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.