Future students

Title: A Primal-Dual Extension of the Goemans and Williamson Algorithm for Weighted Fractional Cut Cover

Abstract: A cut in a graph G = (V, E) is a set of edges which has one endpoint in S, for a given subset S of V. The fractional cut-covering number is the optimal value of a linear programming relaxation for the problem of covering each edge by a set of cuts. Beyond its role as part of Šámal's work on cut continuous functions, this graph parameter also arises as the gauge dual of the maximum cut problem.

Title: Quantum Algorithms and Mutually Unbiased Bases

Abstract: Mutually unbiased bases are fundamental to quantum information theory, showing up in quantum key distribution, quantum error correction, and quantum entanglement measures. They likewise might be familiar to algebraic graph theorists, thanks to their connection to equiangular lines and association schemes.

Title: Sets that Support a Joint Distribution

Abstract: Given a closed set on the plane and two probability distributions on the real line, when are there random variables with the given distributions whose joint distribution is supported by the given set?

Title: The heroes of digraphs: coloring digraphs with forbidden induced subgraphs

Abstract: The chromatic number is one of the most studied graph invariants in graph theory. $\chi$-boundedness, for instance, studies the induced subgraphs present in graphs with large chromatic number and small clique number. Neumann-Lara introduced an analog directed version of this graph invariant: the dichromatic number of digraphs.

Title: Four types of integrality on mixed Cayley graphs over abelian groups

Abstract: A mixed graph is called H-integral (resp. HS-integral) if the eigenvalues of its Hermitian-adjacency matrix (resp. Hermitian-adjacency matrix of second kind) are integers.

Title: Arrangements of Pseudolines, Tropical Grassmannians, and Generalized Scattering Amplitudes

Abstract: For each arrangement of (pseudo)lines on the projective plane, it is possible to construct a differential form that captures its combinatorial structure. The forms have simple poles whenever triangles shrink to a point in the arrangement, and share the same residue when two arrangements are connected via a "triangle flip".

Title: Distance-Regular and Distance-Biregular Graphs

Abstract: For a given diameter d and valency k, what is the maximum number of vertices a k-regular graph of diameter d can have, and what graphs meet that bound? Although there is a straightforward counting argument to bound the number of vertices using the structural information, the problem of characterizing the graphs that meet the bound turns out to be a problem in algebraic graph theory, and helps gives rise to the notion of distance-regular graphs.

Title: Inverse eigenvalue problem of a graph

Abstract:  We often encounter matrices whose pattern (zero-nonzero, or sign) is known while the precise value of each entry is not clear. Thus, a natural question is what we can say about the spectral property of matrices of a given pattern. When the matrix is real and symmetric, one may use a simple graph to describe its off-diagonal nonzero support.

Title: Quasisymmetric varieties, excedances, and bases for the Temperley--Lieb algebra

Abstract:  This talk is about finding a quasisymmetric variety (QSV): a subset of permutations which (i) is a basis for the Temperley--Lieb algebra TL_n(2), and (ii) has a vanishing ideal (as points in n-space) that behaves similarly to the ideal generated by quasisymmetric polynomials.   While this problem is primarily motivated by classical (co-)invariant theory and generalizations thereof, the course of our investigation uncovered a number of remarkable combinatorial properties related to our QSV, and I will survey these as well. 

Title : A Closure Lemma for tough graphs and Hamiltonian degree conditions

Abstract: A graph G is hamiltonian if it exists a cycle in G containing all vertices of G exactly once. A graph G is t-tough if, ,for all subsets of vertices S, the number of connected components in G − S is at most |S| / t.