Clifford Algebra: A Marvelous Machine Offered By the Devil

Geometry Topology Student Seminar
Wednesday, March 27, 2024 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 006
Jaden Wang – Georgia Tech
Thomas Rodewald

Clifford algebra was first developed to describe Maxwell's equations, but the subject has found applications in quantum mechanics, computer graphics, robotics, and even machine learning, way beyond its original purpose. In topology and geometry, Clifford algebra appears in the proofs of the celebrated Atiyah-Singer Index Theorem and Bott Periodicity; it is fundamental to the understanding of spin structures on Riemannian manifolds. Despite its algebraic nature, it somehow gives us the power to understand and manipulate geometry. What a marvelous machine offered by the devil! In this talk, we will investigate the unreasonable effectiveness of Clifford algebra by exploring its algebraic structure and constructing the Pin and Spin groups. If time permits, we will prove that Spin(p,q) is a double cover of SO(p,q), complementing the belt trick talk of Sean Eli.