The proposed research consists of three related projects at the intersection of noncommutative algebra and algebraic geometry. The first project concerns noncommutative analogs of algebraic varieties called maximal orders; these are coherent sheaves of noncommutative algebras on varieties whose generic stalk is a central simple algebra. The project involves studying modules over some maximal orders, developing a minimal model program for maximal orders on 3-folds, and studying Brauer groups of supersingular K3 surfaces in characteristic p. The second project focuses on Clifford algebras and Ulrich sheaves on varieties. This project uses Clifford algebra representations to construct Ulrich sheaves on algebraic varieties and examines stability in the category of Ulrich sheaves on varieties. The third project links Clifford algebras and Brauer groups of fields by investigating the relationship between cyclicity of division algebras and representations of Clifford algebras, and by using weighted Clifford algebras to study relative Brauer groups of curves.
The fruitful interactions between mathematics and theoretical physics in the past several decades have resulted in great interest in noncommutative algebras and Clifford algebras in particular. Noncommutative algebras are similar to polynomials in which the order of multiplication matters. Clifford algebras occur in the definition of the Dirac equation and are an integral part of quantum mechanics and quantum field theory. Noncommutative algebras can be studied using the sophisticated methods of algebraic geometry and, conversely, have been used to answer questions is algebraic geometry. Algebraic geometry was developed and has proved to be a powerful tool for addressing some old problems in algebra. The subject of noncommutative algebraic geometry has been progressing rapidly, and the projects in this proposal further develop some deep geometric and algebraic aspects of noncommutative algebras (and Clifford algebras in particular) and related algebraic geometry.
