The Minimal Model Program in Positive and Mixed Characteristics

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Algebraic geometry is an active and influential field of mathematics. Its goal is to understand algebraic varieties, which are geometric shapes described by algebraic equations. Not only are such objects essential in many fields of mathematics, but they also come up naturally in other disciplines such as engineering, computer science, or biology. The principal investigator's research centers around the ultimate goal of classifying all algebraic varieties. The PI plans to capitalize on recent breakthroughs in other fundamental fields of mathematics, namely number theory and commutative algebra, to significantly extend the scope of this classification.The principal investigator intends to apply new techniques, related to F-regularity, +-regularity, and mixed characteristic perfections, in order to provide a better understanding of the geometry and commutative algebra of schemes in positive and mixed characteristics. A second aspect of the project is to further the Minimal Model Program in positive and mixed characteristics by focusing on dimensions three and four. The PI plans to apply these results to obtain insight into the liftability of singularities and projective varieties from positive characteristic to characteristic zero, as well as the existence of rational points on varieties over finite fields.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.