Syllabus

Download a full course syllabus here.

Course Description

This course presents mathematical methods that are useful in the physical sciences. While proofs and demonstrations are a core part of the course, we will put the primary emphasis on applications. In a wonderful article the theoretical physicist Eugene Wigner explored what he called the ''unreasonable effectiveness of mathematics in the physics sciences''. The article concludes

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

Our aim will be to explore some of the many branches that this miracle has already extended to: how to think of probabilities as a degree of rational belief, how to take derivatives and calculate integrals in many dimensions and the surprising relations that arise between regions and their boundaries, the utility of complex numbers in algebra and in wave physics, and the remarkable versatility of complex numbers in mapping flows and calculating integrals. In this course we will use the rich relationships that arise between collections of more than one variable as the mountain pass to traverse to get to all of the ideas. Not only are these methods of great utility in applications, but their practice in physics has also often led to new discoveries in mathematics!