Interactions between p-adic arithmetic geometry and commutative algebra

PI 建议研究算术几何（尤其是 p 进数方面）、代数几何和交换代数的共同边界上的问题。例如，梅尔·霍克斯特（Mel Hochster）的直接被加数猜想假设了正则环的一些基本性质的存在，四十多年来一直是交换代数中一个重要的开放问题；等特征情况的解（由 70 年代的 Hochster 提出）是现代交换代数的重要组成部分，而 p-adic 情况仍然是诱人的开放性。 PI 最近发现 p-adic Hodge 理论（源自 Faltings）的一些想法可以用来解决该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 的完美类空间理论）进一步追求这个方向，从快速发展的 p 进 Hodge 理论主题到解决直接被加数猜想和其他纯代数问题。相反，PI 之前关于直接被加数猜想的工作，加上一些派生的代数几何，最近被证明有助于实现 p 进 Hodge 理论的某些几何方面的显着简化； PI 计划更彻底地开发派生方面，以更好地概念化图片。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术是首先研究“近似”解，即模素数的解，然后研究素数的模幂。这个想法是，随着素数幂的增加，近似值会变得更好。格洛腾迪克在过去半个世纪对数学的革命性变革不仅使人们不仅能够为先前的陈述赋予精确的含义，而且还提供了一个美丽的几何语境——p进几何的世界——来研究这种近似解。这种背景一直是数学领域许多最新进展的核心（例如怀尔斯对费马大定理的证明以及朗兰兹纲领中其他最近的里程碑）。 PI 计划进一步为基础几何理论做出贡献，并开发纯代数问题的应用。

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.