Invariant rational functions on semisimple Lie algebras and the Gelfand–Kirillov conjecture

The talk focuses on the recent solution (by J.-L. Colliot-Thélène, B. Kunyavskii, V. L. Popov, and Z. Reichstein) of the rationality problem of the fields of rational functions on semisimple Lie algebras and on the heavily based on this solution constructing (by A. Premet) counterexamples to the Gelfand–Kirillov conjecture on the fields of fractions of the universal enveloping algebras of simple Lie algebras.