\indent Let $k$ be of a field of characteristic $0$. The first Weyl algebra $A_1(k) = k/(yx-xy-1)$ is $Z$-graded with deg$(x) = 1$, deg$(y) = -1$, and is a simple ring of $GK$-dimension $2$. Sierra has studied its category of graded modules and shown how to find all $Z$-graded algebras with an equivalent graded module category. Smith has also shown how the geometry of this example is related to a certain stack. Our goal is to study more general classes of $Z$-graded simple rings to find more examples which may have interesting algebraic and geometric properties. Specifically, we study the structure of $Z$-graded simple rings $A$ with graded quotient ring $Q$ such that $Q_0$ is a field with trdeg $Q_0 = GK A - 1$. As a special case, we can classify all $Z$-graded simple rings of $GK$-dimension $2$. This is joint work with Jason Bell.

\indent Generalized barycentric coordinate functions allow for novel, flexible finite element methods accommodating polygonal element geometries. The Sobolev-norm error estimates associated to such methods, however, require varying levels of geometric criteria on the polygons, depending on the definition of the coordinate functions. In this talk, I will discuss these criteria for a variety of coordinate definitions and discuss the practical tradeoffs between enforcing geometric constraints and computing finite element basis functions over polygons.

\indent In a recent paper, Rains and Vazirani used Hecke algebra techniques to develop $(q,t)$-generalizations of a number of well-known vanishing identities for Schur functions. However, their approach does not work directly at $q=0$ (the Hall-Littlewood level). We discuss a technique that is more combinatorial in nature, and allows us to obtain generalizations of some of their results at $q=0$ as well as a finite-dimensional analog of a recent summation formula of Warnaar. We will also briefly explain how
these results are related to $p$-adic representation theory. Finally, we will explain how this method can be extended to give an explicit construction of Hall-Littlewood polynomials of type $BC$.

\indent I will discuss results in analysis on a compact Lie group that can be obtained using Brownian motion. These include a ``Hermite expansion" and an analog of the Segal-Bargmann transform. Both results can be understood by lifting Brownian motion in the group to Brownian motion in the Lie algebra. I will also briefly discuss an open problem concerning the infinite-dimensional limit of these results.

\indent We present the context and theorems for noncommutative maps, or dimension free maps evaluated on tuples of matrices. This turns out to be much more rigid than functions from the classical commutative case of several complex variables or several real variables. We present some theorems of Helton et al on free analytic maps in the context of change of variables.

\indent Ionic size effects are significant in many biological systems. Mean-field descriptions of such effects can be efficient but also challenging. When ionic sizes are different, explicit formulas in such descriptions are not available for the dependence of the ionic concentrations on the electrostatic potential, i.e., there are no explicit, Boltzmann type distributions. This work begins with variational formulations of the continuum electrostatics of an ionic solution with such non-uniform ionic sizes as well as multiple ionic valences. An augmented Lagrange multiplier method is then developed and implemented to numerically solve the underlying constrained optimization problem. Extensive numerical tests demonstrate that the mean-field model and numerical method capture qualitatively some significant ionic size effects, particularly those for multivalent ionic solutions, such as the stratification of multivalent counterions near a charged surface. The ionic valence-to-volume ratio is found to be the key physical parameter in the stratification of concentrations. All these are not well described by the classical Poisson--Boltzmann theory, or the generalized Poisson--Boltzmann theory that treats uniform ionic sizes. Finally, various issues such as the close packing, limitation of the continuum model, and generalization to molecular solvation are discussed. This is joint work with Zhongming Wang and Bo Li.