Perfectoid Spaces, Diamonds, and the Langlands Program

朗兰兹计划就像一个宏大的统一数论。这是一组猜想，包含了欧拉和高斯发现的经典的数模式(称为“互易定律”)，以及费马最后定理等现代结果。简而言之，朗兰兹程序统一了两种对称性：一种来自连续实体(想想球体在空间中可能旋转的所有方式)，另一种来自代数方程的根(想想二次公式中的“正负”)。PI建议对朗兰兹计划进行研究，特别是关于P-进制数的部分，P-进制数是实数的奇怪表亲。(这里的字母p代表一个质数。实数形成一个连通的连续体，而p元数则是完全不连通的，就像无限的分形尘。)朗兰兹计划适用于真实的数字，这是朗兰兹自己制定的，而p-addy的故事仍然有些神秘。PI打算通过研究过去几年发现的一些迷人的新结构的几何学，即完美拟态空间和钻石，为朗兰兹计划的这一部分做出贡献。这些结构是由彼得·肖尔茨发明的，他因这些发现于2018年获得了菲尔兹奖。该项目还支持PI的研究生玛丽亚·费尔南德斯在相关主题上的工作。自2012年左右推出以来，完美拟态空间已经有了一些意想不到的应用。其中之一是劳伦特·法格格的程序，该程序将朗兰兹程序几何化为p-进制数。也就是说，他使它与几何朗兰兹的并行程序保持一致，这似乎要容易得多。受Fgagger程序的启发，PI(与Tasho Kaletha和David Hansen共同工作)使用可应用于完美拟空间的Lefschetz不动点公式，证明了Kottwitz关于Rapoport-Zink空间上同调的猜想的形式。这可以被认为是p-add群和它的内扭曲之一之间的朗兰兹函数的几何表现。PI将扩展这些方法以应用于其他类型的功能。还有另一个与完美拟态空间的“光滑性”有关的项目(一个可以追溯到2016年的美丽概念)，以及另一个关于函数域上椭圆曲线的模性的项目，该项目是与PI的研究生联合进行的。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。

The Langlands program is like a grand unified theory of numbers. It a suite of conjectures which encompasses classical patterns in numbers (called "reciprocity laws") discovered by Euler and Gauss, as well as modern results like Fermat's Last Theorem. In brief, the Langlands program unites two sorts of symmetries: one coming from continuous entities (think of all the ways a sphere might be rotated in space), and the other from roots of algebraic equations (think of the "plus or minus" in the quadratic formula). The PI proposes research on the Langlands program, specifically the part dealing with the p-adic numbers, which are the strange cousins of the real numbers. (The letter p here stands for a prime number. The real numbers form a connected continuum, whereas the p-adic numbers are totally disconnected, like infinite fractal dust.) The Langlands program as it applies to the real numbers was worked out by Langlands himself, while the p-adic story remains somewhat mysterious. The PI intends to contribute to this portion of the Langlands program by studying the geometry of some fascinating new structures discovered in the last several years, namely perfectoid spaces and diamonds. These structures were invented by Peter Scholze, who received the Fields Medal in 2018 for their discovery. The project also supports work of the PI's graduate student, Maria Fernandez, on related topics.Since their introduction around 2012, perfectoid spaces have had some unexpected applications. One of these is Laurent Fargues' program to geometrize the Langlands program over the p-adic numbers. That is, he has brought it in line with the parallel program of geometric Langlands, which seems rather more tractable. Inspired by Fargues' program, the PI has proved (in joint work with Tasho Kaletha and David Hansen) a form of Kottwitz' conjecture on the cohomology of Rapoport-Zink spaces, using a version of the Lefschetz fixed-point formula which can apply to perfectoid spaces. This can be recognized as a geometric manifestion of Langlands functoriality between a p-adic group and one of its inner twists. The PI will expand these methods to apply to other sorts of functorialities. There is a further project concerning the "smoothness" of perfectoid spaces (a beautiful concept dating to 2016), and another on the modularity of elliptic curves over function fields, which is joint with the PI's graduate student.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.