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)

In the 1980s, van den Berg speculated that for all parallelepipeds the gap between the first two Dirichlet eigenvalues is bounded below by a constant. Yau subsequently formulated the fundamental gap conjecture:

For all convex domains in $\R^n$, the gap between the first two Dirichlet eigenvalues is bounded below by $\frac{3 \pi^2}{d^2}$, where $d^2$ is the diameter of the domain.

This talk concerns the spectral gap between Dirichlet eigenvalues of convex domains in $\R^n$, and in particular, the fundamental gap of simplices and triangles. I will discuss recent progress with Z. Lu on the fundamental gap conjecture for triangles and simplices, new connections between Neumann eigenvalues and Dirichlet gaps, and demonstrate a relationship between the fundamental gap and Bakry-Emery geometry. In conclusion, I will offer ideas and open problems.

It is well known that the discretization of a finite element method results in a linear system of equations, in which the matrix and the load vector are usually computed by numerical integration. The inexact integration may lead to a different linear system, and consequently, produce a different finite element solution. In this talk, we will first discuss the existing results on the impact of quadrature rules on the finite element approximation in the energy norm. Then, a sharp estimate on the convergence rate of the finite element approximation with numerical integration for linear functionals will be presented. This is the joint work with Ivo Babuska and Uday Banerjee.

Relations in tautological ring of moduli spaces of stable curves can produce universal equations for Gromov-Witten invariants for all compact symplectic manifolds. A typical example of such equations is the WDVV equation, which is a genus-0 equation and gives the associativity of the quantum cohomology. Finding such relations in higher genera is a very difficult problem. I will talk about some topological recursion relations for all genera which was proved in a joint paper with R. Pandharipande. Some of these relations can be used to prove a conjecture of Kefeng Liu and Hao Xu.