Autumn 2025

30 October

Jackson Lewis

Hyperbolic Angular Momentum from the ground up

Abstract:  We present the construction of “hyperbolic angular momentum” as given by Schwinger (1951), starting with the basic one-dimensional oscillator of classical mechanics. Then we introduce harmonic oscillators in quantum mechanics, angular momentum in classical and quantum mechanics, and the Jordan-Schwinger Map. Some applications will be discussed, possibly including constructions of spin networks and discrete spacetime operators in loop quantum gravity.  

From "Math for Poets..."

16, 23  Oct 2025

Mathew Gluck

The Method of Moving Spheres

Abstract. The method of moving spheres is a powerful and versatile method for analyzing partial differential equations with conformal symmetry. At the core of this method is the amazing fact that one can classify all suitably nice functions f  defined on Euclidean space for which both of the following properties hold:

1.  For every point x there is a sphere centered at x about which f  has inversion symmetry, and

2.  for every direction e, there is a hyperplane with normal direction e about which f  has reflection symmetry.

I will give some examples of functions for which both properties hold, and I will discuss the historical development of the classification of all such functions. Finally, I will overview the method of moving spheres and provide some applications of the method in the analysis of conformally covariant partial differential equations.