Interactions between p-adic arithmetic geometry and commutative algebra

PI建议研究算术几何（特别是p进方面）、代数几何和交换代数的共享边界上的问题。例如，Mel Hochster的直接和猜想，它假定正则环的一些基本性质的存在，是交换代数中四十多年来一个重要的开放问题；等特征情况的解（由于Hochster从70年代开始）负责现代交换代数的大片，而p进情况仍然是诱人的开放。PI最近发现，p进霍奇理论（由于Faltings）中的一些想法可以用来解决这个猜想的某些未知情况。PI打算通过使用强大的最新技术——主要是Scholze的完美空间理论——从p进Hodge理论的快速发展的主题进一步探索这个方向，以接近直接和猜想和其他纯代数问题。相反，PI先前在直接和猜想上的工作，加上一些派生的代数几何，最近被证明有助于对p进霍奇理论的某些几何方面进行重要的简化；PI计划更彻底地发展派生方面，以便更好地概念化图像。对整数系数多项式解的研究可以追溯到古代。这里一个非常有用的技巧是首先研究“近似”解，即，模素数的解，然后是素数的模幂。其思想是，随着质数的幂增加，近似变得更好。格罗腾迪克在过去半个世纪里对数学的革命，不仅使人们能够给前面的陈述附加一个精确的意义，而且还提供了一个美丽的几何背景——p进几何的世界----来研究这种近似解。这一背景一直是最近许多数学进步的核心（比如怀尔斯对费马大定理的证明和朗兰兹纲领中的其他里程碑）。PI计划进一步对基础几何理论做出贡献，并开发纯代数问题的应用程序。

The PI proposes to investigate problems lying at the shared boundaries of arithmetic geometry (especially p-adic aspects), algebraic geometry, and commutative algebra. For example, Mel Hochster's direct summand conjecture, which posits the existence of some fundamental properties of regular rings, has been an important open problem in commutative algebra for over four decades; the solution to the equal characteristic case (due to Hochster from the 70s) is responsible for large swathes of modern commutative algebra, while the p-adic case remains tantalizingly open. The PI recently discovered that some ideas from p-adic Hodge theory (due to Faltings) can be used solve certain unknown cases of this conjecture. The PI intends to pursue this direction further by using powerful recent techniques --- chiefly Scholze's beautiful theory of perfectoid spaces --- from the fast evolving subject of p-adic Hodge theory to approach the direct summand conjecture and other purely algebraic problems. Conversely, previous work of the PI on the direct summand conjecture, coupled with some derived algebraic geometry, has recently proven instrumental in arriving at a significant simplification of certain geometric aspects of p-adic Hodge theory; the PI plans to develop the derived aspects more thoroughly to conceptualize the picture better.The study of solutions of polynomials with integer coefficients dates back to antiquity. An extremely useful technique here is to study "approximate" solutions first, i.e., solutions modulo primes, and then modulo powers of primes. The idea is that as the power of prime increases, the approximation becomes better. Grothendieck's revolutionization of mathematics in the last half century not only allows one to not only attach a precise meaning to the previous statement, but also provides a beautiful geometric context --- the world of p-adic geometry ---- to study such approximate solutions. This context has been at the heart of numerous recent advances in mathematics (such as Wiles' proof of Fermat's last theorem and other recent milestones in the Langlands program). The PI plans to contribute further to underlying geometric theory as well as develop applications to purely algebraic problems.