Graduate Student Organization Seminar

Date: February 5, 2020

Time: 4:00PM - 5:00PM

Location: BLOC 628

Speaker: Taylor Brysiewicz

Title: A funny thing about HSO(4)

Abstract: The special orthogonal group SO(4) is an affine algebraic variety of dimension 6 and degree 40 and intersecting it with 4 very particular hyperplanes gives the "hollow special orthogonal matrices" HSO(4), an algebraic variety of dimension 2 and degree 40. Surprisingly, the degree is the same despite the special intersections and even more surprisingly, HSO(4) decomposes into 8 spheres and 6 tori whose intersections are encoded in the polyhedral geometry of the rhombic dodecahedron. A similar phenomenon occurs for SO(5). We will explain what we know about this behavior for SO(4) and SO(5) as well as the computational tools we used to investigate this problem. We hope that this talk serves as an invitation for others to find a more general explanation which extends to other SO(n).