\indent We report on recent work joint with Jason Bell. Makar-Limanov showed that the quotient division ring of the first Weyl algebra contains a copy of a free algebra on 2 generators. We prove the natural analog of this result for arbitrary iterated skew polynomial rings of fields.

\indent We study the singular set of a singular Levi-flat real-analytic
hypersurface. We prove that the singular set of such a hypersurface is Levi-flat in the appropriate sense. We also show that if the singular set is small enough, then the Levi-foliation extends to a singular codimension one holomorphic foliation of a neighborhood of the hypersurface.

We examine the problem of data assimilation in two different ways:

(1) as a special case of optimization where one attempts to minimize the parameters and state variables of a model set of equations to a time series of observations. To put this problem in context, we look at an example from neuroscience where we optimize spiking neuron models to noisy experimental data.

2) In a path integral formulation using an example from partial differential equations (the barotropic vorticity equations used as a model) as a method for obtaining means and distributions from high level integrals. This approach does not rely on optimization,but on the evaluation of a high dimensional integral.

Both approaches lend themselves to parallel implementation on GPUs using NVIDIA's CUDA C language. These algorithms vary in complexity, with some taking advantage of phenomena from nonlinear dynamics to improve their behavior. We discuss some practical limitations to parallelization due to the hardware architecture and concerns surrounding memory management.

\indent In recent years, kinetic transport models have received a lot of attention in various fields, including semiconductor device modeling, plasma physics, etc. This talk will focus on the Boltzmann equation, which is one of the most important equations in statistical physics. The Boltzmann equations describe the time evolution of the probability density functions, and are generally composed of a transport part and a collision part. Those equations have a lot of interesting structures that comes from applications and are computationally challenging to solve. In this talk, we will look into two classes of Boltzmann equations: one being the Boltzmann-Poisson systems in semiconductor device simulations, and the other being the linear Vlasov-Boltzmann transport equations. The goal is to design computationally efficient schemes that can preserve the important structures of the physical systems. I will motivate the choice of the discontinuous Galerkin (DG) finite element methods for treating those equations. The DG schemes enjoy the advantage of conservative formulation, flexibility for arbitrarily unstructured meshes, with a compact stencil, and with the ability to easily accommodate arbitrary h-p adaptivity. Numerical issues such as implementation, algorithm design and analysis for suitable applications will be addressed. Benchmark numerical tests will be provided to demonstrate the performance of the scheme compared to existing solvers such as Monte-Carlo and finite difference solvers.

\indent Spectral Collocation methods are a powerful set of methods used to solve partial differential equations and ordinary differential equations numerically. This talk will present some of the basic theory behind Spectral Collocation methods, and provide several examples of how they can be utilized. Along the way, we will encounter approximation theory, orders of convergence, wave and soliton equations, and even some nice pictures. No prior knowledge of numerical analysis is assumed; if you are curious about numerical methods but don't have much background, this talk should provide a gentle introduction.

\indent Covariance matrix is of fundamental importance in many aspects of statistics. Recently, there is a surge of interest on regularized covariance matrix estimation using banding, tapering and thresholding methods, in high dimensional statistical inference, where multiple iid copies of the random vector from the underlying multivariate distribution are required.

\indent In the context of time series analysis, however, it is typical that only one realization is available, so the current results are not applicable. In this talk, we shall exploit the connection between covariance matrices and spectral density functions using the idea in Toeplitz~(1911) and Grenander and Szeg\"o~(1958)