The rationality problem in algebraic geometry

Asher Auel (Yale)

Wednesday, January 27, 2016 2:30 pm
MSB 109A

The rationality problem---whether a given finitely generated field

extension is purely transcendental---is one of the oldest problems in

algebraic geometry. In this talk, I will discuss two important cases

that have seen significant recent advances as well as surprising

upsets. The first, Noether's problem, posed in 1913, concerns the

field of invariants of a finite permutation group acting on a set of

algebraically independent variables. The second, the rationality

problem for hypersurfaces in projective space, concerns polynomial

rings defined by a single relation. Here, I will specifically consider

the case of cubic hypersurfaces of dimension 4, which has itself been

an open problem in algebraic geometry since the 1940s and the subject

of several recent conjectures. Applying techniques from derived

categories, moduli of K3 surfaces, and lattice theory, my own current

work settles some of the first instances of these conjectures and

offers new approaches.