¨ Mingmin Shen, 3:30pm-5:00pm, June 30 (then on July 1, 7, 8, 14, 15)

Title: The rationality problem in algebraic geometry

Abstract: One fundamental problem in algebraic geometry is to determine whether a variety is rational or not. Here being rational can be understood as being isomorphic to the projective space modulo lower dimensional subvarieties on both sides. In the lectures, I will explain how the problem gets more complicated as dimension increases. After reviewing the classical solution in low dimensional case, I will explain the work of Clemens—Griffiths on cubic threefold. Then a major part will be devoted to the recent results obtained by Voisin and Totaro via cycle-theoretical approach. I will also discuss the case of cubic fourfolds where the question is still open.