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. 

Since the early 1970's, it has been known in both, the mathematical physics and in the physics communities, that propagation of information in quantum spin chains cannot exceed the so-called Lieb-Robinson bound (effectively providing the quantum analog of the light cone from the relativity theory). Typically these bounds depend on the parameters of the model (interaction strength, external field). The recent Hamza-Sims-Stolz result demonstrates exponential localization (a la Anderson localization) of information propagation in most spin chains (in the sense of a given probability distribution with respect to which interaction and external field couplings are drawn). A natural question arises: what can be said about lower bounds on propagation of information in spin crystals (i.e. the case far from the one in which localization is expected), as well as in the intermediate case--the spin quasicrystals. This problem can be reduced to solving a linear ODE given by a Hermitian matrix, the solutions of which live on finite-dimensional complex spheres.

In this talk we shall discuss the history, give a general overview of the field, reduce the problem to an ODE problem as mentioned above, and look at some open problems. We shall also present some numerical computations with animations.

We recall some classical results on self-similar Markov processes and in particular explain how those relate to a very particular class of stochastic differential equations. Those SDEs have an infinite dimensional counter part which arise naturally in interacting diffusions and can be interpreted as generalized voter process.

In this talk we will present various results on the size of the nodal (zero) set for solutions of partial differential equations of elliptic and parabolic type. In particular, we will establish a sharp upper bound for the (n-1)-dimensional Hausdorff measure of the nodal sets of the eigenfunctions of regular analytic elliptic problems. We will also show certain more recent results concerning semilinear equations (e.g.  Navier-Stokes equations) and equations with non-analytic coefficients.