New lower bounds for cap sets

To appear in Discrete Analysis

September 2022, summer research project supervised by Thomas Bloom

The admissible sets and code used in the paper can be found here



Mathematical Interests

I enjoy thinking about problems which are related to one or both of combinatorics and number theory - whether in their motivation, methods or applications. For example, in my recent research on the cap set problem, I have been studying subsets of a finite field without arithmetic progressions, and I proved a new lower bound on the maximum size of  a cap set. This problem exists as part of the wider branch of maths known as additive combinatorics, which is one of my main areas of interest. Additive combinatorics has the pleasant feature of combining both combinatorics and number theory, and is currently where most of my experience is. I plan to continue to pursue research within this area, both related to my previous work and on other problems in the field, and I am writing my Part C dissertation on 'Almost Periodicity in Additive Number Theory' with Thomas Bloom.

In addition to additive combinatorics, I am also intrigued by many parts of the wider field of combinatorics. Generally speaking, I am interested in problems which ask a question about how a relatively complicated local property can depend on a rather simple global structure. This includes, but is certainly not limited to: Ramsey theory type problems, probabilistic and polynomial methods and the broad areas of extremal combinatorics, arithmetic combinatorics and probabilistic combinatorics. Furthermore, I am interested in computational techniques and methods in combinatorics, as well as applications of combinatorics to other areas, especially computer science. 

Within number theory, my tastes include many of the major branches of modern number theory. As well as techniques and problems from analytic, combinatorial and algebraic number theory, I have interests in applied and computational number theory, including applications of number theory to cryptography, as well as the development of computational algorithms and methods for use within the more theoretical branches of number theory.