Title:Cubic Fourfolds: Rationality and Derived Categories

Abstract: The question of determining if a given algebraic variety is rationalis a notoriously difficult problem in algebraic geometry, and attempts to solverationality problems have often produced powerful new techniques.  Awell-known open rationality problem is the determination of a criterion forwhen a cubic hypersurface of five-dimensional projective space is rational. After discussing the history of this problem, I will introduce the twoconjectural rationality criteria that have been put forth and then discuss apackage of tools I have developed with my collaborators to bring these twoconjectures together.  Our theory of Relative Bridgeland Stability has anumber of other beautiful consequences such as a new proof of the integralHodge Conjecture for Cubic Fourfolds and the construction of full-dimensionalfamilies of projective HyperKahler manifolds.  Time permitting I’lldiscuss applications of the theory of relative stability conditions to problemsother than cubic fourfolds.