


CMS Plenary Speakers [PDF]
 KEITH DEVLIN, Stanford University
How much mathematics can be for all?
[PDF] 
In my book The Math Gene [Basic Books, 2000], I presented an
evolutionary argument to show that the basic capacity for mathematical
thinking is present in everyone as part of our genetic inheritance.
But how much mathematics comes in this way? Is there a point beyond
which most people will simply never get it? I believe there is
sufficient evidence to suggest that the answer may be yes, and that
among those parts of mathematics that can possibly be mastered only by
a few is at least one topic taught in the middle school.
 DAN FREED, University of Texas at Austin
Correspondences, Ktheory, and loop groups
[PDF] 
Ideas from quantum field theory and string theory continue to
influence many fields of mathematics. We consider the Verlinde ring,
which originally arose in two dimensional conformal field theory. Let
G be a compact Lie group and LG the free loop space of G, viewed
as a group by pointwise multiplication. There is a special class of
unitary representations of LG in terms of which the Verlinde ring is
mathematically defined. In joint work with Michael Hopkins and
Constantin Teleman we locate this ring in topology, specifically a
twisted version of Ktheory. Furthermore, we construct the ring
structureand a Frobenius ring structurein terms of
correspondence diagrams. The talk will include some expository
material on correspondences and quantum field theory.
 ROBERT McCANN, University of Toronto
Fluid flow in the semigeostrophic oceans and atmosphere
[PDF] 
The semigeostrophic approximation to Euler's equation is used to
caricature the largescale, longtime evolution of the atmosphere and
oceans, and to model such phenomena as the formation and evolution of
pressure fronts. Even in 3D, it can be formulated as an active scalar
transport modellike the 2D incompressible Euler equationsbut
with the Hessian of the stream function related to the conserved
scalar quantity by a Determinant instead of a Trace. This talk
surveys some mathematical developments and open questions concerning
this nonlinear model, exposing in particular a family of exact
solutions to the equations, representing 2D circulations of an ideal
fluid in a elliptical ocean basin. For these special solutions, the
fluid pressure and stream function remain quadratic functions of space
at each instant in time, whose fluctuations are described by a single
degree of freedom Hamiltonian system depending on two conserved
parameters: domain eccentricity and the constant value of potential
vorticity. These parameters determine the presence or absence of
periodic orbits with arbitrarily long periods, fixed points of the
dynamics, and aperiodic homoclinic orbits linking hyperbolic saddle
points. The energy relative to these parameters selects the frequency
and direction in which isobars nutate or precess, as well as the
steady circulation direction of the fluctuating flow. Canonically
conjugate variables are given, which describing the complete evolution
of an elliptical inversepotentialvorticity patch in dual space.
 ANDREI OKOUNKOV, Princeton University
Enumerative geometry of curves in threefolds
[PDF] 
This will be a review of some mostly conjectural ideas about
enumerative geometry of curves of given degree and genus in a smooth
projective threefold such as, e.g., projective space. It will
be based on my joint work with Davesh Maulik, Nikita Nekrasov, and
Rahul Pandharipande.
 GILLES PISIER, Texas A&M and Paris VI
Similarity problems and amenability
[PDF] 
In 1955, Kadison formulated the following conjecture: any bounded
homomorphism u : A® B(H), from a C^{*}algebra into the
algebra B(H) of all bounded operators on a Hilbert space H, is
similar to a *homomorphism, i.e., there is an invertible
operator x: H® H such that x® xu(x) x^{1}
satisfies xu(x^{*}) x^{1} = (xu(x) x^{1})^{*} for all x in
A. This conjecture remains unproved, although many partial results
are known. We will survey those as well as more recent results on the
closely related notion of length of an operator algebra. In
particular, we will explain why length equal to 2 characterizes
amenable groups or nuclear C^{*}algebras.
 KEN RIBET, University of California at Berkeley
The modularity of some mod p Galois representations
[PDF] 
I will sketch the main ideas of recent preprints of
KhareWintenberger and Dieulefait that allow one to establish certain
cases of Serre's conjectures on mod p irreducible twodimensional
representations of the Galois group of Q. Recall that the 1994
proof of Fermat's Last Theorem for exponent p associates an elliptic
curve E to each putative nontrivial solution of Fermat's equation.
The mod p representation E[p] associated to E is incompatible
with Serre's conjecture; it cannot be modular. The classical method
of Wiles is to prove that E[p] is modular by proving that E is
modular. In the new millennium, KhareWintenberger and Dieulefait
work directly with E[p] and forget where it came from. While the
new method may not strike everyone as a simplification of the proof of
FLT, it will very likely lead to striking new results.

