Recorded 10 February 2025. Galen Dorpalen-Barry of Texas A&M University - College Station presents "Yoshinaga's Criteron & the Topology of a Complexified Complement of a Real Arrangement" at IPAM's Computational Interactions between Algebra, Combinatorics, and Discrete Geometry Workshop. Abstract: The cohomology ring of the complement of a complexified real arrangement has a wonderful combinatorial description via the Orlik--Solomon algebra. The homotopy groups, however, are harder to pin down. In the 1980s, Salvetti introduced a homotopy model for this complexified complement. This model-- which you can view either as the order complex of a poset or a bunch of copies of the dual zonotope glued together-- is a useful tool for computing topology of these complements, but it doesn't immediately give a combinatorial description of the groups. One of the major open questions in this area is to give a simple combinatorial test to determine if all homotopy groups above the first one vanish. Last year, Yoshinaga introduced a new critreron for determining when this property fails. While it doesn't solve the open problem, we can use his work-- along with results from Edelman--Reiner, Bailey, Falk--Proudfoot, and others-- to characterize which subarrangements of the Type B reflection arrangement satisfy this property. In order to generate the conjectures that we ultimately proved by hand, we showed that Yoshinaga's criterion is equivalent to an algebraic one, implemented the algebraic criterion in Macaulay2, and ran parallelized calculations on a compute cluster to do exhaustive searches on large families of hyperplane arrangements. This is joint work with Graham Denham and Nick Proudfoot. Learn more online at: https://www.ipam.ucla.edu/programs/workshops/computational-interactions-between-algebra-combinatorics-and-discrete-geometry/