A year ago I gave a talk raising questions about Faraday shielding which stimulated discussion with John Ockendon and others and led to a collaboration with Jon Chapman and Dave Hewett.  The problem is one of harmonic functions subject to constant-potential boundary conditions.  A year later, we are happy with the solution we have found, and the paper will appear in SIAM Review.  Though many assume as we originally did that Faraday shielding must be exponentially effective, and Feynman even argues this explicitly in his Lectures, we have found that in fact, the shielding is only linear.  Along the way to explaining this we make use of Mikhlin's numerical method of series expansion, homogenization by multiple scales analysis, conformal mapping, a phase transition, Brownian motion, some ideas recollected from high school about electrostatic induction, and a constrained quadratic optimization problem solvable via a block 2x2 KKT matrix.

In particular, some nice things about branch groups, whose subgroup structure "sees" all actions on rooted trees.

Motivated by an open conjecture in anabelian geometry, we investigate which arithmetic properties of the rationals are encoded in the absolute Galois group G_Q. We give a model-theoretic framework for studying absolute Galois groups and discuss in what respect orderings and valuations of the field are known to their first-order theory. Some questions regarding local-global principles and the transfer to elementary extensions of Q are raised.

Suppose we have a finite graph. We can view this as a resistor network where each edge has unit resistance. We can then calculate the resistance between any two vertices and ask questions like `which graph with $n$ vertices and $m$ edges minimises the average resistance between pairs of vertices?' There is a `obvious' solution; we show that this answer is not correct.

This problem was motivated by some questions about the design of statistical experiments (and has some surprising applications in chemistry) but this talk will not assume any statistical knowledge.

This is joint work with Robert Johnson.

Compact F-spaces play an important role in the area of compactification theory, the prototype being w*, the Stone-Cech remainder of the integers. Two curious topological characteristics of compact F-spaces are that they don’t contain convergent sequences (apart from the constant ones), and moreover, that they often contain points that don’t lie in the boundary of any countable subset (so-called weak P-points). In this talk we investigate the space of self-maps S(X) on compact zero-dimensional F-spaces X, endowed with the compact-open topology. A natural question is whether S(X) reflects properties of the ground space X. Our main result is that for zero-dimensional compact F-spaces X, also S(X) doesn’t contain convergent sequences. If time permits, I will also comment on the existence of weak P-points in S(X). This is joint work with Richard Lupton.