Schedule for: 23w5003 - Interactions Between Topological Combinatorics and Combinatorial Commutative Algebra

In this talk, we will discuss several new algebraically-motivated directions in the study of simplicial complexes. They are: minimal Cohen-Macaulay complexes (which include many interesting old and new constructions in combinatorial topology), higher nerve complexes (which capture numerous algebraic invari-ants) and acyclicity results suggested by Kodaira vanishing. We hope to bring many unanswered questions to the attention of the audience.

The Garland method comes from geometric group theory and works for simplicial complexes $\Delta$ as follows: Goal: Show that $H_i(\Delta,\mathbb{Q}) = 0$ (only works for homology with coefficients in a field of characteristic $0$). Method: Consider the $1$-skeleta (graphs) of the links of all $(i-1)$-simplices in $\Delta$. If the smallest non-zero Eigenvalue of the normalized graph Laplacian of all those $1$-skeleta is $> i/(i+1)$, then $H_i(\Delta,\mathbb{Q}) = 0$. In recent and ongoing work with Eric Babson we have extended the method to chain complexes satisfying certain rather weak conditions. We have some applications in topological combinatorics, but we are looking for new ones arising from chain complexes in commutative algebra.

Introduction and presentations of problems by Adam Van Tuyl; Mina Bigdeli-Sara Faridi, Satoshi Murai-Isabella Novik.

Problem presentation by Susan Cooper-Sara Faridi-Susan Morey-Liana Sega, Sara Faridi, Russ Woodroofe, Victor Reiner, Ezra Miller, Alexandru Constantinescu, Christos Athanasiadis.

Any other problem (if any boday has)

The first cotangent cohomology module $T^1$ describes the first order deformations of a commutative ring. For Stanley-Reisner rings, this module has a purely combinatorial description: its multigraded components are given as the relative cohomology of some topological spaces associated to the defining simplicial complex. When the Stanley-Reisner ring is associated to a matroid, I will present a very explicit formula for the dimensions of these components. Furthermore, I will show that $T^1$ provides a new complete characterization for matroids. This talk is based on the joint work with William Bitsch: https://arxiv.org/pdf/2204.05777.pdf https://arxiv.org/pdf/2204.05777.pdf

I’ll review the topological view of the classical Alexander dual of a simplicial complex. I’ll give an alternative construction that may be well suited for the independence complex of a graph with a perfect matching.

This is an open talk slot. If nobody propose to talk, then this slot becomes a group work.

In 2022, Jinha Kim proved a conjecture by Engstrom that states the independence complex of graphs with no induced cycle of length divisible by 3 is either contractible or homotopy equivalent to a sphere. These graphs are called ternary. A direct corollary is that the minimal free resolution of the edge ideal of these graphs is characteristic-free. We apply this result to give a combinatorial description of projective dimension and depth of the edge ideals of ternary graphs. As a consequence, we give a complete description of the multigraded betti numbers of edge ideals of ternary graphs in terms of its combinatorial structure and classify ternary graphs whose independence complex is contractible.