Breaking boundaries

As well as the PCC, last week I was at a conference at the University of Sussex entitled Breaking Boundaries between Analysis, Geometry and Topology. With a title like that, how could I resist?

There were some lovely and wide-ranging talks. Here is a whistle-stop tour.

David Edmunds made some “Remarks on Approximation Numbers”. These numbers, for maps on Banach spaces, turned out to be a kind of generalisation of eigenvalues in the positive self-adjoint case, so interesting even if I don’t expect to have an immediate use for them myself. He remarked at one point that nuclear operators are sometimes called “operators of trace class”, even though they may not actually have a trace!

David Applebaum talked about “Generalised spherical functions on groups and symmetric spaces”. This was a blend of harmonic analysis and probability theory, bringing in, among other things, the Lévy–Khintchine formula for the Fourier transform of an infinitely divisible element (one which has a convolution nth root for any n with the Harish-Chandra formula. Lévy–Khintchine works on Euclidean space, and can be extended to locally compact abelian groups, but to go further to non-abelian groups and symmetric spaces you have a much harder job, and they ended up with a formulation involving infinite matrices, but still in the spirit of the original!

Dale Rolfsen gave two talks, on the theme of generalised torsion in groups. A group has generalised torsion if the product of some n conjugates of a non-identity element is equal to the identity (this reduces to ordinary torsion if the conjugating elements are all the identity). It is known that “biorderable” (having an order invariant under both left and right translation) implies “locally indicable”, which implies “left-orderable” (a left-translation-invariant order). In addition, “biorderable” implies “no generalised torsion”. In the first talk he concentrated on knot groups: if all the roots of the Alexander polynomial of a knot are real and positive, then the knot group is biorderable; but the Alexander polynomial does not detect generalised torsion. The second talk was about generalised torsion in homeomorphism groups of manifolds (especially cubes) which fix the boundary pointwise, and involved some ingenious constructions moving cubes around.

Michiel van den Berg talked about heat flow in Riemannian manifolds. You start with a region Ω at unit temperature, with either the rest of the manifold at zero temperature, or the boundary of Ω fixed at zero temperature; you are interested in how much heat remains in Ω at arbitrary later time, and in particular its asymptotics.

Neils Jacob talked about “Symbol and geometry related to Lévy processes”. From a function of two variables, you can define a pseudo-differential operator (where the second variable is “replaced” by differentiation) by means of the Fourier transform. The talk took me back to the course on partial differential equations I took as a final-year undergraduate, with lots of stuff about the geometry of wavefronts and characteristic manifolds. I regret to say that I didn’t understand much back then, and didn’t really get much further this time!

Roger Fenn presented us with a challenge. Take Gauss’ encoding of a knot, and turn it into a nice drawing of the knot; not a job just for computers since aesthetic considerations are involved.

I also gave two talks, one about the ADE affair (which I have discussed at length on this blog, beginning here), and the other about recent work with Collin Bleak on the outer automorphism groups of the Higman–Thompson groups, in which we have to count foldings of de Bruijn graphs, which I also posted about recently here. The slides are in the usual place.