Title: Combinatorics of the Descent Set Polynomial and the Diamond Product

Abstract: In this talk, we will first examine the descent set polynomial, which was defined by Chebikin, Ehrenborg, Pylyavskyy, and Readdy in terms of the descent set statistics of the symmetric group.  We will explain why large classes of cyclotomic polynomials are factors of the descent set polynomial, focusing on instances of the 2pth cyclotomic polynomial for a prime p.  Next, the diamond product of two Eulerian posets will be discussed, particularly the effect this product has on their cd-indices.  A combinatorial interpretation involving weighted lattice paths will be introduced to describe the outcome of applying the diamond product operator to two cd-monomials.

Title: Ehrhart from Hilbert

Abstract: Many discrete problems in various mathematical areas arise from linear systems, thus they ask about integer points of polytopes in disguise. Ehrhart theory tries to develop tools to encode information about integer points of polytopes. One of the most important objects in Ehrhart theory is the so-called Ehrhart function. We will show that Ehrhart theory is closely related to commutative algebra. In particular, we will show how graded modules and the Hilbert function can be used to prove interesting results about the Ehrhart function of a lattice polytope.

Title:  My favorite unsolved math problems, interesting new developments in psychology, and various reasons why infinity makes me nervous

Abstract:  Mathematicians and scientists spend much of their time thinking about how to solve problems that no one knows how to solve. There are lots of cool unsolved math problems that are easy to explain and think about (even for elementary and high school students), but that are nevertheless incredibly hard for professional mathematicians to make any progress on. In this talk, I'll share a few of my favorite unsolved math problems, and discuss why they are my favorites. Along the way, we'll talk about some recent research in psychology and sociology that comes into play when people struggle to understand the mathematical unknown.

Title: "A Matrix Analysis of Centrality Measures"



Abstract:  When analyzing a network, one of the most basic concerns is identifying the "important" nodes in the network. What defines "important" can vary from network to network, depending on what one is trying to analyze about the network. In this paper by Benzi and Klymko several different centrality measures, methods of computing node importance, are introduced and compared. We will see that some centrality measures give more information about the network on a local scale, while others help to analyze on a more global scale. In particular, the paper analyzes the behavior of these measures as we let the parameters defining them approach certain limits that appear to be problematic.