Interactions between p-adic arithmetic geometry and commutative algebra

PI建议研究存在于算术几何(特别是p-进方面)、代数几何和交换代数的共同边界的问题。例如，Mel Hochster的直和猜想，它假设了正则环的一些基本性质的存在，40多年来一直是交换代数中的一个重要的公开问题；等价特征情况的解决(由Hochster从70年代开始)是现代交换代数的主要原因，而p-进情况仍然是诱人的开放的。PI最近发现，p-进Hodge理论的一些想法(由于Faltings)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-主要是肖尔茨的完美拟态空间的美丽理论--进一步追求这个方向-从快速发展的p-进Hodge理论到直接求和猜想和其他纯粹的代数问题。相反，PI以前关于直和猜想的工作，再加上一些派生的代数几何，最近已被证明有助于实现p-进Hodge理论的某些几何方面的显著简化；PI计划更深入地发展派生方面以更好地概念化画面。对具有整数系数的多项式的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，即以素数为模数的解，然后是素数的模幂。这个想法是，随着素数的幂的增加，近似变得更好。Grothendieck在过去半个世纪中对数学的革命性研究不仅使人们不仅可以给前面的陈述赋予精确的含义，而且还提供了一个美丽的几何背景-p进几何的世界-来研究这样的近似解。这一背景一直是许多最近数学进步的核心(例如威尔斯对费马大定理的证明和朗兰兹计划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.