Noncommutative Algebras and Their Interactions With Algebraic and Arithmetic Geometry

数学和理论物理之间卓有成效的相互作用，增加了人们对非交换代数的兴趣。非交换代数与更熟悉的结构（如多项式）有相似之处，但关键的区别在于乘法的顺序很重要。非交换代数与代数几何之间存在共生关系，代数几何是研究多项式方程解的形状的学科。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中的问题。此外，这两个领域之间的相互作用在物理学和纠错码中有潜在的应用。非交换代数几何的学科发展迅速，本课题进一步发展了非交换代数和相关代数几何的一些深层次的代数和算术方面。该项目还包括培训研究生，为他们在未来几年提供充足的研究机会。研究项目的统一主题是非交换代数及其与代数几何和算术几何的相互作用。激发性的问题主要来自非交换代数和在它们的探索中使用的技术范围从代数到代数和算术几何。第一个项目研究非交换代数，称为极大阶。这些是连贯的代数捆，它们的一般柄是一个中心的简单代数。该项目包括使用多正则映射和Kodaira维研究任意维代数变体上最大阶的两种分类，某些阶的派生类别称为del Pezzo阶，以及特征p中最大阶的分支。第二个项目侧重于使用雅可比曲线的Brauer群研究1属曲线的模堆。研究了投影变体上的未分枝Brauer类及其在Azumaya代数上的表示，并进一步发展了与Clifford代数相关的阿贝尔变体的构造。第三个项目是关于光滑射影变异和Clifford代数表示上的Ulrich束。该项目使用Clifford代数的表示来探索光滑射影变种上Ulrich束的存在性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.