Other notes
- A classification of Gaussian primes
- This is my second year essay from Warwick wherein I classify which of the Gaussian integers are also prime in that ring. It uses only elementary number theory, hence why it is 5000 words long.
- Transcendence and irrationality proofs
- This, meanwhile, is my fourth year project from Warwick. Supervised by Dr Samir Siksek, it proves an important theorem in transcendental number theory, the Lindemann–Weierstraß theorem, and the surprising result that ζ(3) is irrational. It's not that this fact is surprising, what surprised people was that no one proved it until the late 1970s when, as one observer noted, the proof was something Euler could have done.
- On a transcendence type lemma
- This short note proves a standard result in transcendental number theory, demonstrating a lower bound for the transcendence type of a transcendental field extension.
- Strassmann's theorem
- One of our shorter-lived reading groups was on the p-adic numbers. Unfortunately our book of choice omitted Strassmann's theorem, so, using Cassel's Local Fields, I wrote this proof of the theorem and highlighted one use of it in solving a diophantine equation.
- On sieves
- This is a short essay about Selberg's sieve, written for Tim Browning's TCC course on analytic number theory. The tight word limit and my desire to use lots of baking puns means the actual maths fluctuates between tenuous and non-existent. Enjoy!
- How to write numbers
- In June 2008 we had a small workshop intended to allow Jeffrey Stopple and Lynne Walling to transfer their skills to us. The particular skills they transferred were the ones on how to give a talk to undergraduates or even younger students. The idea was that younger students get wiggly after a short time so we had to give a brief talk and then provide an activity for the other workshopees to work on. I talked about continued fractions before letting everyone figure out a few continued fractions of their own, and prove that √2 is irrational in the process.
- Some Pfaffian functions of degree two
- This short note demonstrates twice that the Bessel functions are in fact Pfaffian on suitable intervals of the real line. Which is nice.
- Two minutes of talk
- At the start of each year we mathematicians in Howard House each give a strictly timed two minute talk to everyone else. These are the slides from my effort at the start of the 2009-2010 year.
- Linfoot seminar
- Discussing the lamentable lack of opportunities for Bristol number theorists to talk about their work at their own institute, Tim Browning and I somehow talked our way into organising a new weekly seminar series—the Linfoot seminars, named after Heilbronn's collaborator Edward Linfoot. Agreeing to split the work-load straight down the middle, Tim organised everything and I gave the inaugural seminar. Here are the
LATEX -ed notes from it. - The square root of two is irrational
- It's a little known fact that Professor Trevor Wooley is writing a book containing a thousand different proofs that the square root of two is irrational. Since he had a birthday this past year I wrote up a little proof of my own. It uses a technique from a recent preprint of mine wherein I proved the irrationality of ζ(3). It's a bit like using a nuclear-powered-sledgehammer to crack an egg, but oh well.
- Twenty minutes of talk
- Part two of my series of "Talking about my research in an arbitrary number of minutes" talks is a twenty minute talk I gave to various members of the Bristol maths department showing how to apply Brun's little-known irrationality criterion to Apéry's sequences for ζ(3). It skimps on the details (there were applied mathematicians in the audience) and makes little-to-no sense without the accomanying rambling monologue, but here it is nonetheless.
