Birational Algebraic Geometry in Characteristic Zero and Positive Characteristic

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Algebraic Geometry is one of the oldest and most active areas of mathematics. It plays an important role in many neighboring fields such as Commutative Algebra, Number Theory, Differential Geometry, and String Theory to name a few. Algebraic Geometry studies the solution sets of polynomial equations. These solution sets define geometric objects and it is important to classify them and understand their qualitative behavior. Birational Geometry aims to classify these geometric objects (up to birational isomorphism). This is one of the principal areas of Algebraic Geometry which in recent years has been at the center of spectacular progress. Many important central questions remain. The PI hopes to capitalize on these recent successes to make progress towards the remaining key open problems and conjectures.The principal investigator will conduct research in Algebraic Geometry and especially in Higher Dimensional Birational Algebraic Geometry over algebraically closed fields of characteristic 0 and characteristic p 0. In particular this project is focused on questions related to the minimal model program for threefolds over an algebraically closed field of characteristic p = 2,3,5 including related questions on the inversion of adjunction, and to the minimal model program and the singularities of threefolds over an algebraically closed field of characteristic p 0. The PI also plans to investigate generic vanishing theorems and the birational geometry of varieties defined over an algebraically closed field of characteristic p 0 as well as questions related to the birational geometry of generalized log canonical pairs and to the termination of log-flips in dimension greater or equal than 4 (in characteristic 0).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.