An isometric action of a Lie group is called polar if it admits
sections, i.e. submanifolds which meet all orbits and always
perpendicularly. Polarity is a very restrictive condition. For example,
in case of linear actions on *R^*n polarity characterizes the isotropy
representations of symmetric spaces (Dadok).

The aim of this talk is to report on work in progress to prove an
infinite dimensional analogue of Dadok's theorem. C.-L. Terng has
constructed interesting examples of polar actions on Hilbert spaces by
affine isometries, the so called P(G,H) actions. Here G is a compact Lie
group, H a closed subgroup of G \times G, and P(G,H) consists of all
paths in G with end points in H. The action of P(G,H) on the Hilbert
space of L^2-curves in the Lie algebra of G is by gauge transformations.
Surprisingly the actions correspond also to isotropy actions of
symmetric spaces which are now infinite dimension and quotients of a
Kac-Moody group by the fixed point set of an involution. We conjecture
that the P(G,H) actions exhaust all polar actions on a Hilbert space.

Type 1 diabetes (T1D) is an autoimmune disease in which immune cells
target and kill the insulin-secreting pancreatic beta cells.
Recent investigation of diabetes-prone (NOD) mice reveals large cyclic
fluctuations in the levels of T cells (cells of the adaptive
immune system) weeks before the onset of the disease. We extend
a previous mathematical model for T-cell dynamics to account for the
gradual killing of beta cells, and show how such cycles can arise
as a natural consquence of feedback between self-antigen and T-cell
populations. The model has interesting nonlinear dynamics
including Hopf and homoclinic bifurcations in biologically reasonable
regimes of parameters. The model fits into a larger program of
investigation of type 1 diabetes, and suggests experimental tests.