After some generalities on étale fundamental groups and anabelian geometry, I will explore some of the current results on the section conjecture, including those of Koenigsmann and Pop on the birational section conjecture, and a recent unpublished result of Mohamed Saidi which reduces the section conjecture for finitely generated fields over the rationals to the case of number fields.

Many hard problems in Diophantine geometry have analogues over function fields which are less hard. I will give some examples.

CANCELLED - CANCELLED - CANCELLED

With the general election looming upon I will discuss the various different kinds of voting system that one could implement in such an election. I will show that these can give very different answers to the same set of voters. I will then discuss Arrow's Impossibility Theorem which shows that no voting system is compatible with 4 simple axioms which may be desireable.

In recent years, several preconditioning strategies have been proposed for control problems in fluid dynamics. These are a special case of the general saddle point problem in optimisation. Here, we will extend a preconditionier for distributed Stokes control problems, developed by Rees and Wathen, to the case of boundary control. We will show the usefulness of low-rank structures in constructing a good approximation for the Schur complement of the saddle point matrix. In the end, we will discuss the applicability of this strategy for Navier-Stokes control.

When the Computing Laboratory discarded its hardcopy journals around 2008, I kept the first ten years or so of each of six classic numerical analysis journals, starting from volume 1, number 1.  This will not be a seminar in the usual sense but a mutual exploration.  Come prepared to look through a few of these old volumes yourself and perhaps to tell the group of something interesting you find.  Bring a pen and paper.  All are welcome.

Mathematics of Computation, from 1943
SIAM Journal, from 1953
Numerische Mathematik, from 1959
BIT, from 1961
SIAM Journal on Numerical Analysis, from 1964
Journal of the IMA, from 1965

We develop a spectral method for solving univariate singular integral equations over unions of intervals and circles, by utilizing Chebyshev, ultraspherical and Laurent polynomials to reformulate the equations as banded infinite-dimensional systems. Low rank approximations are used to obtain compressed representations of the bivariate kernels. The resulting system can be solved in linear time using an adaptive QR factorization, determining an optimal number of unknowns needed to resolve the solution to any pre-determined accuracy. Applications considered include fracture mechanics, the Faraday cage, and acoustic scattering. The Julia software package https://github.com/ApproxFun/SIE.jl implements our method with a convenient, user-friendly interface.