Current students

Title: Filtered deformations of commutative algebras

Note: There will be no pre-seminar.

Abstract: We’ll look at different ways of deforming the multiplicative structure of “classical” algebras to obtain new algebras and explain how this algebraic structure can often be understood combinatorially.  We’ll then look at a special class of algebras one can produce this way called filtered deformations and we’ll discuss a conjecture of Etingof which asserts that in positive characteristic that filtered deformations of commutative rings should be in some natural sense very close to being commutative themselves. Not much background will be assumed.

Title: Diagonal coefficients, graph invariants with the symmetries of Feynman integrals, and the proof of the c_2 completion conjecture

Abstract: In a scalar field theory the contribution of a Feynman diagram to the beta function of the theory, the Feynman period, can be written as an integral in terms of the (dual) Kirchhoff polynomial of the graph.

Title: Hypergraph Matchings Avoiding Forbidden Submatchings

Abstract: We overview a general theory for finding perfect or almost perfect matchings in a hypergraph G avoiding a given set of forbidden submatchings (which we view as a hypergraph H where V(H)=E(G)).

Title: Linear relations and Lorentzian property of chromatic symmetric functions

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:00 pm.

Abstract: The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian Wachs q-analogue (q-CSF) have important connections to Hessenberg varieties, diagonal harmonics and LLT polynomials. In the, so called, abelian case they are related to placements of non-attacking rooks by results of Stanley-Stembridge (1993) and Guay-Paquet (2013).

Title: Distance-regular graphs that support a uniform structure

Abstract: Given a connected bipartite graph $G$, the adjacency matrix $A$ of $G$ can be decomposed as  $A=L+R$, where $L=L(x)$ and $R=R(x)$ are respectively the  lowering and the raising matrices with respect to a certain vertex $x$. The graph $G$ has a \textit{uniform structure} with respect to $x$ if the matrices $RL^2$, $LRL$, $L^2R$, and $L$ satisfy a certain linear dependency.

Title: Extended Schur Functions and Bases Related by Involutions

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:00 pm.

Abstract: The extended Schur basis and the shin basis generalize the Schur functions to the dual algebras of the quasisymmetric functions and the noncommutative symmetric functions. We define a creation operator and a Jacobi-Trudi rule for certain shin functions and show that a similar matrix determinant expression does not exist for every shin function.

Title: Kemeny’s constant and random walks on graphs

Abstract: Kemeny's constant is an interesting and useful quantifier of how well-connected the states of a Markov chain are. This comes to the forefront when the Markov chain in question is a random walk on a graph, in which case Kemeny's constant is interpreted as a measure of how `well-connected' the graph is. Though it was first introduced in the 1960s, interest in this concept has recently exploded. This talk will provide an introduction to Markov chains, an overview of the history of Kemeny’s constant, discussion of some applications, and a survey of recent results, with an emphasis on those that are extensions or generalizations of simple random walks on graphs, to complement Sooyeong’s talk from two weeks ago.

Title: Algorithmic Tools for Congressional Districting: Fairness via Analytics

Abstract: The American winner-take-all congressional district system empowers politicians to engineer electoral outcomes by manipulating district boundaries. To date, computational solutions mostly focus on drawing unbiased maps by ignoring political and demographic input, and instead simply optimize for compactness and other related metrics.

Title: The Tutte Polynomial, Bipartite Representations of Graphs, and Grid Walking 

Abstract: The Tutte Polynomial has many equivalent definitions. It can be defined by a deletion-contraction relation with the terms determined by the sequence of contractions, deletions, loops, and isthmi. This definition is independent of edge order. Another definition relies on a fixed edge ordering and examines the edge activities over maximal spanning forests. There is a direct relationship between edge activity and deletion/contraction for a given edge ordering. Furthermore, the monomials of the Tutte polynomial can be interpreted as grid walks. This allows for an approach to the Tutte polynomial on hypergraphs by examining the grid walks of the bipartite representation of the graph.

Title: Graphs that Admit Orthogonal Matrices

Abstract: Given a simple graph $G=(\{1,\ldots, n},E), we consider the class $S(G)$ of real symmetric $n \times n$ matrices $A=[a_{ij}]$ such that for $i\neq j$, $a_{ij}\neq 0$ iff $ij \in E$. Under the umbrella of the inverse eigenvalue problem for graphs (IEPG), $q(G)$ - known as the minimum number of distinct eigenvalues of $G$ - has emerged as one of the most well-studied parameters of the IEPG. Naturally, characterizing graphs $G$ for which $q(G) \leq, =, \geq k$ is an important step for studying the IEPG.