HUJI Combinatorics Seminar
When: Monday June 20th, 2022, at 11AM (Israel time)
Where: Sprinzak 202
Link to live session:
Speaker: Moshe White (HUJI)
Title: Representability by convex sets in R^d as an embedding problem
An abstract simplicial complex K is said to be d-representable if it records the intersection patterns of a collection of convex sets in R^d. We start by defining a dual complex K', and prove that K is d-representable if and only if there exists a linear map from K' into Rd, which avoids certain unwanted intersections. This equivalence can be seen as a wide generalization of a 2011 result by Tancer, and suggests a framework for proving (and disproving) d-representability of simplicial complexes using topological methods such as applications of the Borsuk-Ulam theorem.
In addition to general results, when we restrict the problem to low dimensions we reclassify 1-representable complexes, as well as d-representability for complexes such that K' has dimension 1. The first unsolved case is 2-representability of graphs, where the Borsuk-Ulam framework fails to differentiate 2-representable graphs from string graphs.
We will also discuss other properties related to d-representability, such as d-Leray and d-collapsibility.