


Conférenciers pléniers [PDF]
 BRENT DAVIS, University of British Columbia, 2125 Main Mall, Vancouver,
BC, V6T 1Z4
MathematicsforTeaching: Where are we with all of this?
[PDF] 
Over the past decade, mathematicsforteaching (MfT) has emerged as
the most prominent and popular topic in mathematics education
research. In this presentation, I offer a partial review of research
orientations and findings that give shape to current discussions and
investigations. In the process, I flag what I think to be some of the
more promising avenues. I also attempt to highlight a few of the
pragmatic implications for Departments of Mathematics and Faculties of
Education.
 DMITRY DOLGOPYAT, Pennsylvania State University, University Park, PA, 16802,
USA
Fermi acceleration
[PDF] 
Fermi acceleration is a mechanism, first suggested by Enrico Fermi in
1949, to explain heating of particles in cosmic rays. Fermi studied
charged particles being reflected by the moving interstellar magnetic
field and either gaining or losing energy, depending on whether the
"magnetic mirror" is approaching or receding. In a typical
environment, Fermi argued, the probability of a headon collision is
greater than a headtail collision, so particles would, on average, be
accelerated. Since then Fermi acceleration has been used to explain a
number of natural phenomena and several simple mathematical models
demonstrating Fermi acceleration have been proposed. We describe
these models and explain why they do or do not exhibit Fermi
acceleration. We also mention some models where the answer is not
known.
 DIMITRI SHLYAKHTENKO, University of California at Los Angeles
L^{2} Invariants, Free Probability and Operator Algebras
[PDF] 
The CheegerGromov L^{2} Betti numbers of a discrete group are
numerical invariants, going back to Atiyah's work on the equivariant
AtiyahSinger index theorem. On the other hand, Voiculescu has
introduced another discrete group invariant, coming from his free
probability theory, called the free entropy dimension. Very roughly,
this number measures the "asymptotic dimensions" of the sets of
approximate embeddings of a group into unitary matrices. We describe
our joint work with A. Connes and I. Mineyev that has provided a
connection between these numbers. Both of these approaches (operator
algebra versions of L^{2} Betti numbers and Voiculescu's free entropy
dimension) are attempts to arrive at the von Neumann algebra analog of
the theory of L^{2} Betti numbers (Gaboriau) and of free entropy
dimension for measurable equivalence relations.
 KAREN SMITH, University of Michigan, Ann Arbor, MI
Uniform Results in Algebra and Geometry via Multiplier
Ideals
[PDF] 
In recent years, there has been a flurry of questions about
uniform behavior in commutative algebra and algebraic geometry.
Multiplier ideals have had a tremendous impact in solving many of
these questions. Multiplier ideals can be defined in three rather
different ways. Originally defined by analysts, they are functions in
some L^{2}space. For algebraic geometers, they are defined via
resolutions of singularities. For commutative algebraists, they are
defined in rings of prime characteristic using tight closure. In this
talk, we will discuss a few of the diverse problems about
uniform behavior in algebra and geometry that have been solved
using multiplier ideals, as well as the different perspectives from
which multiplier ideals can be viewed.
 SUSAN TOLMAN, University of Illinois at UrbanaChampaign
Group actions and the symplectomorphism group
[PDF] 
Much progress has been made in recent years studying the group of
diffeomorphisms which preserve a symplectic form. We will discuss
recent advances in this question, focussing on the relationship
between this group and its compact subgroups. In particular, we will
consider how to use these subgroups to gain information about the
group itself, and ways in which it behaves (or does not behave) like a
compact group.
 SHMUEL WEINBERGER, University of Chicago
Quantitative Aspects of Topology
[PDF] 
In the popular imagination, there is a strict dichotomy between the
quantitative and the qualitative and Topology is the epitome of the
qualitative. However, many problems, both theoretical and applied,
have suggested a need for topology to encompass estimates and to grow
to include discontinuous maps. I will try to explain some of these
via examples taken from problems like estimating solutions to
equations, the influence of the fundamental group on the
geometry/topology of manifolds, and recent methods of data analysis.

