I will discuss a recent proof of a weak upper bound on the
coarsening rate of the discrete-in-space version of an
ill-posed, nonlinear diffusion equation. The continuum
version of the equation violates parabolicity and lacks
a complete well-posedness theory. In particular,
numerical simulations indicate very sensitive
dependence on initial data. Nevertheless, models based
on its discrete-in-space version, which I will discuss,
are widely used in a number of applications, including
population dynamics (chemotactic movement of bacteria),
granular flow (formation of shear bands), and computer
vision (image denoising and segmentation). The bounds
have implications for all three applications. This is
joint work with Selim Esedoglu (U. of Michigan
Mathematics).

In this talk I will introduce the study of finite
dimensional division algebras and the Brauer group, and I will discuss
the fundamental problem of computing the index of a Brauer class. It
turns out that algebraic geometry can play an important role in this
problem. In particular I will describe the theory of twisted sheaves
and how it allows one to solve this problem in certain cases.

The study of cycles on homogeneous varietes has seen a great deal of
activity in the past few years. In particular, new results about
cycles on quadric hypersurfaces has resulted in fundamental
breakthroughs in the theory of quadratic forms.

The goal of this talk will be to answer a basic question about cycles
on homogeneous of more general types. In particular, we will address
the problem of calculating the group of zero dimensional cycles on
such varieties. In the case of quadrics, this was first done by Swan
in 1989 (and also independently by Karpenko). Similar computations
were made later for certain classes of other homogeneous varieties by
Merkurjev and Panin.

By using the geometry of Hilbert schemes of points on homogeneous
varieties, we will describe how to extend the previous results and to
compute the group of zero cycles for some homogeneous varieties of
each of the classical types.

A new proof of the hook formula
Abstract: The hook-length formula is a well known result expressing the
number of standard tableaux of shape $\lambda$ in terms of the lengths
of the hooks in the diagram of $\lambda$. Many proofs of this fact have
been given, of varying complexity. I'll give a new and simple proof
which uses only some power series and partial fractions expansions.
Other versions of the hook formula will also be discussed.

Suppose a domain has a defining function which is plurisubharmonic at each boundary point.
Does it have a plurisubharmonic defining function?
I will report on some recent progress on this question. This is joint work with Anne Katrin Herbig.

Exotic spheres, manifolds homeomorphic but <italic>not diffeomorphic</italic> to the standard sphere, had eluded mathematicians since the dawn of differential topology. In 1956, John Milnor found one in seven dimensions. His short paper on the result stunned the math world and won him the Field's medal. In this talk, we'll survey the ideas of smooth structures on manifolds and smooth vector bundles, and with a few smooth moves we'll construct Milnor's exotic sphere.

Note: This is meant to be a highly accessible talk; some familiarity with algebraic topology will enhance the experience, but is not strictly necessary!

Elliptic cohomology is a field at the intersection of number theory,
algebraic geometry and algebraic topology. Its definition is very
technical and highly homotopy theoretic. While its geometric definition
is still an open question, elliptic cohomology exhibits striking formal
similarities to string theory, and it is strongly expected that a
geometric interpretation will come from there.

To illustrate the interaction between the two fields, I will speak about
my work on orbifold genera and product formulas:
After a very informal introduction to elliptic cohomology, I will
discuss string theory on orbifolds and explain how a formula by
Dijkgraaf, Moore, Verlinde and Verlinde on the orbifold elliptic genus
of symmetric powers of a manifold motivated my work in elliptic
cohomology. I will proceed to explain why elliptic cohomology provides a
good framework for the study of orbifold genera.

Superspecial abelian varieties are positive characteristic
phenomenon. These are abelian varieties enjoying many special, even
super special l;-) properties, which I'll attempt to explain. After some
background about stratifications of moduli spaces and how, in that
context, the superspecial locus is the smallest, I will describe some
more recent work on superspecial abelian varieties. In particular, work
of M.-H. Nicole and A. Ghitza that makes use of the superspecial locus
to construct modular forms and mention very briefly some beautiful
graphs that can be constructed from the superspecial locus.

The relation between modular forms and Galois representations
is more popular than ever, thanks to recent progress on the
Fontaine-Mazur conjecture by Kisin, and on Serre's conjecture by Khare
and Wintenberger. After a short introduction to this circle of ideas, I
will discuss ongoing work on higher-dimensional generalizations, in the
context of Gross' philosophy of modular Galois representations.
I will attempt to make the talk as self-contained and accessible as
possible.