## Mathematics

From October 2009 to April 2013, I worked on a Ph.D in Mathematical Logic at the University of Leeds.

My supervisors were the late S. Barry Cooper (obituary) and Andy Lewis.

### Computability theory and algorithmic randomness

The topic of my thesis was computability theory, and more specifically algorithmic randomness.

I've written a quick introduction to algorithmic randomness for this website.

My full thesis is titled "Notions and applications of algorithmic randomness" (pdf, 801 kB).

### Papers and other mathematical downloads:

**A new total injection betting strategy** (September 2011) is my paper in which I give a direct construction of a partial computably random sequence which is not total injection random.

I presented an earlier version of the paper at the conference CiE 2011 in Sofia, Bulgaria. **The Angel Problem** was the subject of a couple of talks that I gave. Here are the slides: **Problem 10 of the 2010 LIMO mathematics competition** ask you to prove that certain matrices are nilpotent. The model solutions use the Cayley-Hamilton theorem. I give a purely combinatorial solution: **Sequences and nets in topology** is an expository article in which I show how sequences might fail to characterize topological properties such as openness, continuity and compactness correctly. Moreover, I define nets and show how they succeed where sequences fail. - In
**Embeddings into the countable atomless Boolean algebra** I prove that every countable distributive lattice embeds into the lattice of computable subsets of the natural numbers. This is a well known result, but a proof is nowhere to be found. I give a proof and explain some related mathematics. **Realizability Toposes** is the essay I wrote as part of Certificate of Advanced Study in Mathematics, commonly known as *Part III*, at the University of Cambridge in 2009.