In this talk, I will present a numerical method for solving tracking-type optimal control problems subject to scalar nonlinear hyperbolic balance laws in one and two space dimensions. The approach is based on the formal optimality system and requires numerical solutions of the hyperbolic balance law forward in time and its nonconservative adjoint equation backward in time. To this end, we develop a hybrid method, which utilizes advantages of both the Eulerian finite-volume scheme (for solving the balance law) and the Lagrangian discrete characteristics method (for solving the adjoint transport equation). Experimental convergence rates as well as numerical results for optimization problems with both linear and nonlinear constraints and a duct design problem will also be presented and discussed.

We talk about about a Lioville type result for the sigma-2 equation: any entire (semi-)convex solution must be quadratic. This is joint work with Alice Chang.

In this talk, I will use the shortest path problem as an example to
illustrate how one can connect optimization, stochastic differential
equations and partial differential equations together to solve some
challenging real world problems. On the other end, I will show what
new and challenging mathematical problems can be raised from those applications. The talk is based on a joint work with Shui-Nee Chow
and Jun Lu.

We will discuss the question of defining a p-adic L-function and formulating a main conjecture for an Artin representation. The case where the Artin representation is totally even (or odd) is classical. The corresponding main conjecture has been proven by Wiles.  This talk will discuss the special case where the representation is 2-dimensional, but not totally even or odd. As we will explain, under certain assumptions, there are two p-adic L-functions, two Selmer groups, and two main conjectures. This talk is about joint work with Nike Vatsal.