TATERS

The classical Erdős-Ko-Rado theorem characterizes the maximum intersecting families of $k$-sets in $[n]$. Several analogs of these results arise from varying the ambient family and notion of intersection. Examples of such variations (in the ambient family) include characterizing the largest intersecting families of: permutations of a set, subspaces of a vector space, and triangulations of a convex $n$-gon.

In a recent paper, Frankl et al. characterize maximum $t$-intersecting families of spanning trees of $K_n$, for $n$ sufficiently large. They use a probabilistic tool in conjunction with an extramal set theoretic one, namely, the Lopsided Lovász Local Lemma (LLLL) and the recently developed spread approximation technique.

In this talk we present results on maximum $t$-intersecting families of $K_{n,n}$ where we replace the probabilistic tools (LLLL) of Frankl et al. with a spectral graph theoretic one - Laplacian eigenvalues of graphs.    

This is based on joint work with Gordian Bruns, Alexander Gavrilyuk, Josias Gomez and Nathan Lindzey.  

I will describe a random model for an n-fold branched cover of a finite acceptable 2-complex X. This includes presentation 2-complexes for finitely presented groups satisfying some mild conditions. In joint work with Rachel Skipper (U. Utah) and Hyeran Cho (Tufts U.), we showed that as n goes to infinity, a random branched cover asymptotically almost surely is homotopy equivalent to a 2-complex satisfying geometric small cancellation. I will spend the bulk of the time discussing the random model, explaining the statement of the theorem, and motivating why these notions (e.g. small cancellation) are interesting. Time permitting, I will also give a sketch of the main steps in the proof of the theorem. 

Schubert polynomials, introduced by Lascoux and Schutzenberger in 1982 as models for Schubert classes in the cohomology of flag manifolds, now count among the key characters in algebraic combinatorics.  These polynomials S_w(x1, x2, .. ) are indexed by permutations w; in addition to their origins in enumerative geometry, they admit many wonderful combinatorial descriptions.

In this talk, I’ll first explain these polynomials in elementary terms and then discuss the integers N_w = S_w(1, 1, .. ) obtained by setting all variables equal to 1 in a Schubert polynomial.  About 10 years ago, in two apparently unrelated contexts, Merzon-Smirnov and Stanley asked about the maximum value of N_w, for w in the group of permutations of [n], and about the asymptotics of this value as n grows.  Morales-Pak-Panova addressed these questions for a special class of “layered permutations”, and gave precise asymptotic formulas in this case.  In general, however, computational barriers make exhaustive search intractable already around n=15.  I’ll explain how recent work with Panova and Petrov on (bumpless) pipe-dream sampling led us to find non-layered permutations with large N_w, why we think that the asymptotics may nonetheless be captured by layered permutations, and the extent to which new AI tools did or didn’t help us.

Much work in algebraic combinatorics involves the comparison of two different bases of a vector space, and pictorial formulas for entries of the change-of-basis matrix that relates the two bases. Such formulas tell us how to compute a matrix entry by drawing certain pictures and playing a game with them. In one vector space known as the Hecke algebra, a certain basis defined by Kazhdan and Lusztig and another natural basis are related by a change-of-basis matrix having no known elementary formula. We will look at pictures and games which describe some entries of this matrix, and at the open problem of describing the remaining entries.

This is joint work with Tommy Parisi, Ben Spahiu, and Jiayuan Wang.