# Mathematical Sciences

## Two Math Senior Thesis

**Date:** 3:30pm - 4:30pm PDT April 22, 2015
**Location:** John Howard 254

#### John Howard 254

**Maxwell’s Equations in Curved Spacetime**

Ali Brauer ’15

Major in Math and Physics

Maxwell’s equations are the foundation of classical electrodynamics, describing the propagation of the electromagnetic wave: light. Together, they form arguably the most beautiful set of equations in all of physics. In the notation of special and general relativity–that is to say, in four-dimensional spacetime–Maxwell’s equations can be expressed even more compactly and elegantly. In this presentation, we will study Maxwell’s equations in the context of general relativity, where gravitation manifests itself in the curvature of spacetime. Specifically, we will focus on the Schwarzschild metric, which describes the spacetime outside a massive spherical object, such as a black hole. Using tensor calculus and the notation of special and general relativity, we will show that Maxwell’s equations are wave equations. Next, we will study the behavior of light near black holes by analyzing the scalar wave equation. Lastly, we will determine how this behavior depends on initial conditions, such as distance from the black hole, and on the angular momentum of photons.

**A Mathematical Investigation of Optimal Control Theory**

Katie Keith ’15

Major in Mathematics

Optimal control theory, an extension of calculations of variations, was developed by applied mathematicians during the Cold War when they were studying minimum flight time interception of rockets. This presentation will give a brief explanation of optimal control theory, describe the mathematical proof of the Pontryagin Maximum Principle, the primary solution technique of optimal control problems, and extend the Pontryagin Maximum Principle to problems involving discounting and inequality constraints.