Burt Totaro (University of Cambridge)
B.J.Totaro@dpmms.cam.ac.uk We start by discussing the role of invariants in geometry, including Hilbert's work on invariant theory. Hilbert's fourteenth problem asks whether rings of invariants are always finitely generated. Nagata gave the first counterexamples, making brilliant use of the geometry of plane curves. We present the recent advances on the problem by Mukai. We conclude with some new results.