Masters Exam

Speaker:  Ford McElroy, University of Kentucky

Title:         The Eulerian Transformation and Real-Rootedness

Abstract:

Many combinatorial polynomials are known to be real-rooted. Many others are conjectured to be real-rooted. The Eulerian Transformation is a map from A:R[t] --> R[t] generated by A(t^n)= A_n(t), the nth Eulerian polynomial. Brenti (1989) conjectured that the Eulerian Transformation preserves real-rootedness. In the 2022 paper The Eulerian Transformation by Brändén and Jochemko, they disprove Brenti's conjecture and make one of their own. In the talk, we will look at

 (i) polynomial properties related to real-rootedness, 

(ii) Brändén and Jochemko's counterexample to Brenti's conjecture

(iii) evidence Brändén and Jochemko provide to support their conjecture.

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: "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.