Mathematical Methods

I am currently the tutor for Mathematical Methods in Mansfield College.

Course Guide

This course is a course on functional analysis for aspiring (theoretical) physicists. The reason why we are doing this is two-fold – firstly, to build a (semi-)complete mathematical understanding of how functions should be formally analysed in mathematics, and two, how these concepts naturally arise in physics, and how they should be interpreted.

It might be a bit daunting to see Banach spaces and all these defintions appear at the first instance as some horrifyingly written up theorems, lemmae and corollaries. The course is set up this way to allow you to investigate how functions come into play in theoretical physics - we start with Banach spaces and Hilbert spaces which is where we can a notion of distance between functions (in something known as measure spaces). This will allow us to define some notion of a norm (some size of functions in this space), an inner product (the ‘angle’ between two functions), etc. Most importantly, this will allow us to define linear operators on Hilbert spaces which has a spectrum of eigenvalues.

This set-up will allow us to proceed and analyse how we can decompose functions into simple units for anlaysis in different Hilbert spaces. Fourier analysis and orthogonal functions are basically ways of decomposing functions into fundamental units in different Hilbert spaces, and we will then see how we can use these techniques to solve differential equations. In particular, in Sturm-Liouville problems, it is precisely the operator formalism that allows us to completely analyse the spectrum of eigenvalues in a differential operator.

If you cannot really comprehend all the mathematics the first time you go through the course, this is completely fine. I would encourage you to take your time and just understand some basic ideas of the course. The most important thing to take away from this course is how these methods arise and how to apply them in problems in physics. The mathematical intuition will come along as you go further into physics.

Resources

Below you can find some useful resources that I have compiled.

• Prof. David Skinner’s notes (in DAMTP) is very good and is a bit less mathematically-demanding so if you want a different perspective on things it might be a good start. link
• For some introductory text on measure theory, Banach spaces, Hilbert spaces and more Axler’s book is extremely well written. It is written for undergraduate mathematicians so it is quite a nice read. link.
• more will be updated soon.