Abstract: The spectral action principle was introduced by Connes and Chamseddine to formulate the notion of action for spectral triples. This talk will be an introduction to this principle and to the so called spectral action functional. After a quick review of the theory of Dirac operators and spectral triples, we will show how one can produce classical actions, such as Einstein-Hilbert or Yang-Mills action, by considering the spectral action functional for a suitable Dirac operator.

Abstract: I will sketch Adams' original proof of the Hopf invariant one problem.

Abstract: We investigate the rationality of four-dimensional algebraic tori and the associated equivariant birational linearisation problem. We connect these problems to the question of determining when an algebraic group is (stably) Cayley - that is (stably) equivariantly birationally isomorphic to its Lie algebra and earlier joint work on the (stably) Cayley problem with Popov, Reichstein, Borovoi and Kunyavskii.