Future students

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.

Title: Snake decompositions of lattice path matroids

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

Abstract: We study Ehrhart theory of lattice path matroid polytopes motivated by a conjecture by De Loera, Haws and Köppe. More specifically, we aim to understand the h*-vector of this family of matroid polytopes. To do so, we subdivide them into smaller matroid polytopes such that each piece is a snake, which are matroids such that their matroid polytope coincides with the order polytope of a fence poset.

Title: The Chambolle-Pock algorithm revisited: splitting operator and its range with applications

Abstract: Primal-dual hybrid gradient (PDHG) is a first-order method for saddle-point problems and convex programming introduced by Chambolle and Pock. Recently, Applegate et al. analyzed the behavior of PDHG when applied to an infeasible or unbounded instance of linear programming, and in particular, showed that PDHG is able to diagnose these conditions.

Title: Higher-order correlation inequalities for random spanning trees

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

Abstract: The connection between enumerating spanning trees of a graph and the theory of (linear) electrical networks goes all the way back to Kirchhoff's 1847 paper. It is physically sensible that if one increases the conductance of one wire in an electrical network, then the overall conductance of the network can not decrease. This corresponds to the less obvious fact that any two distinct edges are non-positively correlated, when one chooses a random spanning tree. Covariance is the 2-point "Ursell function'', and expectation is the 1-point Ursell function. For any subset of edges there is an associated Ursell function, and these are related to occupation probabilities by Möbius inversion. I will discuss some situations in which the signs of these Ursell functions can be predicted, yielding higher-order correlation inequalities for random spanning trees.

Title: Chromatic Number of Random Signed Graphs

Abstract: A signed graph is a graph where edges are labelled {+1,-1}. A signed colouring in 2k colours maps the vertices of a signed graph to {-k,...,-1,1,...,k}, such that neighbours joined by a positive edge do not share the same colour, and those joined by a negative edge do not share opposite colours. It is a classical result that the chromatic number of a G(n,p) Erdos-Renyi random graph is asymptotically almost surely n/(2log_b(n)), where p is constant and b=1/(1-p). We extend the method used there to prove that the chromatic number of a G(n,p,q) random signed graph, where q is the probability that an edge is labelled -1, is also a.a.s. n/(2log_b(n)), if p is constant and q=o(1).

Title: Fast Combinatorial Algorithms for Efficient Sortation

Abstract: Modern parcel logistic networks are designed to ship demand between given origin, destination pairs of nodes in an underlying directed network. Efficiency dictates that volume needs to be consolidated at intermediate nodes in typical hub-and-spoke fashion. In practice, such consolidation requires parcel-sortation. In this work, we propose a mathematical model for the physical requirements, and limitations of parcel sortation.

Title: A Simple Sparsification Algorithm for Maximum Matching with Applications to Graph Streams 

Abstract: In this talk, we present a simple algorithm that reduces approximating the maximum weight matching problem in arbitrary graphs to few adaptively chosen instances of the same problem on sparse graphs.

Title: Forest polynomials and harmonics for the ideal of quasisymmetric polynomials

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

Abstract: The type A coinvariant algebra, obtained by quotienting the polynomial ring by the ideal of positive degree symmetric polynomials, is a rich and active object of study. A distinguished basis for this quotient is given by Schubert polynomials. There is a dual to this story involving degree polynomials studied in depth by Postnikov-Stanley who shed light on their combinatorics.