CAREER: New Frontiers for Frobenius, Singularity Theory, Differential Operators, and Local Cohomology

职业生涯：弗罗贝尼乌斯、奇点理论、微分算子和局部上同调的新领域

• 批准号: 1945611
• 负责人: Emily Witt
• 金额: $ 40万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-04-01 至 2025-03-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1945611&HistoricalAwards=false
• 关键词: CAREER; New; Frontiers; Frobenius; Singularity

The field of commutative algebra provides a framework in which to study polynomial equations and their set of solutions. Given the prominent role of such equations in our society (e.g., in pure and applied math, computer science and technology, engineering, physics, biology, and chemistry), the applications of commutative algebra are broad and impactful. For example, commutative algebra is fundamental to cryptography, which many of us rely on daily, but also to genomics. This project advances the field of commutative algebra, and also trains junior scientists to develop skills relevant to a variety of scientific careers. This project also includes three initiatives focused on education, outreach and scientific leadership. These initiatives support research collaboration among women algebraists, implement an REU training program serving students from groups that are underrepresented in the STEM fields, and create new computer algebra software for research and education.This project aims to further our understanding of commutative rings and algebraic varieties using prime characteristic methods, differential operators, and local cohomology. In prime characteristic, one goal is to use the Frobenius map to study hypersurfaces by providing a better understanding of test ideals and Frobenius jumping exponents. The project also seeks effective algorithms for computing the Bernstein- Sato polynomial over the complex numbers. Another goal of the project is to investigate differential operators, and modules over them, in non-regular settings, and then apply this new theory (for example, to study multiplier ideals). Finally, the project proposes to investigate new geometric and topological properties determined by local cohomology, with an emphasis on connectedness properties of the irreducible components of a ring's spectrum.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.

交换代数领域提供了一个研究多项式方程及其解集的框架。鉴于这些方程在我们社会中的突出作用（例如，在纯数学和应用数学、计算机科学和技术、工程、物理、生物和化学中），交换代数的应用是广泛而有影响力的。例如，交换代数是我们许多人日常使用的密码学的基础，也是基因组学的基础。该项目推动了交换代数领域的发展，也培养了初级科学家发展与各种科学事业相关的技能。该项目还包括三项侧重于教育、推广和科学领导的举措。这些举措支持女性代数学家之间的研究合作，实施REU培训计划，为STEM领域中代表性不足的群体的学生提供服务，并为研究和教育创建新的计算机代数软件。本项目旨在利用素特征方法、微分算子和局部上同调进一步加深我们对交换环和代数变异的理解。在素数特征中，一个目标是通过提供对测试理想和Frobenius跳跃指数的更好理解来使用Frobenius映射来研究超曲面。该项目还寻求在复数上计算Bernstein- Sato多项式的有效算法。这个项目的另一个目标是研究微分算子，以及它们上面的模，在非规则环境下，然后应用这个新理论（例如，研究乘数理想）。最后，研究了由局部上同调决定的新的几何和拓扑性质，重点研究了环谱不可约分量的连通性。该奖项反映了美国国家科学基金会的法定使命，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，被认为值得支持。

项目成果

Lower bounds on the F-pure threshold and extremal singularities

F 纯阈值和极值奇点的下界

• DOI: 10.1090/btran/106
• 发表时间: 2022
• 期刊: Series B
• 作者: Kadyrsizova, Zhibek;Kenkel, Jennifer;Page, Janet;Singh, Jyoti;Smith, Karen;Vraciu, Adela;Witt, Emily
• 通讯作者: Witt, Emily

Frobenius powers

弗罗贝尼乌斯幂

• DOI: 10.1007/s00209-019-02442-2
• 发表时间: 2020
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Hernández, Daniel J.;Teixeira, Pedro;Witt, Emily E.
• 通讯作者: Witt, Emily E.

The FrobeniusThresholds package for Macaulay2

Macaulay2 的 FrobeniusThresholds 包

• DOI: 10.2140/jsag.2021.11.25
• 发表时间: 2021
• 期刊: The journal of software for algebra and geometry
• 作者: Hernández, Daniel;Schwede, Karl;Teixeira, Pedro;Witt, Emily
• 通讯作者: Witt, Emily

Bernstein–Sato functional equations, V-filtrations, and multiplier ideals of direct summands

Bernstein-Sato 函数方程、V 过滤和直接被加数的乘数理想

• DOI: 10.1142/s0219199721500838
• 发表时间: 2022
• 期刊: Communications in Contemporary Mathematics
• 影响因子: 1.6
• 作者: Àlvarez Montaner, Josep;Hernández, Daniel J.;Jeffries, Jack;Núñez-Betancourt, Luis;Teixeira, Pedro;Witt, Emily E.
• 通讯作者: Witt, Emily E.

Frobenius powers of some monomial ideals

一些单项式理想的 Frobenius 幂

• DOI: 10.1016/j.jpaa.2019.04.015
• 发表时间: 2020
• 期刊: Journal of pure and applied algebra
• 影响因子: 0.8
• 作者: Hernández, Daniel J;Teixeira, Pedro;Witt, Emily E.
• 通讯作者: Witt, Emily E.