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卓有成效的相互作用导致了对非交换代数的兴趣增加。非交换代数与我们更熟悉的结构（如多项式）有相似之处，但一个关键的区别是乘法的顺序很重要。 非交换代数与研究多项式方程解的形状的代数几何有着共生的关系。 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。非交换代数几何的主题一直进展迅速，这个项目进一步发展了一些深刻的代数和算术方面的特定类别的非交换代数和相关的代数几何。该项目还涉及研究生的培训，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。激励问题主要来自非交换代数和利用其探索范围从代数代数和算术几何的技术。第一个项目研究称为极大阶的非交换代数。这些是代数的凝聚层，其通用茎是中心单代数。该项目涉及使用多正则映射和科代拉维数研究任意维代数簇上最大阶的双有理分类，称为del Pezzo阶的某些阶的导出类别，以及特征p中最大阶的衍生。第二个项目侧重于使用Jacobian曲线的Brauer群研究亏格1曲线的模堆叠，研究了投射簇上的非分歧Brauer类及其用Azumaya代数的表示，并进一步发展了与Clifford代数相关的交换簇的构造。第三个项目涉及光滑投射簇上的乌尔里希丛和Clifford代数的表示。 该项目使用Clifford代数的表示法来探索光滑投射簇上的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知识价值和更广泛的影响审查标准进行评估而被认为值得支持。