After motivating why we would like to find examples of simple totally disconnected locally compact groups, I will describe a construction due to Banks, Elder and Willis which yields infinitely many such examples when given certain groups acting on a tree.

I will describe a generalization of toric symplectic geometry to a new class of Poisson manifolds which are
symplectic away from a collection of hypersurfaces forming a normal crossing configuration.  Using a "tropical
moment map",  I will describe the classification of such manifolds in terms of decorated log affine polytopes,
in analogy with the Delzant classification of toric symplectic manifolds. 

 I will report on recent work on a tropical/symplectic approach to the Horn inequalities. These describe the possible spectra of Hermitian matrices which may be obtained as the sum of two Hermitian matrices with fixed spectra. This is joint work with Anton Alekseev and Maria Podkopaeva.

Quantum Mechanics presents a radically different perspective on physical reality compared with the world of classical physics. In particular, results such as the Bell and Kochen-Specker theorems highlight the essentially non-local and contextual nature of quantum mechanics. The rapidly developing field of quantum information seeks to exploit these non-classical features of quantum physics to transcend classical bounds on information processing tasks.

In this talk, we shall explore the rich mathematical structures underlying these results. The study of non-locality and contextuality can be expressed in a unified and generalised form in the language of sheaves or bundles, in terms of obstructions to global sections. These obstructions can, in many cases, be witnessed by cohomology invariants. There are also strong connections with logic. For example, Bell inequalities, one of the major tools of quantum information and foundations, arise systematically from logical consistency conditions.

These general mathematical characterisations of non-locality and contextuality also allow precise connections to be made with a number of seemingly unrelated topics, in classical computation, logic, and natural language semantics. By varying the semiring in which distributions are valued, the same structures and results can be recognised in databases and constraint satisfaction as in probability models arising from quantum mechanics. A rich field of contextual semantics, applicable to many of the situations where the pervasive phenomenon of contextuality arises, promises to emerge.

Cramer conjectured a random model for the distribution of the primes, which would suggest that, on the scale of the average prime gap, the primes can be modelled by a Poisson process. In particular, the set of limit points of normalized prime gaps would be the whole interval $[0,\infty)$. I will describe joint work with Banks and Freiberg which shows that at least 1/8 of the positive reals are in the set of limit points. 

Given a commutative ring A with ideal m, we consider the formal completion of A at m, and we ask when algebraic structures over the completion can be approximated by algebraic structures over the ring A itself. As we will see, Artin's approximation theorem tells us for which types of algebraic structures and which pairs (A,m) we can expect an affirmative answer. We will introduce some local notions from algebraic geometry, including formal and etale neighbourhoods. Then we will discuss some algebraic structures and rings arising in algebraic geometry and satisfying the conditions of the theorem, and show as a corollary how we can lift isomorphisms from formal neighbourhoods to etale neighbourhoods of varieties.

We will discuss the relation between perturbative gauge theory and
perturbative gravity, and look at how this relation extends to some exact
classical solutions. First, we will review the double copy prescription that
takes gauge theory amplitudes into gravity amplitudes, which has been
crucial to progress in perturbative studies of supergravity. Then, we will
see how the relation between the two theories can be made manifest when we
restrict to the self-dual sector, in four dimensions. A key role is played
by a kinematic algebraic structure mirroring the colour structure, which can
be extended from the self-dual sector to the full theory, in any number of
dimensions. Finally, we will see how these ideas can be applied also to some
exact classical solutions, namely black holes and plane waves. This leads to
a relation of the type Schwarzschild as (Coulomb charge)^2.