Talk 1. Justin Shaw, Pure Mathematics Department, University of Waterloo
“Spheres, the quaternions, and rotations”
The ideas developed in last week’s lecture on division algebras will be used to show which spheres are groups and to define the Hopf fibrations. We will then focus on the quaternions and discuss their usefulness in describing rotations in three dimensional space.
Talk 2. Spiro Karigiannis, Pure Mathematics Department, University of Waterloo
“Generalized cross products and geometry”
This will be a continuation of the previous lecture on real normed division algebras. We will define the notion of a k-fold real vector cross product, and state a classification theorem of Brown and Gray. The case k=1 is intimately related to real normed division algebras and this will be explained. Finally, we will discuss explicit examples of Riemannian manifolds admitting such vector cross product structures. If time permits, we will briefly define the notion of calibrations and calibrated submanifolds.