Algebraic Geometry Close to Characteristic p

这个项目支持代数几何的研究(研究多变量多项式方程组的解)。这门学科已经有数千年的历史了，它提供了一种系统的语言，用于将几何中的语句(如“一个甜甜圈比一个球更弯曲”)翻译成代数中的那些语句(如“椭圆函数域不是有理的”)。这个项目的一个主要主题是使用这本词典来更好地理解当一个人扰动定义方程的系数时，某一方程组的解集的几何结构的变化，特别是在算术意义上(即从通常的算术到模算术或‘发条’算术)。更好地理解模算术中方程解集的几何结构是许多数学应用的基础。PI应研究来自数论的技术(如p-adic Hodge理论中的上同调理论)和代数拓扑(如p-adady域上的代数簇)之间的相互作用。这两者之间应该有很深的关系：它们应该像基序上同调和代数K-理论一样相关(即通过Atiyah-Hirzebruch型谱序列)。PI还应使用来自数论和p进几何的工具来处理交换代数中的问题。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 project supports research in algebraic geometry (which studies solutions to systems of polynomial equations in many variables). This subject is thousands of years old, and provides a systematic language for translating statements in geometry (such as "a doughnut is more curved than a ball") to those in algebra (such as "an elliptic function field is not rational"). A dominant theme of this project is to use this dictionary to better understand the variation in the geometric structure of a solution set of a certain system of equations as one perturbs the coefficients of the defining equations, especially in an arithmetic sense (i.e., in passing from usual arithmetic to modular or ``clockwork'' arithmetic). A better understanding of the geometric structure of solution sets of equations in modular arithmetic is fundamental to many applications of mathematics. The PI shall study the interaction between techniques coming from number theory (such as cohomology theories in p-adic Hodge theory) and algebraic topology (such as topological Hochschild homology) in the context of algebraic varieties over a p-adic field. A deep relationship between the two is expected: they should be related in the same way that motivic cohomology and algebraic K-theory are related (i.e., via an Atiyah-Hirzebruch type spectral sequence). The PI shall also use tools coming from number theory and p-adic geometry (such as perfectoid geometry) to approach problems in commutative algebra.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.