We will discuss some of the local theory of rigid-analytic spaces including Tate's algebra, affinoid algebras, Washnitzer's algebra and dagger algebras. After we provide enough motivation we will discuss the results of research completed by myself and Professor Daqing Wan. The results of this research form a basis for generalizing Washnitzer's algebra.

The Continuum Problem, or Hilbert's first problem, asks whether the Continuum Hypothesis is true. It's arguably the most famous unsolved
problem from Hilbert's list. In this talk, I'll present recent progress made in set theory related to the Continuum Problem.
I'll point out the metamathematical significance of the Continuum Hypothesis through a
stunning theorem of Hugh Woodin which roughly states that the Continuum Hypothesis is a universal \Sigma^2_1 statement for generic absoluteness. If time permits, I'll talk about \Omega-logic, a strong logic used to analyze truth in the structure (H(\omega_2), \in) which could settle the Continuum Problem.

I will present numerical methods for two recent optimal control projects.

The first of these (joint work with J. Andrews) deals with deterministic optimal control of processes with probabilistically specified fixed-horizon. Subject to additional technical assumptions on cost & dynamics, this problem can be converted to an infinite-horizon obstacle problem. Despite the occurrence of non-trivial free boundary, we show that causal numerical algorithms (e.g., Fast Marching, Ordered Upwind) are still applicable. We illustrate our method using examples from optimal idle-time processing.

The second project (joint with A. Kumar) deals with multiple criteria for optimality (e.g., fastest versus shortest trajectories) and optimality under integral constraints. We show that an augmented PDE on a higher-dimensional domain describes all Pareto-optimal trajectories. Our numerical method uses the causality of this PDE to approximate its discontinuous viscosity solution efficiently. The method is illustrated by problems in robotic navigation (e.g., minimizing the path length and exposure to an enemy observer simultaneously)