Since 2005 a class of algebras, the {\it Leavitt path algebras} $L_K(E)$ (for $K$ any field and $E$ any directed graph), has been a focus of investigation by both algebraists and C$^*$-analysts. In this talk I'll define these algebras, and give some of their general properties. Then I'll describe some of the current lines of investigation in the area. In particular, I'll show a connection between ideas from symbolic dynamics (``flow equivalence") and the Grothendieck group $K_0(L_K(E))$. With that connection in mind, I'll explain one of the most compelling open problems in Leavitt path algebras, the {\it Algebraic Kirchberg Phillips Question}, which can be paraphrased as: can we recover $L_K(E)$ from $K_0(L_K(E))$? While the answer to the corresponding question for graph C$^*$-algebras is {\it Yes}, there remains a barrier to a complete answer on the algebra side.

If $n$ is a nonnegative integer, a partition of $n$ is a sequence of weakly decreasing positive integers which sum to $n$. Partitions arise in the study of the symmetric group of permutations of the set $\{1, 2, \ldots, n\}$, the geometry of the Grassmannian of $k$-dimensional subspaces of an $n$-dimensional vector space, and in numerous combinatorial areas such as finite field theory. I will show how a visualization of partitions obtained by shoving boxes into a corner can be used to define a polynomial refinement of the binomial coefficients called the Gaussian polynomials and will discuss various properties of this polynomial refinement (and present at least one open problem).

It has been a long history of utilizing interactions in regression
analysis to investigate interactive effects of covariates on response
variables. In this paper we aim to address two kinds of new challenges
resulted from the inclusion of such high-order effects in the regression
model for complex data. The first kind arises from a situation where
interaction effects of individual covariates are weak but those of
combined covariates are strong, and the other kind pertains to the
presence of nonlinear interactive effects. Generalizing the single index
coefficient regression model, we propose a new class of semiparametric
models with varying index coefficients, which enables us to model and
assess nonlinear interaction effects between grouped covariates on the
response variable. As a result, most of the existing semiparametric
regression models are special cases of our proposed models. We develop a
numerically stable and computationally fast estimation procedure utilizing
both profile least squares method and local fitting. We establish both
estimation consistency and asymptotic normality for the proposed
estimators of index coefficients as well as the oracle property for
thenonparametric function estimator. In addition, a generalized likelihood
ratio test is provided to test for the existence of interaction effects or
the existence of nonlinear interaction effects. Our models and estimation
methods are illustrated by both simulation studies and an analysis of body
fat dataset.

Recent work of Eugene Gorsky and Andrei Negut have put the finishing touches of what may be viewed as the symmetric function side of the Extended Shuffle conjectures. Their work, together with Hikita's construction of the combinatorial side combine into a truly remarkable family of conjectures which beautifully extend the classical Shuffle conjecture. These developments have opened up a vast research area of Algebraic Combinatorics considerably enlarging classical symmetric function theory as well as the Theory of Parking functions created by Computer Scientists. This talk is an elementary introduction to this new research area.

It is well-known that adding extra capacity to queues in networks where individuals choose their own route can sometimes severely degrade performance, rather than improving it. We will discuss some simple examples of queueing networks where this is the case under probabilistic routing, but where under state-dependent routing the worst case performance is no longer seen. This raises the question of whether giving arrivals more information about the state of the network leads to better performance more generally.

This is joint work with Heti Afimeimounga, Lisa Chen, Mark Holmes, Bill Solomon, and, latterly, Niffe Hermansson and Elena Yudovina.
(Jointly sponsored by the Mathematics and ECE Departments.)

In this talk I will try to give some idea of what originally got me interested in the theory of noncommutative rings. The Weyl algebras are fundamental examples in noncommutative ring theory. They have many interesting properties which illustrate the many ways that noncommutative rings are more complicated (and intriguing) than commutative ones. After introducing these examples, I will discuss how Weyl algebras have served as a starting point for one avenue in my recent research.

In this work, we study singular solutions and pattern formation in aggregation swarming models in two dimensions. This class of models involve pairwise interactions and an active scalar equation in the continuum limit. We show the connection between this model and the classical vorticity equation from fluid dynamics. The aggregation model can lead to a rich family of patterns. We discuss the stability of the singular patterns formed with this model.