Current students

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: 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: Cyclic sieving with focus on open problems

Abstract:

The cyclic sieving phenomenon (CSP) connects a cyclic group action on a family of combinatorial objects with some q-analog of that set. We discuss some recent results and open problems for standard and semistandard tableaux, as well as some other families of combinatorial objects.
Several open problems with various levels of difficulty will be presented.

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: Faster Algorithms for Isogeny Problems using Torsion Point Images. 

Abstract:

In this talk, we will discuss cryptanalysis of some SIDH-type protocols due to Christophe Petit: https://eprint.iacr.org/2017/571.pdf. While finding isogenies between supersingular elliptic curves remains computationally hard to solve, knowledge of images of specific torsion points by the unknown isogeny helps build faster attacks.

Title: On Donuts and Quasigraphic matroids

Abstract:

Quasigraphic matroids are graph-like objects that give a common generalization of lift and frame matroids. Donuts are edible topological surfaces. I will talk about a surprising link between these two types of object, assuming no prior knowledge of quasigraphic matroids or donuts.

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: 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.

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: P-partition power sums

Abstract:

The Hopf algebra of symmetric functions is spanned by several important bases, including by power sum symmetric functions, which encode the class values of the characters of the symmetric group under the Frobenius characteristic map.