3/4/26 - Introduction to Special Relativity (Notes)
Abstract: Introduction to special relativity; Maxwell's equations and the constant speed of light; the Lorentz group and Lorentz transformations; some consequences
10/11/26 - Sturm-Liouville Theory and its Application (Notes)
Abstract: We learn in elementary calculus that it is possible to express a “well-behaved” function f as an infinite series known as the Taylor series, a sum of powers of x. In similar fashion, it is possible to express a well-behaved f as an infinite series constructed from other sets of functions, and these expressions have different applications depending on the sets of functions utilized. For instance, we can use a Fourier series to build a function using sine waves, and this was originally utilized in order to solve the heat equation, the differential equation modeling heat flow. The sets of functions that can be used for such constructions are special, and while all of these sets of functions may appear different from one another, there is a general framework called Sturm-Liouville theory that unites them all and describes the conditions placed upon each set. We will discuss this framework, and along the way, we will discuss what it means for a set of functions to be orthogonal and complete.
10/14/26 - Introduction to Morse Theory (Notes)
Abstract: Introduction to Morse Theory, Morse functions, and a worked example with CP^n.