Events

Title: Distance magic labeling on directed graphs

Abstract:

This talk will explore two new types of distance magic labelings on directed graphs. Results on some specific classes of directed graphs will be discussed. We will also look at various properties of these two types of labelings and some construction techniques.

Title: Linear arboricity of sparse multigraphs via orientations

Abstract:

The linear arboricity $la(G)$ of a loopless multigraph $G$ is the minimum number of colors required to edge-color $G$ into linear forests, that is, forests whose components are all paths. The Linear Arboricity Conjecture of Akiyama, Exoo, and Harary asserts that the linear arboricity of a simple graph $G$ is at most $\lceil (\Delta(G)+1)/2 \rceil$.

Title: Generalized Subspace Subcode with Application in Cryptology

Abstract:

Most codes with an algebraic decoding algorithm are derived from Reed-Solomon codes. They are obtained by taking equivalent codes, for example Generalized Reed-Solomon codes, or by using the so-called subfield subcode method, which leads to Alternant codes over the underlying prime field, or over some intermediate subfield. The main advantage of these constructions is to preserve both the minimum distance and the decoding algorithm of the underlying Reed-Solomon code.

Title: Virtual characters of permutation statistics

Abstract:

Functions of permutations are studied in a wide variety of fields including probability, statistics and theoretical computer science. I will introduce a method for studying such functions using representation theory and symmetric functions. As a consequence, one can extract detailed information about asymptotic behavior of many permutation statistics with respect to non-uniform measures that are invariant under conjugation. The key new tool is a combinatorial formula called the path Murnaghan-Nakayama rule that gives the Schur expansion of a novel basis of the ring of symmetric functions. This is joint work with Brendon Rhoades.

Title: Algebraic Graph Theory

Abstract:

A graph is distance-regular if we can write the distance adjacency matrices as polynomials in the adjacency matrix. Distance-regular graphs are a class of graphs of significant interest to algebraic graph theorists for their structural and algebraic properties. The notion of distance-regularity can be weakened to a local property on vertices, but when every vertex in the graph is locally distance-regular, the graph will either be distance-regular or in the closely related class of distance-biregular graphs.

Title: Combinatorial atlas for log-concave inequalities

Abstract:

The study of log-concave inequalities for combinatorial objects have seen much progress in recent years. One such progress is the solution to the strongest form of Mason’s conjecture (independently by Anari et. al. and Brándën-Huh).

Title: Maximal cliques in strongly regular graphs

Abstract: In this talk, I will introduce a cubic polynomial that can be associated to a strongly regular graph Γ. The roots of this polynomial give rise to upper and lower bounds for the size of a maximal clique in Γ. I will explain how we can use this cubic polynomial to rule out the existence of strongly regular graphs that correspond to an infinite family of otherwise feasible parameters. This talk is based on joint work with Jack Koolen and Jongyook Park.

Title: Theorems and Exchange Graph Algorithms concerning Paths, Cycles and Trees

Abstract: Carsten Thomassen and I proved that in any graph G, the number of cycles containing a specified edge as well as all the odd-degree vertices is odd if and only if G is eulerian. Where all vertices have even degree this is a theorem of Sunichi Toida and where all vertices have odd degree it is Andrew Thomason's generalization of Smith's Theorem.

Title: 1-skeleton posets of Bruhat interval polytopes

Abstract:  Bruhat interval polytopes are a well-studied class of generalized permutohedra which arise as moment map images of various toric varieties and totally positive spaces in the flag variety. I will describe work in progress in which I study the 1-skeleta of these polytopes, viewed as posets interpolating between weak order and Bruhat order. In many cases these posets are lattices and the polytopes, despite not being simple, have interesting h-vectors.

Title: An Introduction to Nonnegativity and Polynomial Optimization

Abstract:

In science and engineering, we regularly face polynomial optimization problems, that is: minimize a real, multivariate polynomial under polynomial constraints. Solving these problems is essentially equivalent to certifying of nonnegativity of real polynomials -- a key problem in real algebraic geometry since the 19th century.