A central problem in modern mathematical
finance is that of estimating the volatility
of financial time series, whether they are
equity prices, exchange rates, interest rates
or something else, such as options. A recent trend is to try to
estimate the implied volatility of an asset from
the fluctuations in the price of derivatives
whose underlying it is. This is the volatility
surface estimation problem. I will review briefly the
background and status of this problem, including
computational issues, and I will present a variational
theory for volatility surface estimation within
stochastic volatilty models. I will show the form
this theory takes under a fast mean reverting hypothesis
and I will conclude with a calibration of the theoretical
framework using SP500 options data.

This talk will serve as an introduction to the use of pairings
(especially Weil pairings on elliptic curves or abelian varieties)
in cryptography. We will mention some open questions that have
practical interest for cryptographers and should be more fully
explored by number theorists. We also show how abelian varieties
and the Weil restriction of scalars can (sometimes) be used to
"compress" points on elliptic curves.

This will be an introduction to the speaker's joint
FOCS (04) paper with Q. Cheng. I will outline the
main ideas linking the decoding of Reed-Solomon codes
to discrete logarithms and character sums over finite
fields.

Let G be a finite subgroup of SL(2, C) and let X be a
"nice" resolution of singularities of the singular space C^2/G. The classical McKay correspondence gives a bijection between the irreducible representations of G and the components of exceptional divisor in X (which give a basis of its second homology).

We explain how this correspondence follows from a more general statement on categories of sheaves, and give an overview of known generalizations to subgroups of SL(n, C)