Interactions between noncommutative algebra, algebraic geometry and representation theory

本研究由非交换代数与代数几何交叉的三个相关课题组成。第一个项目涉及称为极大阶的代数变体的非交换类似物；它们是变种上的非交换代数的相干束，其属柄是一个中心简单代数。本项目主要研究了一些极大阶上的模，开发了3-fold上极大阶的最小模型程序，研究了特征p上的超奇异K3曲面的Brauer群。第二个项目主要研究了Clifford代数和变种上的Ulrich束。本课题利用Clifford代数表示构造了代数变种上的乌尔里希捆，并检验了变种上乌尔里希捆范畴的稳定性。第三个项目通过研究划分代数的循环性与Clifford代数表示之间的关系，以及使用加权Clifford代数来研究曲线的相对Brauer群，将Clifford代数与场的Brauer群联系起来。在过去的几十年里，数学和理论物理之间卓有成效的相互作用引起了人们对非交换代数特别是Clifford代数的极大兴趣。非交换代数类似于多项式，其中乘法的顺序很重要。Clifford代数出现在狄拉克方程的定义中，是量子力学和量子场论的重要组成部分。非交换代数可以用代数几何的复杂方法来研究，反过来，也被用来回答代数几何的问题。代数几何得到了发展，并被证明是解决代数中一些老问题的有力工具。非交换代数几何学科发展迅速，本课题进一步发展了非交换代数（特别是Clifford代数）和相关代数几何的一些深层次的几何和代数方面。

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.