It is well known that there is a finite colouring of the natural numbers such that there is no infinite set X with X+X (the pairwise sums from X, allowing repetition) monochromatic. It is easy to extend this to the rationals. Hindman, Leader and Strauss showed that there is also such a colouring of the reals, and asked if there exists a space 'large enough' that for every finite colouring there does exist an infinite X with X+X monochromatic. We show that there is indeed such a space. Joint work with Imre Leader.

Kim's iterative non-abelian reciprocity laws carve out a sequence of subsets of the adelic points of a suitable algebraic variety, containing the global points. Like Ellenberg's obstructions to the existence of global points, they are based on nilpotent approximations to the variety. Systematically exploiting this idea gives a sequence starting with the Brauer-Manin obstruction, based on the theory of obstruction towers in algebraic topology. For Shimura varieties, nilpotent approximations are inadequate as the fundamental group is nearly perfect, but relative completions produce an interesting obstruction tower. For modular curves, these maps take values in Galois cohomology of modular forms, and give obstructions to an adelic elliptic curve with global Tate module underlying a global elliptic curve.

.. showing that a field K is isomorphic to Q if it has the same absolute Galois group and if it satisfies a very small additional condition (very similar to my talk 2 years ago).

The Robin-Schensted-Knuth insertion algorithm provides a bijection between non-negative integer matrices and pairs of semistandard Young tableau. However, by relaxing the conditions on the correspondence, it allows us to define the Poirer-Reutenauer bialgebra, which exactly describes the algebra of symmetric functions viewed as generated by the Schur polynomials. This gives an interesting combinatorial decomposition of symmetric products of Schur polynomials, called a Littlewood Richardson rule, which we will discuss. We will then power through as many generalisations as I have time for: Hecke insertion and stable Grothendieck polynomials, shifted insertion and Schur P-functions, and shifted Hecke insertion and weak shifted stable Grothendieck polynomials

Deficiency is a measure of how complicated the presentations of a particular group need to be; it is defined as the maximum of the number of generators minus the number of relators (over all finite presentations of the group). This talk will introduce the basics of deficiency, give a deft example of Swan which illustrates why our understanding of deficiency is deficient, and conclude with some new examples that defy this defeatism: finite $p$-groups can have any deficiency you could (reasonably) wish for.

Starting from an example in quantum chemistry, we explain the techniques of Numerical Tensor Calculus with particular emphasis on the convolution operation. The tensorisation technique also applies to one-dimensional grid functions and allows to perform the convolution with a cost which may be much cheaper than the fast Fourier transform.

It is a standard heuristic that sums of oscillating number theoretic functions, like the M\"obius function or Dirichlet characters, should exhibit squareroot cancellation. It is often very difficult to prove anything as strong as that, and we generally expect that if we could prove squareroot cancellation it would be the best possible bound. I will discuss recent results showing that, in fact, certain averages of multiplicative functions exhibit a bit more than squareroot cancellation.