C5

I will review several known problems on the automorphism group of finite $p$-groups and present a sketch of the proof of the the following result obtained jointly with Jon Gonz\'alez-S\'anchez:

For each prime $p$ we construct a family $\{G_i\}$ of finite $p$-groups such that $|Aut (G_i)|/|G_i|$ goes to $0$, as $i$ goes to infinity. This disproves a well-known conjecture that $|G|$ divides $|Aut(G)|$ for every non-abelian finite $p$-group $G$.

Let F be a field. A symplectic alternating algebra over F

consists of a symplectic vector space V over F with a non-degenerate

alternating form that is also equipped with a binary alternating

product · such that the law (u·v, w)=(v·w, u) holds. These algebraic

structures have arisen from the study of 2-Engel groups but seem also

to be of interest in their own right with many beautiful properties.

We will give an overview with a focus on some recent work on the

structure of nilpotent symplectic alternating algebras.

By using the Weil-Gel'fand-Zak transform of Faddeev's quantum dilogarithm,

Andersen and Kasheav have proposed a new state-integral model for the

Andersen--Kashaev TQFT, where the circle valued state variables live on

the edges of oriented levelled shaped triangulations. I will look at a

couple of examples which give an idea of how the theories are coupled.

This talk will be a gentle introduction to the main ideas behind some of the obstructions to the Hasse principle. In particular, I'll focus on the Brauer-Manin obstruction and on the descent obstruction, and explain briefly how other types of obstructions could be constructed.

Mixed motives turn up in number theory in various guises. Rather than discuss the rather deep foundational questions involved, this talk will aim

to give several illustrations of the ubiquity of mixed motives and their realizations. Along the way I hope to mention some of: the Mordell-Weil

theorem, the theory of height pairings, special values of L-functions, the Mahler measure of a polynomial, Galois deformations and the motivic

fundamental group.

It is well known that one can attach Galois representations to certain modular forms, it is natural to ask how one might generalise this to produce more Galois representations. One such approach, due to Gross, defines objects called algebraic modular forms on certain types of reductive groups and then conjectures the existence of Galois representations attached to them. In this talk I will outline how for a particular choice of reductive group the conjectured Galois representations exist and are the classical modular Galois representations, thus providing some evidence that this is a good generalisation to consider.

We will discuss some geometric methods to study Diophantine equations. We focus on the case of elliptic curves and their natural generalisations: Abelian varieties, Calabi-Yau manifolds and hyperelliptic curves.