SR1

The AJ conjecture relates two different knot invariants, namely the coloured Jones polynomial and the A-polynomial. The approach we will use will be that of 2+1 dimensional Topological Quantum Field Theory. Indeed, the coloured Jones polynomial is constructed in Reshetikhin and Turaev's formulation of a TQFT using quantum groups. The A-polynomial is defined by a subvariety of the moduli space of flat SL(2,C) connections of a torus.  Geometric quantization on this moduli space also gives a TQFT, and the correspondence between these provides a framework where the knot invariants can be compared. In the talk I will sketch the above constructions and show how we can do explicit calculations for simple knots. This is work in progress joint with J. E. Andersen.

In the previous talks we have seen the formulation of the Birch--Swinnerton-Dyer conjecture. This talk will focus on a fundamental question in diophantine geometry. Namely, given an algebraic curve \textit{C} defined over $\mathbb{Q}$ possessing at least one rational point, what is

the probability that \textit{C} has infinitely many rational points?

For curves of genus 0, the answer has been known ever since the ancient Greeks roamed the earth, and for genus > 1 the answer is also known (albeit for a much shorter time). The remaining case is genus 1, and this question has a history filled with tension and

conflict between data and conjecture.

I shall describe the heuristics behind the conjectures, taking into account the Birch--Swinnerton-Dyer Conjecture and the Parity Conjecture. I shall go on to outline the contrary numeric data, both in families of elliptic curves and for all elliptic curves of increasing conductor.

If one instead considers elliptic curves over function fields $\mathbb{F}_{q} (t)$, then, via a conjecture of Artin and Tate, one can compute the rank (and more) of elliptic curves of extremely large discriminant degree. I shall briefly describe the interplay between Artin--Tate and

Birch--Swinnerton-Dyer, and give new evidence finally supporting the conjecture.

In this talk I will outline the two constructions of the Brauer group Br($X$) of a scheme $X$, namely via etale cohomology and Azumaya algebras and briefly describe how one may compute this group using the Hochschild-Serre spectral sequence. In the early '70s Manin observed that one can use the Brauer group of a projective variety $X/k$ to define an obstruction to the existence of rational points on $X$. I will discuss this arithmetic application and time permitting, outline an example for $X$ a K3 surface.

This is the second (of two) talks concerning the Birch--Swinnerton-Dyer Conjecture.

This is the first (of two) talks which will be given concerning the Birch--Swinnerton-Dyer Conjecture.

A parabolic bundle on a marked curve is a vector bundle with extra structure (a flag) in each of the fibres over the marked points, together with data corresponding to a choice of stability condition Parabolic bundles are natural generalisations of vector bundles when the base comes with a marking (for example, they partially generalise the Narasimhan-Seshadri correspondence between representations of the fundamental group and semistable vector bundles), but they also play an important role in the study of pure sheaves on nodal curves (which are needed to compactify moduli of vector bundles on stable curves). Consider the following moduli problem: pairs $(C,E)$ of smooth marked curves $C$

and semistable parabolic bundles $E\rightarrow C$. I will sketch a construction of projective moduli spaces which compactify the above moduli problem over the space of stable curves. I'll discuss further questions of interest, including strategies for understanding the cohomology of these moduli spaces, generalisations of the construction to higher-dimensional base schemes, and possible connections with Torelli theorems for parabolic vector bundles on marked curves.