It has become possible in recent years to provide unconditional lower bounds on the time needed to perform a number of basic computational operations. I will briefly discuss some of the main techniques involved and show how one in particular, the information transfer method, can be exploited to give  time lower bounds for computation on streaming data.

I will then go on to present a simple looking mathematical conjecture with a probabilistic combinatorics flavour that derives from this work.  The conjecture is related to the classic "coin weighing with a spring scale" puzzle but has so far resisted our best efforts at resolution.

An edge (vertex) coloured graph is rainbow-connected if there is a rainbow path between any two vertices, i.e. a path all of whose edges (internal vertices) carry distinct colours. Rainbow edge (vertex) connectivity of a graph G is the smallest number of colours needed for a rainbow edge (vertex) colouring of G. We propose a very simple approach to studying rainbow connectivity in graphs. Using this idea, we give a unified proof of several new and known results, focusing on random regular graphs. This is joint work with Michael Krivelevich and Benny Sudakov.

This will be an overview of Prof Stroffolini's research and precursor to the eight-hour mini-course Prof Stroffolini will be giving later in October.

Milnor has shown that three-dimensional Lie groups with left invariant Riemannian metrics furnish examples of 3-manifolds with principal Ricci curvatures of fixed signature --- except for the signatures (-,+,+), (0,+,-), and (0,+,+).  We examine these three cases on a Riemannian 3-manifold, and prove global obstructions in certain cases.  For example, if the manifold is closed, then the signature (-,+,+) is not globally possible if it is of the form -µ,f,f, with µ a positive constant and f a smooth function that never takes the values 0,-µ (this generalizes a result by Yamato '91).  Similar obstructions for the other cases will also be discussed.  Our methods of proof rely upon frame techniques inspired by the Newman-Penrose formalism.  Thus, we will close by turning our attention to four dimensions and Lorentzian geometry, to uncover a relation between null vector fields and exact symplectic forms, with relations to Weinstein structures. 

I will discuss theta-stability, a framework for analyzing moduli problems in algebraic geometry by finding a special kind of stratification called a theta-stratification, a notion which generalizes the Kempf-Ness stratification in geometric invariant theory and the Harder-Narasimhan-Shatz stratification of the moduli of vector bundles on a Riemann surface.