Perfectoid Spaces, Diamonds, and the Langlands Program

朗兰兹纲领就像一个大统一的数论。它是一套包含了欧拉和高斯发现的经典数字模式（称为“互易定律”）以及费马大定理等现代结果的体系。简而言之，朗兰兹纲领统一了两种对称性：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“加或减”）。PI建议研究朗兰兹程序，特别是处理p-adic数的部分，这是真实的数的奇怪表亲。(The这里的字母p代表质数。 真实的数形成一个连通的连续统，而p-adic数则完全不连通，就像无限分形尘埃。） 适用于真实的数的朗兰兹纲领是朗兰兹自己制定的，而p-adic的故事仍然有些神秘。 PI打算通过研究在过去几年中发现的一些迷人的新结构的几何学，即完美空间和钻石，为朗兰兹计划的这一部分做出贡献。 这些结构由Peter Scholze发明，他因其发现而于2018年获得菲尔兹奖。该项目还支持PI的研究生Maria费尔南德斯在相关主题上的工作。自2012年左右引入以来，完美空间已经有了一些意想不到的应用。其中之一是洛朗法尔盖的计划，几何朗兰兹计划的p进数。 也就是说，他使它与几何朗兰兹的平行程序相一致，后者似乎更容易处理。 受法尔盖的启发，PI（与Tasho Kaletha和大卫汉森）证明了关于Rapoport-Zink空间上同调的Kottwitz猜想的一种形式，使用了一个可以应用于完美空间的Lefschetz不动点公式。这可以被认为是一个p进群和它的一个内部扭曲之间的朗兰兹泛函的几何表现。PI将扩展这些方法以应用于其他类型的函数。此外，还与PI的研究生共同进行了关于完美空间的“光滑性”（2016年提出的美丽概念）和函数域上椭圆曲线的模块化的研究。该奖项反映了NSF的法定使命，通过基金会的智力价值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.