A place cell is a neuron corresponding to a subset of Euclidean space known as a place field, that will fire if and only if the individual to which the neuron belongs is within that place field. The firing patterns of a collection of n place fields can be represented by a neural code C on n neurons, which is a subset of 2n . Determining whether C is convex, meaning that there is an arrangement of convex place fields for which C is the code, remains an open problem. A sufficient condition for convexity is being max intersection complete: any intersection of maximal codewords is also a codeword. Currently, the only way to determine this property is to evaluate all such intersections. We present a new method to determine max intersection completeness by introducing a simplicial complex for a code C called the factor complex ∆O(C) of C. We show how to construct ∆O(C) using Stanley-Reisner theory, describe how ∆O(C) encodes information about C, and give an algorithm to check whether C is max intersection complete using the factor complex of a closely related code.

The coordinate ring $S = \mathbb{C}[x_{i,j}]$ of space of $m \times n$ matrices carries an action of the group $\mathrm{GL}_m \times \mathrm{GL}_n$ via row and column operations on the matrix entries. If we consider any $\mathrm{GL}_m \times \mathrm{GL}_n$-invariant ideal $I$ in $S$, the syzygy modules $\mathrm{Tor}_i(I,\mathbb{C})$ will carry a natural action of $\mathrm{GL}_m \times \mathrm{GL}_n$. Via BGG correspondence, they also carry an action of $\bigwedge^{\bullet} (\mathbb{C}^m \otimes \mathbb{C}^n)$. It is a recent result by Raicu and Weyman that we can combine these actions together and make them modules over the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. We will explain how this works and how it enables us to compute all Betti numbers of any $\mathrm{GL}_m \times \mathrm{GL}_n$-invariant ideal $I$. The latter part will involve combinatorics of Dyck paths

Intuitively, "dense" graphs contain any "desired substructure". In this talk, we will use a unified tool to prove few conjectures and open questions proposed since 1980s with this flavor, where the "desired substructure" is a set of cycles whose lengths satisfy certain conditions. They include two conjectures of Thomassen about minimum degree, a conjecture of Dean about connectivity, a conjecture of Sudakov and Verstraete about chromatic number, and an optimal answer of a question of Bondy and Vince about minimum degree. Joint work with Jun Gao, Qingyi Huo and Jie Ma.