# Discrete Mathematics Seminar

This seminar is held on Thursdays at 11am. The Zoom Meeting number is 981 1773 0523

### Spring Quarter, 2021

- April 15,
**Marissa Masden** (UO)

Bent hyperplane arrangements and sign sequence complexes in neural networks
**Abstract**: We introduce neural networks as a composite of particular functions called layers, and briefly discuss how networks are used and trained. We then focus on the combinatorics of ReLU neural networks, referring to E. Grigsby & K. Lindsey’s 2020 work constructing canonical polyhedral complexes corresponding to a network. We will see that hyperplane arrangements play a role in the combinatorics of these polyhedral complexes: A neural network has a corresponding hyperplane arrangement for each layer, which together give rise to a bent hyperplane arrangement of the network, which almost always forms the 1-skeleton of the canonical polyhedral complex of the network. Consequently, basic ideas in hyperplane arrangements translate nicely to this setting: we describe a “sign sequence complex” which is almost always dual to the polyhedral complex and gives rise to some algebraic structure. We are interested in understanding further directions these results can be taken.

- May 13,
**Grace Stadnyk** (Furman University)

A Recursive Formulation of CC-Shellability
**Abstract**: Lexicographic shellability is a popular combinatorial technique used to study the topology of the order complex of a poset. The most commonly used flavor of lexicographic shellability, CL-shellability, was introduced by Björner and Wachs, who subsequently developed the notion of recursive atom orderings as a useful and alternative method for identifying posets that are CL-shellable. In this talk, I will introduce generalized recursive atom orderings, which can be viewed as a recursive formulation of the seemingly more general version of lexicographic shellability called CC-shellability. I will then discuss how these generalized recursive atom orderings lead to some surprising results about lexicographic shellability and the relationship between CL-shellable posets and CC-shellable posets. This is joint work with Patricia Hersh.