Commutative Algebra: F-Regularity in Algebraic Geometry and Non-Commutative Algebra

这个项目研究多项式和它们定义的几何形状，称为代数簇。 这些问题属于数学领域，称为代数几何。 代数几何在工业和国家安全领域有许多应用，包括纠错码、机器学习、密码学、计算机辅助设计和3D打印。其中一些应用涉及研究多项式为零模某些素数的集合，这是该项目的特别重点。该项目是理解有限域上定义的代数簇的奇异性的基础研究，包括理解簇离光滑有多远的标准，以及在某些情况下避免光滑失败的技术。该项目包括旨在为下一代数学家提供培训的学生准备的研究。更准确地说，该项目研究了非交换代数，组合数学和双有理代数几何中的强F-正则性。研究了对素特征的强F-正则局部环R，F*R的R-线性自同态代数A（F是Frobenius映射）是否是货车den Bergh意义下的奇点的非交换分解.此外，微分算子的导出函子的理论将被研究，着眼于使用它来证明特征为零的域上的微分算子环在某些条件下可以被还原为特征p。最后，建立在最近的工作中的估值环，该项目渴望表明，某些部分环的品种总是生成的，在最小模型程序的特征p的重要一步。这个奖项反映了NSF的法定使命，并已被认为是值得支持的，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。

This project investigates polynomials and the geometric shapes they define, called algebraic varieties. The problems belong to the field of mathematics called algebraic geometry. Algebraic geometry has many applications throughout industry and national security, including to error-correcting codes, machine learning, cryptography, computer aided design, and 3D printing. Some of these applications are concerned with studying the sets where polynomials are zero module certain prime numbers, the particular focus of this project. This project is basic research into understanding the singularities of algebraic varieties defined over finite fields, including criteria for understanding how far the varieties are from being smooth, and techniques for circumventing the failure of smoothness in some cases. The project includes research designed to be student-ready to provide training for the next generation of mathematicians.In more precise terms, the project investigates strong F-regularity in several contexts in non-commutative algebra, combinatorics, and birational algebraic geometry. It investigates whether or not, for a strongly F-regular local ring R of prime characteristic, the algebra A of R-linear endomorphisms of F*R, where F is the Frobenius map, is a non-commutative resolution of singularities in the sense of Van den Bergh. In addition, a theory of derived functors of differential operators will be investigated, with an eye toward using it to show that rings of differential operators over fields of characteristic zero can be reduced to characteristic p under some conditions. Finally, building on recent work in valuation rings, the project aspires to show that certain section rings of varieties are always finitely generated, an important step in the minimal model program in characteristic p.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.