Noncommutative Algebras and Their Interactions With Algebraic and Arithmetic Geometry

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The fruitful interactions between mathematics and theoretical physics have resulted in increased interest in noncommutative algebras. Noncommutative algebras have similarities with more familiar constructs, like polynomials, but a key difference is that the order of multiplication matters. There is a symbiotic relationship between noncommutative algebras and algebraic geometry, which is the study of shapes of solutions to polynomial equations. Noncommutative algebras can be studied using sophisticated methods of algebraic geometry and, conversely, have been used to answer questions in algebraic geometry. In addition, the interactions between the two areas have potential applications in physics and in error correcting codes. The subject of noncommutative algebraic geometry has been progressing rapidly, and this project further develops some deep algebraic and arithmetic aspects of specific classes of noncommutative algebras and related algebraic geometry. The project also involves training of graduate students, providing them with ample opportunities for research in the coming years.The unifying theme of the research projects is noncommutative algebras and their interactions with algebraic and arithmetic geometry. The motivating questions arise primarily from noncommutative algebras and the techniques utilized in their exploration range from algebra to algebraic and arithmetic geometry. The first project investigates noncommutative algebras called maximal orders. These are coherent sheaves of algebras whose generic stalk is a central simple algebra. The project involves studying birational classification of maximal orders on algebraic varieties in arbitrary dimensions using the pluri-canonical map and the Kodaira dimension, the derived categories of certain orders called the del Pezzo orders, and the ramification of maximal orders in characteristic p. The second project focuses on investigating moduli stack of genus 1 curves using Brauer groups of their Jacobian curves, studying unramified Brauer classes on projective varieties and their representation by Azumaya algebras, and further developing the construction of abelian varieties associated to Clifford algebras. The third project concerns Ulrich bundles on smooth projective varieties and representations of Clifford algebras. The project uses representations of Clifford algebras in exploration of existence of Ulrich bundles on smooth projective varieties.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.