| DATE |
SPEAKER |
TOPIC |
| Sep 20 |
Stanislav Smirnov
University of Geneva |
Towards conformal invariance of two-dimensional lattice models
|
| Sep 27 |
Richard Taylor
Harvard University |
The Sato-Tate conjecture
A fixed elliptic curve over the rational numbers is known to have
approximately p points modulo p for any prime number p. In about 1960 Sato
and Tate gave a conjectural distribution for the error term. Laurent
Clozel, Michael Harris, Nick Shepherd-Barron and I recently proved this
conjecture in the case that the elliptic curve has somewhere
multiplicative reduction.
In this talk I will describe the Sato-Tate conjecture and the ideas Tate
and Serre had for proving it. I will also sketch how we were able to prove
sufficient higher dimensional modularity results to complete the proof.
|
| Oct 4 |
Luis Caffarelli
University of Texas at Austin |
Nonlinear problems involving fractional diffusion |
| Oct 11 |
Klaus Schmidt
University of Vienna / Schrodinger Institute |
Mahler Measure and
Entropy
One can associate canonically and very simply with every ideal $I$ in the integral group ring $\mathbf{Z}[\mathbf{Z}^d], d\ge 1$, a measure preserving action $\alpha = \alpha _I$ of $\mathbf{Z}^d$ by automorphisms of a compact abelian group $X=X_I$ (or an 'algebraic' $\mathbf{Z}^d$-action for short). There are many interesting correspondences between algebraic properties of the ideal $I$ and dynamical properties of $\alpha _I$. This lecture will focus on one of these connections: if the ideal $I$ is principal and generated by an element $f\in \mathbf{Z}[\mathbf{Z}^d]$, then the entropy of $\alpha _I$ is the logarithm of the Mahler measure of $f$.
Mahler measures of multivariate polynomials $f\in \mathbf{Z}[\mathbf{Z}^d]$ and, in particular, certain values of $L$-functions, also occur as entropies of certain lattice models (especially dimer models) in statistical mechanics, and the connection between these lattice models and the algebraic $\mathbf{Z}^d$-actions associated with the principal ideals generated by these polynomials is still somewhat mysterious.
The connection between ideals in the integral group ring $\mathbf{Z}[\Gamma ]$ of a discrete group $\Gamma $ and algebraic $\Gamma $-actions also extends in a straightforward way to arbitrary discrete (amenable) groups. For principal ideals the entropy of these actions is a quantity which extends the notion of logarithmic Mahler measure to elements of $\mathbf{Z}[\Gamma ]$. The last part of the lecture will discuss what little is currently known about this extension.
Much of this material is joint work with Christopher Deninger and/or Doug Lind. |
| Oct 18 |
Robert Seiringer
Princeton University |
Dilute Quantum Gases
We present an overview of mathematical results on the low temperature properties of dilute quantum gases, which have been obtained in the past few years. The discussion includes, for instance, results on the free energy in the thermodynamic limit, and on Bose-Einstein condensation, Superfluidity and quantized vortices in trapped gases. All these properties are intensely being studied in current experiments on cold atomic gases. We will give a brief description of the mathematics involved in understanding these phenomena, starting from the underlying many-body Schroedinger equation. |
| Oct 25 |
|
|
| Nov 8 |
Richard Schwartz
Brown University |
Irrational triangular billiards
It is an old and open problem whether or not every triangular
shaped billiard table has a periodic billiard path. The answer is known
to be yes for acute, right, and rational triangles but unknown in the
obtuse irrational case. Over several years, Pat Hooper and I have
developed a graphical user interface, called McBilliards, with a
view towards resolving the triangular billiards problem. The huge
experimental output from the progran illustrates the extreme and
previously unexpected complexities of the problem.
In my talk I will survey the experimental evidence from McBilliards
and also explain some of our rigorous results which were inspired
by the experiments. |
| Nov 15 |
Mircea Mustaţă
University of Michigan; IAS |
Invariants of singularities in positive characteristic
In characteristic zero one defines invariants of singularities
using the order of vanishing along various divisors. In practice, one can
compute them using a resolution of singularities. I will discuss some related
invariants defined in positive characteristic. While their definition is very
elementary, using the Frobenius morphism, they seem to encode
subtle arithmetic information. I will discuss rationality properties of these
invariants, as well as known and conjectural connections between the
invariants in characteristic zero and those in characteristic p.
|
| Nov 29 |
Chris Skinner
Princeton University |
Zeros of L-functions and ranks of Selmer groups
It has long been recognized that certain `special'
zeros of L-functions (motivic, automorphic, pick your favorite flavor)
have an arithmetic significance. For example, the order of the zero of
the Dedekind zeta function of a number field is equal to the rank of
the group of units of the integer ring of this field (so the fact that
the Riemann zeta function does *not* vanish at s=0 reflects the fact
that the units of the integers are a finite group). Similarly, the
order of vanishing at s=1 of the L-function of an elliptic curve over
the rationals is expected to be the rank of the group of rational
points on the curve.
In this talk I will describe some recent efforts to provide theoretical
evidence for this expectation and its generalizations to other
L-functions.
|
| Dec 6 |
Yakov Sinai
Princeton University |
Blow ups of complex solutions of 3D-Navier-Stokes system and
Renormalization Group Method.
In this talk I shall explain the following result of Dong Li and
mine:there exists an open set in the space of 10-parameter families of
initial conditions such that for each family from this set there are
values of parameters such that the corresponding solution develops blow up
in finite time. |
| Dec 13 |
Paul Baum
Penn State University |
Trees, elliptic operators, and K-theory for group C*-algebra
Let G be a locally compact Hausdorff second countable topological group. Examples are Lie groups, discrete groups, p-adic groups and adelic groups.
The regular representation of G gives rise to a C* algebra known as the reduced C* algebra of G. Twenty five years ago P.Baum and A.Connes
conjectured an answer to the problem of calculating the K-theory of this C* algebra. When true, this conjecture has corollaries in various branches
of mathematics. Among these corollaries are the Novikov conjecture (topology) and the stable Gromov-Lawson-Rosenberg conjecture (differential
geometry). In essence, the conjecture asserts that every element in the K-theory of the reduced C* algebra of G is the index of some G-equivariant
elliptic operator, and that the only relations on these indices are the "obvious" index preserving relations. This is made precise by using the
universal example for proper actions of G. In low dimensions this universal example is a tree.
Due to the work of a number of mathematicians, the conjecture is now known to be true for certain classes of groups (e.g. connected Lie groups,
discrete hyperbolic groups, discrete a-t-menable groups, algebraic p-adic groups, algebraic adelic groups). The search for a counter-example
(to a somewhat generalized version of the conjecture) has led to some intriguing questions involving the expander graphs of Lubotzky-Sarnak
and a random group (which probably exists) of Gromov.
The talk is intended for a general mathematical audience. The basic definitions (C* algebra, K-theory etc) will be carefully stated in the talk. |