Title: What is algebraic statistics?
Abstract: Much can be learned about statistical models by studying their corresponding algebraic varieties. Combinatorial objects including trees, permutations, matroids, and Grassmannians have a habit of appearing along the way. I will give two examples of models and their corresponding varieties.
Title: Counting colored solutions to linear equations
Abstract: A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations $\E$ there is a threshold value $R_k(\E)$ (the Rado number of $\E$) such that for any $k$-coloring of the integers in the interval $[1,n]$, with $n \ge R_k(\E)$, there exists at least one monochromatic solution. But one can further ask, \emph{how many monochromatic solutions is the minimum possible in terms of $n$?} Several authors have estimated this function before, here we offer new tools from integer and semidefinite optimization that help find either optimal or near optimal 2-colorings minimizing the number of monochromatic solutions of several families of 3-variable non-regular homogeneous linear equations. In the last part of the paper we further extend to three and more colors for the Schur equation, improving earlier work.
Title: Counting Vertices on Hyperplane Slices of Polytopes
Abstract: The study of slicing convex sets, especially polytopes, has been an active area of research in geometry and combinatorics, inspiring numerous investigations. In this talk, we will focus on the number of vertices in slices of polytopes. We will define the sequences of slices for a polytope and analyze the gaps within these sequences, providing new insights into their structure and properties.
Title: The $e$-positivity of twinned paths and cycles
Abstract: The operation of twinning a graph at a vertex was introduced by Foley, Ho\`ang, and Merkel (2019), who conjectured that twinning preserves $e$-positivity of the chromatic symmetric function. A counterexample to this conjecture was given by Li, Li, Wang, and Yang (2021). We show that $e$-positivity is preserved by one application of the twinning operation on path and cycle graphs, using recurrence relations and the triple deletion formulas of Orellana and Scott (2014). Based on joint work with E. Banaian, K. Celano, M. Chang-Lee, L. Colmenarejo, O. Goff, J. Kimble, L. Kimpel, J. Liang, and S. Sundaram.
Abstract: The descent and inversions statistics of two statistics of interest from combinatorics that are defined for the permutation group $S_n$. MacMahon introduced the major statistic which has the same generating function as the inversion statistic. Foata and Zeilberger (1995) characterized digraphs to extend the notion of inversion and descent statistics onto them. They defined the inversion and descent polynomials for digraphs. Celano et al (2023) observed that descent and inversion polynomials could be viewed as special cases of D-descents on different graphs. Celano et al further investigates the evaluations of the D-descents at -1. We extend existing results to signed permutations on bidirected graphs, with special cases giving Euler-Mahonian statistics for the hyper octahedral group.
Title: Charge monomials and Garsia--Procesi rings
Abstract: Charge is a statistic on tableaux (or more generally, words) that appear in the study of Hall-Littlewood polynomials. For each permutation w, we can construct a monomial called its charge monomial using this statistic. We give a construction for a set of charge monomials that are a monomial basis for the Garsia--Procesi rings.
Title: Splicing skew Schubert varieties.
Abstract: We study the relation between two classes of varieties admitting a cluster structure: \textbf{skew Schubert varieties} in the Grassmannian, introduced by Serhiyenko--Sherman-Bennett--Williams; and \textbf{double Bott-Samelson varieties} introduced by Elek--Lu--Yu. We simplify the cluster structures given by Galashin-Lam for the case of skew Schubert varieties, and compare this cluster structure by Shen-Weng for double Bott-Samelson varieties. Interestingly, the cluster structures on skew Schubert varieties can be described explicitly in a combinatorial way. This yields an explicit realization of skew Schubert as a double Bott-Samelson varieties. As an application of this connection, we provide a cluster compatible splicing map for skew Schubert varieties, generalizing the splicing map of Gorsky-Scroggin for the maximal positroid cell. More precisely, we show that double Bott-Samelson varieties admit splicing maps, which are predicted by results in link homology. This is a joint work with Eugene Gorsky, Tonie Scroggin and José Simental.
Title: Affine Springer Fibers and Link Homology
Abstract: We’ll briefly talk about what algebraic geometry (homology of affine Springer fibers) has to do with knot theory (Khovanov-Rozansky homology of links) and the combinatorics that arise from both.