Updated: 09/01/24
I have a YouTube channel
- you probably are aware of this because it's the only way I really advertise this website, but if you weren't, you are now!My content is quite varied - some videos are lecture courses in topics varying from A-level calculus to Path Integration for quantum field theory, while I also run a vlog, "Sam's Maths Adventures", where I make videos discussing what maths and physics I've been up to recently, whether that's an interesting result or just a summary of some topic I've been focussing on.
This page will act as a sort of blog - I'll write about upcoming content here, as well as a list of videos currently uploaded and I'll link any relevant notes/additional materials as well.
Episode 1: The First One (here) The beginning of Sam's Maths Adventures - I talk about my degree and what I'd studied so far, and introduce the series.
Episode 2: Nathan Jacobson and Murray Gell-Mann (here and also here) A long episode (in two parts) about two under-appreciated geniuses in maths and physics. The first part is about Nathan Jacobson and Jacobson rings, the second talks about Murray Gell-Mann and the invention of the quark model.
Episode 3: If It Looks Like A Tensor, And It Quacks Like A Tensor...(here) "What is a tensor?" is a very common question in physics degrees, and "Something that transforms like a tensor" is the usual answer. In this video I give a hopefully more satisfying answer.
Episode 4: How To Differentiate \(e^x\) (Not The Way You Learn In School!)(here) I differentiate \(e^x\) completely rigorously in a way that doesn't assume I already know the answer, first by properly defining \(a^x\), then by differentiating from first principles and determining properties of the mysterious \(k_a\) function. Notes to accompany the episode are available here.
A series of lectures on path integral methods and their applications in quantum field theory. Lecture 1: Gaussian Integrals. I demonstrate how to calculate very general Gaussian integrals, i.e. \(\int_{\mathbb{R}^d}\mathrm{d}^d\mathbf{x}\:Ae^{-\frac{1}{2}\mathbf{x}^T M\mathbf{x}+\mathbf{J}\cdot\mathbf{x}}\) for any positive integer \(d\), symmetric positive-definite matrix \(M\), and vector \(\mathbf{J}\). Lecture 2: Sources and Feynman Diagrams. I use the results of the previous lecture to calculate cumulant expansions, i.e. integrals \(\int_{\mathbb{R}^d}\mathrm{d}^d\mathbf{x}\:Ax_{i_1}\cdots x_{i_n}e^{-\frac{1}{2}\mathbf{x}^T M\mathbf{x}}\), and introduce the method of Feynman diagrams to calculate these. Lecture 3: Fourier Theory. I establish the central results of the Fourier transform in \(d\) dimensions, which will be used to move between position- and momentum- space. This can probably be skipped if you already know the content, but does establish the conventions I'll use.