problems that have been resolved in the past few years,
such as Catalan's Conjecture. Of particular interest are new
techniques, such as those arising from the modularity of Galois
representations. I will conclude with a list of what I hope are
challenging open problems in this area.
 PERSI DIACONIS, Stanford
A Mathematician Flips a Coin

In joint work with Susan Holmes and Richard Montgomery we have proved
that naturally flipped coins are biased to come up the same way they
started. The analysis is a mix of mechanics and Tauberian Theorems
coupled with image analysis and new statistical techniques for
highspeed slow motion cameras.
 ROSTISLAV GRIGORCHUK, Texas A&M University
Algebraic, algorithmic, and spectral properties of automata
groups

Automata groups were defined 40 years ago and started to play an
important role two decades later when it was discovered that they
provide solutions to such famous problems as Burnside problem on
torsion groups or Milnor problem on growth of finitely generated
groups.
The development of the theory of automata groups during the last two
decades showed that the number of difficult problems to which the
methods and the ideas involving groups generated by finite automata
can be applied is rather large and that they could be used in many
areas of mathematics such as algebra, analysis, geometry, probability,
dynamics, computer science, mathematical logic, and other.
Automata groups were used to solve several difficult problems in
analysis centered around the notion of amenability introduced by von
Neumann in 1929 (as a result of his study of the algebraic roots
behind the BanachTarski paradox), in Riemannian geometry (Atiyah
Conjecture on L^{2} invariants), in theory of profinite groups
(Zelmanov Conjecture on groups of finite width), etc.
There are indications that automata groups can be used to attack
KaplanskyKadison Conjecture on Idempotents, Kaplansky Conjecture on
Jacobson radical, Dixmier Unitarizibility Problem, BaumConnes and
Novikov Conjectures, FontaineMazur Conjecture in Number theory and
other problems in mathematics.
We will try to convince the audience that this is indeed the case.
 FRANÇOIS LALONDE, CRM
How pure mathematical aspects of String theory can solve deep
problems of geometric group theory

In any simple group, the subgroup consisting of finite products of
commutators must obviously be either the identity or the whole group.
Thus, in a nonabelian simple group, any element is the product of a
finite number of commutators. Said crudely: in such a group, the lack
of commutativity is enough to generate the whole group! The
"commutator length" of an element is the minimal number of
commutators in such a product.
It turns out that persistent efforts of topologists in the '70s and
'80s led to the following theorem: the group of area preserving
transformations of the 2sphere (which is obviously nonabelian) is
simple! (This is the simplest example of a much broader class that
includes all algebraic manifolds and much more.) The methods used in
this theorem are "soft", and give no idea on the behaviour of the
commutator length of a given transformation (with respect, say, to
iterations of this transformation).
On the other hand, a simple nonabelian group has no nontrivial
homomorphism to the reals (or to any abelian group). But there might
be quasimorphisms (maps that are almost homomorphisms) and I will
explain how to construct such quasimorphims, according to ideas of
collaborators in Israel (Entov, Polterovich, Biran) that are based on
a clever use of the semisimplicity of quantum homology,
i.e., on ideas based on Floer and String theories. I will
then explain how these quasimorphisms give the first significant
informations on the commutator length problem.
 RAINER STEINWANDT, Universität Karlsruhe, 76131 Karlsruhe, Germany
Nonabelian groups in public key cryptography

While finite cyclic groups are a wellestablished tool in public key
cryptography, there are not many practical proposals for cryptographic
schemes based on nonabelian groups. One interesting line of research
in this context, which has received a lot of attention in the
cryptographic community, explores the cryptographic potential of braid
groups. Other conceptually interesting proposals rely on "wild"
factorizations of finite nonabelian groups and inspired (not only
cryptographic) questions about the existence of certain factorizations
of finite groups.
The talk surveys some cryptographic schemes based on finite or
finitely presented nonabelian groups and discusses some
(non)mathematical issues that have to be dealt with when trying to
implement practical cryptographic schemes on the basis of nonabelian
groups.

