University of Oxford

In the first half of the talk we explain - in very broad terms - how the objects defined in the previous meetings are linked with each other. We will motivate this 'big picture' by briefly discussing class field theory and the Artin conjecture for L-functions. In the second part we focus on a particular aspect of the theory, namely the L-function preserving construction of elliptic curves from weight 2 newforms via Eichler-Shimura theory. Assuming the Modularity theorem we obtain a proof of the Hasse-Weil conjecture.

This talk is the second in a series of an elementary introduction to the ideas unifying elliptic curves, modular forms and Galois representations. I will discuss what it means for an elliptic curve to be modular and what type of representations one associates to such objects.

I'll start with the definition of the first Grigorchuk group as an automorphism group on a binary tree. After that I give a short overview about what growth means, and what kinds of growth we know. On this occasion I will mention a few groups that have each kind of growth and also outline what the 'Gap Problem' was. Having explained this I will prove - or depending on the time sketch - why this Grigorchuk group has intermediate growth. Depending on the time I will maybe also mention one or two open problems concerning growth.

There are two competing notions for a normal subsystem of a (saturated) fusion system. A recent theorem of mine shows how the two notions are related. In this talk I will discuss normal subsystems and their properties, and give some ideas on why this might be useful or interesting.

The first part of Galois' Second Mémoire, less than three pages of manuscript written in 1830, is devoted to an amazing insight, far ahead of its time. Translated into modern mathematical language (and out of French), it is the theorem that a primitive soluble finite permutation group has prime-power degree. This, and Galois' ideas, and counterexamples to some of

them, will be my theme.