Title: The Geometry and Combinatorics of Convex Union Representable Complexes  

Abstract: The study of convex neural codes seeks to classify the intersection and covering patterns of convex sets in Euclidean space. A specific instance of this is to classify "convex union representable" (CUR) complexes: the simplicial complexes that arise as the nerve of a collection of convex sets whose union is convex. In 2018 Chen, Frick, and Shiu showed that CUR complexes are always collapsible, and asked if the converse holds: is every collapsible complex also CUR? We will provide a negative answer to this question, and more generally describe the combinatorial consequences arising from the geometric representations of CUR complexes. This talk is based on joint work with Isabella Novik.