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使人们对非对易代数产生了越来越大的兴趣。非对易代数与更常见的结构有相似之处，比如多项式，但一个关键的区别是乘法的顺序很重要。非对易代数与代数几何之间存在着一种共生关系，即研究多项式方程的解的形状。非对易代数可以用复杂的代数几何方法来研究，反过来，也可以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和纠错码中都有潜在的应用。非对易代数几何的研究进展很快，本课题进一步发展了特定非对易代数及其相关代数几何的一些深层次的代数和算术方面的研究。该项目还包括培养研究生，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对易代数及其与代数和算术几何的相互作用。激励性的问题主要来自非对易代数，以及从代数到代数和算术几何的探索中所使用的技巧。第一个项目研究称为极大阶的非交换代数。这是一组凝聚的代数，它们的通用柄是中心单代数。利用多重标准映射和Kodaira维研究任意维代数簇的最大阶的二元分类，某些阶的派生范畴称为del Pezzo阶，以及特征p中的最大阶的分支。第二个项目集中于利用亏格1曲线的Jacobian曲线的Brauer群来研究亏格1曲线的模叠叠，研究射影簇上的未分枝Brauer类及其Azumaya代数表示，以及进一步发展与Clifford代数相关的阿贝尔簇的构造。第三个项目涉及光滑射影簇上的Ulrich丛和Clifford代数的表示。该项目使用Clifford代数的表示来探索光滑射影簇上的Ulrich丛的存在性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.