Oxford

Starting from Hilbert's 10th problem, I will explain how to characterise the set of integers by non-solubility of a set of polynomial equations and discuss related challenges. The methods needed are almost entirely elementary; ingredients from algebraic number theory will be explained as we go along. No knowledge of first-order logic is necessary.

I will give a historical overview of some of the theorems proved by the
Ancient Greeks, which are now taken for granted but were, and are,
landmarks in the history of mathematics. Particular attention will be
given to the calculation of areas, including theorems of Hippocrates,
Euclid and Archimedes.

I will discuss a notoriously hard problem in group theory known as the flat closing conjecture. This states that a group with a finite classifying space is either hyperbolic or contains a Baumslag-Solitar Subgroup. I will give some strategies to try and create a counterexample to this conjecture. 

Let p>0 be a prime number. We shall describe a short Frobenius-theoretic proof of the Adams-Riemann-Roch theorem for the p-th Adams operation, when the involved schemes live in characteristic p and the morphism is smooth. This result implies the Grothendieck-Riemann-Roch theorem for smooth morphisms in positive characteristic and the Hirzebruch-Riemann-Roch theorem in any characteristic. This is joint work with R. Pink.

New ambitwistor string models are presented for a variety of theories and older models are shown to work at 1 loop and perhaps higher using a simpler formulation on the Riemann sphere.

Many people talk about properties that you would expect of a group. When they say this they are considering random groups, I will define what it means to pick a random group in one of many models and will give some properties that these groups will have with overwhelming probability. I will look at the proof of some of these results although the talk will mainly avoid proving things rigorously.

In 1964 Golod and Shafarevich discovered a powerful tool that gives a criteria for when a certain presentation defines an infinite dimensional algebra. In my talk I will assume the main machinery of the Golod-Shafarevich inequality for graded algebras and use it to provide counter examples to certain analogues of the Burnside problem in infinite dimensional algebras and infinite groups. Then, time dependent, I will define the Tarski number for groups relating to the Banach-Tarski paradox and show that we can using the G-S inequality show that the set of Tarski numbers is unbounded. Despite the fact we can only find groups of Tarski number 4, 5 and 6.

The study of rational points on K3 surfaces has recently seen a lot of activity. We discuss how to compute the Picard rank of a K3 surface over a number field, and the implications for the Brauer-Manin obstruction.