British Mathematical Colloquium: satellite meetings

The BMC was followed by three satellite meetings, on Semigroups and Applications, Geometric Group Theory, and Ergodic Theory. I had a lot of catching up and other jobs to do, and so was unable to go to more than a few of these talks. I managed three.

Gareth Tracey talked about crowns in finite group theory. If we are studying minimal generating sets in finite groups, it is useful to understand groups G which require more generators than any proper quotient; these are crowns. Given an arbitrary group G, for any normal subgroup N, we know that the number of generators of G/N does not exceed the number for G; so, if we choose N maximal such that equality holds, then G/N is a crown. Gareth has used his results on crowns for various things; in particular, he gave us a preview of his result with Colva Roney-Dougal on the number of subgroups of the symmetric group S_n.

Tom Coleman told us about his work (much of it in his recent PhD thesis) on generation, cofinality and strong cofinality, and the Bergman property for various transformation semigroups on an infinite set, such as all maps, injective maps, surjective maps, bijective maps, partial maps, and bijective partial maps on a countable set, and analogous things for homomorphisms etc of the random graph. The Bergman property for a structure asserts that, if S is a generating set, then any element of the structure is a word of bounded length in elements of S.

Justine Falque talked about her work with Nicolas Thiéry on the orbit algebra of a permutation group in the case where the number of orbits on n-sets is bounded by a polynomial in n. They show that the algebra is finitely generated and Cohen–Macaulay, so that the generating function for the number of orbits on n-sets is a rational function of the form P(x)/Π(1-x^d_i), where P is a polynomial. I have already discussed this here; I have read the short version on the arXiv and eagerly await the full version.