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

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。 考虑到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、工程学、物理学、生物学和化学中），交换代数的应用是广泛和有影响力的。 例如，交换代数是密码学的基础，我们中的许多人每天都依赖密码学，但也是基因组学的基础。 该项目推进了交换代数领域，并培训初级科学家发展与各种科学职业相关的技能。该项目还包括三项举措，重点是教育、外联和科学领导。 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。该项目旨在利用素特征方法，微分算子和局部上同调进一步理解交换环和代数簇。在素特征中，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，使用Frobenius映射来研究超曲面。 该项目还寻求有效的算法来计算伯恩斯坦-佐藤多项式的复数。 该项目的另一个目标是研究非正则设置中的微分算子及其上的模，然后应用这个新理论（例如，研究乘数理想）。 最后，该项目提出了调查新的几何和拓扑性质所确定的局部上同调，重点是连通性的一个环的频谱不可约组件的属性。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.