Singularities in Positive and Mixed Characteristic Commutative Algebra

交换代数研究诺特交换环，其中的例子是多项式和普通整数的集合。 它被用于各种各样的应用环境，从计算机科学和基因组学中的纠错码到工程中的控制理论和建模。其他使用交换代数，特别是交换环的数学领域是代数和算术几何，以及复分析，拓扑学和表示论。在这个补助金的项目执行将既推进交换代数的理论理解，并阐明一些重要的问题，这些相关领域。此外，首席研究员致力于促进数学教育，培养未来几代研究人员，并协助在美国建立强大的STEM劳动力。为了实现这些目标，PI将在整个赠款过程中监督，培训和指导博士生和博士后研究员。PI还将为本科生和研究生举办一些研讨会、讲习班和阅读课程。PI专门研究正和混合特征交换代数中的奇点，并将解决一些自然产生的问题，这些问题来自Frobenius分裂理论、紧闭包、同调闭包和绝对积分闭包的最新发展。PI的目的是利用通过Frobenius映射定义的奇点之间的联系，以及在复杂的代数几何中产生的奇点，以便在三个新的方向上开展研究项目。首先，PI将研究特征p 0中对偶F签名极限的性质，证明其存在性并导出新的性质，并应用于F理性。其次，同样在正特征中，PI将通过探索优秀正则环的等价物和推广附加语句来进一步理解F-有限设置之外的F-奇异性。最后，利用在混合特性的绝对积分封闭的突破性成果，PI将攻击有关分裂条件的开放性问题，并展示新的几何应用。该奖项反映了NSF的法定使命，并已被认为是值得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的支持。

Commutative algebra studies Noetherian commutative rings, examples of which are sets of polynomials and the ordinary integers. It is used in a wide variety of applied settings, ranging from error-correcting codes in computer science and genomics to control theory and modeling in engineering. Other fields of mathematics that use commutative algebra, and commutative rings in particular, are algebraic and arithmetic geometry, as well as complex analysis, topology, and representation theory. Execution of the projects in this grant will both advance theoretical understanding in commutative algebra and shed light on important problems from some of these related fields. In addition, the principal investigator is dedicated to promoting mathematics education, developing future generations of researchers, and assisting in building a strong STEM workforce in the US. Towards these goals, the PI will supervise, train, and mentor doctoral students and postdoctoral fellows throughout the course of the grant. The PI will also facilitate a number of seminars, workshops, and reading courses for undergraduate and graduate students.The PI specializes in the study of singularities in positive and mixed characteristic commutative algebra, and will address a number of questions arising naturally from recent developments in the theory of Frobenius splittings, tight closure, the homological conjectures, and absolute integral closure. The PI aims to exploit the connection between singularities defined via the Frobenius map in positive characteristic and those arising in complex algebraic geometry in order to carry out research projects in three new directions. First, the PI will investigate properties of the dual F-signature limit in characteristic p 0, showing its existence and deriving new properties with applications to F-rationality. Second, also in positive characteristic, the PI will further the understanding of F-singularities outside of the F-finite setting by exploring quotients of excellent regular rings and generalizing adjunction statements. Lastly, leveraging breakthrough results on the absolute integral closure in mixed characteristic, the PI will attack open questions regarding the splinter condition and exhibit new geometric applications.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.