Interactions between noncommutative algebra, algebraic geometry and representation theory

建议的研究包括三个相关的项目在非交换代数和代数几何的交叉点。第一个项目涉及代数簇的非交换类似物称为最大订单;这些是凝聚层的非交换代数品种的通用茎是一个中心的简单代数。该项目涉及研究模在一些最大的订单，开发一个最小的模型程序的最大订单的3倍，并研究Brauer群的超奇异K3表面的特征p。第二个项目的重点是Clifford代数和乌尔里希层的品种。这个项目使用Clifford代数表示来构造代数簇上的Ulrich层，并研究簇上Ulrich层范畴的稳定性。第三个项目通过研究除代数的循环性和Clifford代数的表示之间的关系，以及通过使用加权Clifford代数来研究相对Brauer曲线群，将Clifford代数和Brauer域群联系起来。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。克利福德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一个组成部分。非交换代数可以用代数几何的复杂方法来研究，反过来，也可以用来回答代数几何的问题。代数几何的发展，并已被证明是一个强大的工具，解决一些老问题的代数。非交换代数几何的主题一直进展迅速，本提案中的项目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刻的几何和代数方面。

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.