# ACGT Seminar

The Algebra, Combinatorics, Geometry, and Topology (ACGT) Seminar meets on Tuesdays at 12:40-1:30PM in Room 164 of the Adel Mathematics Building. If you are interested in giving a talk, please contact Mike Falk or Dana C. Ernst, ACGT coordinators.

# Schedule Spring 2022

Note: The ACGT seminar is on zoom while the omicron variant surges.

### Tuesday, February 8, 15, and 22 (via zoom, 12:40 to 1:30)

Speaker: Jim Swift

Title: Polydiagonal subspaces, coupled cell networks, and Cayley digraphs

Abstract: A polydiagonal subspace of $\mathbb R^n$ is a linear subspace defined by equations of the form $x_i = \pm x_j$. Such subspaces form a patially ordered set, which is also a lattice. In coupled cell networks with the linear coupling of the $n$ cells determined by an $n \times n$ matrix $A$, the lattice of $A$-invariant polydiagonal subspaces describes the possible synchrony and anti-synchrony of the cells. Among other examples, we let $A$ be the adjacency matrix of a weighted digraph obtained from a Cayley digraph of the group $Q_8$ (the 8-element group of quaternions).

These talks describe work with Nandor Sieben, Eddie Nijholt, and Jack Eigen.

### Tuesday, March 22

Speaker: Mike Falk

Title: Models for braid-like groups and their pure subgroups

Abstract: We will start by describing a construction that produces a model for the braid group, starting from the graph of the permutahedron, and show how it can be generalized to graphic braid groups. The latter arise from consideration of fundamental groups of graphic hyperplane arrangements, which are “pure graphic braid groups.” We describe some partial structure theorems for those groups, ending with an open problem. This talk is based on joint work with various people including Dan Cohen, Emanuele Delucchi, Dana Ernst, and Sonja Riedel.

### Tuesday, April 5

Speaker: Mike Falk

Title: Coxeter complexes and Davis complexes

Abstract: Episode 1: We construct the Coxeter complex of a Coxeter system.