A Study of Hypersurfaces in General Position Inspired by Frobenius

这是交换代数的一个项目，粗略地说，它是研究多项式方程组的。无论是在纯数学和应用数学中，还是在计算机科学、物理、生物和化学中，多项式方程以及这些方程所描述的几何对象都是基本的。该项目将研究这类方程组的性质，重点是了解它们可能不光滑或规则的微妙方式。首席研究员将继续指导在数学科学中未被充分代表的学生。更准确地说，项目团队将研究代数族的奇点。这项研究试图得出并加强数学的许多领域之间的联系，包括素数特征交换代数、经典奇点理论、二次代数几何、多面体几何和微分算子理论。该项目侧重于一般位置上的多项式方程，以及显式计算。该项目由代数和数论计划(ANT)和既定的激励竞争研究计划(EPSCoR)共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This is a project in commutative algebra, which, roughly speaking, is the study of systems of polynomials equations. Polynomial equations, and the geometric objects that that these equations describe, are fundamental, both in pure and applied mathematics, but also in computer science, physics, biology, and chemistry. The project will study the nature of such systems of equations, with an emphasis on understanding the subtle ways in which they can fail to be smooth, or regular. The principal investigator will continue to mentor students that are underrepresented in the mathematical sciences.More precisely, the project team will study the singularities of algebraic varieties. The research seeks to draw, and strengthen, connections among many areas of mathematics, including prime characteristic commutative algebra, classical singularity theory, birational algebraic geometry, polyhedral geometry, and the theory of differential operators. This project places an emphasis on polynomial equations that lie in general position, as well as on explicit computations.This project is jointly funded by the Algebra and Number Theory program (ANT) and the Established Program to Stimulate Competitive Research (EPSCoR).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.