Noncommutative Algebras and Their Interactions With Algebraic and Arithmetic Geometry

非交换代数及其与代数和算术几何的相互作用

• 批准号: 2101761
• 负责人: Rajesh Kulkarni
• 金额: $ 31.4万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101761&HistoricalAwards=false
• 关键词: Noncommutative; Algebras; Their; Interactions; Algebraic

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.

数学和理论物理之间富有成效的相互作用导致人们对非交换代数的兴趣增加。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。 非交换代数和代数几何之间存在共生关系，代数几何是对多项式方程解的形状的研究。 非交换代数可以使用代数几何的复杂方法来研究，相反，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学和纠错码中具有潜在的应用。非交换代数几何学科一直在快速发展，该项目进一步发展了特定类别的非交换代数和相关代数几何的一些深层代数和算术方面。该项目还涉及研究生培训，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和算术几何的相互作用。激发问题主要来自非交换代数及其探索中使用的技术，范围从代数到代数和算术几何。第一个项目研究称为最大阶的非交换代数。这些是连贯的代数层，其通用茎是中心简单代数。该项目涉及使用多正则映射和 Kodaira 维数研究任意维度代数簇上最大阶的双有理分类、称为 del Pezzo 阶的某些阶的派生类别，以及特征 p 中最大阶的分支。第二个项目的重点是使用雅可比曲线的布劳尔群来研究属 1 曲线的模堆栈，研究射影簇的无分支布劳尔类及其用 Azumaya 代数表示，并进一步发展与 Clifford 代数相关的阿贝尔簇的构造。第三个项目涉及平滑射影簇和 Clifford 代数表示的 Ulrich 丛。 该项目使用 Clifford 代数的表示来探索平滑射影簇上乌尔里希丛的存在。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。