Understanding biomolecules---their structures, dynamics, and interactions
with solvent---is essential to revealing mechanisms and functions of
biological systems. While atomistic simulations that treat both solvent
and solute molecules explicitly are usually more accurate, implicit or
continuum solvent models for biomolecules are far more efficient. With an
implicit solvent, the free energy and structure of an underlying solvation
system is described through the solute particles and the interface that
separates the solutes and solvent.

Dzubiella, Swanson and McCammon [Phys. Rev. Lett.104, 527 (2006) and J. Chem.
Phys. 124, 084905 (2006)] developed a class of variational implicit-solvent
models. Central in these models is a free-energy functional of all admissible
solute-solvent interfaces, coupling both nonpolar and polar contributions
of an underlying system. An energy-minimizing interface then defines an
equilibrium solute-solvent interface. Cheng et al. [J. Chem. Phys. 127,
084503 (2007)] developed a robust level-set method for numerically capturing
such interfaces.

In this talk, I will begin with a brief introduction of the level-set method.
I will then give details of the application of this method to the implicit-solvent
computation of nonpolar molecules. Finally, I will present some new results
on the coupling of the level-set method with molecular mechanics for
implicit-solvent modeling of molecules.

This is joint work with Jianwei Che, Joachim Dzubiella, Bo Li, J. Andy McCammon,
and Yang Xie.

MathStorm is a mathematics consulting group started by
graduate students at UCSD several years ago. It provides an
opportunity for graduate students to help or collaborate
with students and professors from other UCSD departments,
as well as with people from outside the university. I'll
describe some typical problems participants might work on
and discuss why this could be a useful or profitable thing
for graduate students to do.

We develop an arithmetic analogue of linear partial differential
equations in two independent ``space-time'' variables. The
spatial derivative is a Fermat quotient operator, while the time
derivative is a usual derivation. This allows us to ``flow''
integers or, more generally, points on algebraic groups with
coordinates in rings with arithmetic flavor. In particular, we
show that elliptic curves have certain canonical ``arithmetic
flows'' on them that are arithmetic analogues of the convection,
heat, and wave equations. The same is true for the additive and
the multiplicative group and also for modular curves.