Monte Carlo is a computational workhorse for valuation of financial securities and risk. It is directly applicable to almost all types of financial securities and is robust in that it is insensitive to the complexities of a security. On the other hand, Monte Carlo can be terribly slow and inaccurate. This talk will review the basics of Monte Carlo quadrature in the context of finance and methods for its acceleration, including variance reduction, quasi-Monte Carlo and ad hoc methods. American options, for which the exercise time is chosen by the option holder, are a class of securities to which Monte Carlo is not directly applicable. The talk will also describe the recently developed Least Square Monte Carlo (LSM) method for American options, some generalizations of LSM, and methods for estimating the accuracy of Monte Carlo for American options.

Conference on Geometric Analysis -- in honor of Peter Li's 60th Birthday

I will review recent work linking quantum dynamical estimates
and rates of mixing in fluid flow. The main result is a sharp classification
of stationary or time periodic flows that are particularly efficient mixers.
I will also formulate some open questions.

As an alternative to elliptic curve groups, Koblitz (1989) suggested Jacobians of hyperelliptic curves for use in public-key cryptography. Hyperelliptic curves can achieve the same level of discrete log based security as elliptic curves, whilst offering the potential advantage of being defined over much smaller fields. At present however, elliptic curves still outperform hyperelliptic curves in general, because of the significant difference in the complexity of computing group operations. Indeed, when deriving fast explicit formulas for elliptic curve computations, one is aided by the simple geometric "chord-and-tangent" description. In contrast, Cantor's algorithm for arithmetic in Jacobian groups suffers from more computationally heavy operations, such as Euclid's algorithm for finding the gcd of two polynomials, and the chinese remainder theorem. In this talk we discuss recent results which exploit a chord-and-tangent-like analogue for hyperelliptic curves. We give a simple description of higher genus Jacobian arithmetic and show that for genus 2 curves this gives rise to explicit formulas which are significantly faster than their Cantor-based counterparts. This is joint work with Kristin Lauter.