The Orlik-Solomon algebra, the Varchenko-Gelfand ring, and complements of hyperplane arrangements

Given a real hyperplane arrangement, there are two very natural algebras one can define. The first, the Orlik-Solomon algebra, is quite famous and describes the cohomology of the complexified complement of a hyperplane arrangement. The second, the Varchenko-Gelfand ring, is far less studied, but describes the cohomology of the real complement of a hyperplane arrangement. In this talk, I will define these two algebras and explain the relationship between them. All of the concepts listed above will be defined; in particular, I will not assume familiarity with hyperplane arrangements. Note: this talk is technically a “pre”-talk for a talk that already happened, but can be watched here.