Collaborative Research: Algebraic K-Theory, Topological Periodic Cyclic Homology, and Noncommutative Algebraic Geometry

合作研究：代数K理论、拓扑周期循环同调和非交换代数几何

基本信息

批准号：
1812064
负责人：
Andrew Blumberg
金额：
$ 27.53万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812064&HistoricalAwards=false
关键词：
Collaborative Research Algebraic Theory Topological

项目摘要

Algebraic topology began as the study of algebraic invariants of geometric objects which are preserved under certain smooth deformations. Gradually, it was realized that these algebraic invariants (called cohomology theories) could themselves be represented by geometric objects, known as spectra. A central triumph of modern homotopy theory has been the construction of categories of ring spectra (representing objects for multiplicative cohomology theories) which are suitable for performing constructions directly analogous to those of classical algebra. This move has turned out to be incredibly fruitful, both by providing invariants which shed new light on old questions as well as by raising new questions which have unexpected connections to other areas of mathematics and physics. The project funded by this grant carries out this program in the setting of a rich invariant called algebraic K-theory and related theories known as topological Hochschild, cyclic, and periodic homology. The project studies applications of these theories to a broad range of questions in number theory, algebraic geometry, and geometric topology, as well as algebraic topology itself.This research continues a broad research program aimed at applying recent work of the PIs on algebraic K-theory and trace methods to study a wide variety of basic problems in number theory, noncommutative algebraic geometry, and symplectic topology. It also includes a project to develop the foundations of equivariant derived algebraic geometry, which has applications to organizing computational phenomena observed in the study of topological modular forms. The PIs' recent work has resulted in a complete description of the homotopy groups of K(S) (in terms of other known spectra) and a canonical identification of the fiber of the cyclotomic trace via a spectral lift of Tate-Poitou duality. The PIs have a program to apply this work to provide novel evidence for the Kummer-Vandiver conjecture. If successful, this would provide another example of input from algebraic topology addressing questions in number theory. The PIs previously applied their work on the fiber of the cyclotomic trace to resolve conjectures in the p-adic Langlands program about the (co)homology of stable congruence subgroups. The PIs describe a series of projects that would use homotopy theoretic data about the fiber in the study of the p-adic Langlands program. Other recent work of the PIs established a Kunneth theorem for topological periodic cyclic homology (TP) of dualizable dg categories. This result has already had interesting applications in noncommutative algebraic geometry, as a consequence of regarding TP as a kind of noncommutative Weil cohomology theory. The grant includes a project to establish this viewpoint and to apply TP in noncommutative algebraic geometry. Based on conversations with Abouzaid and Kragh, the PIs have started exploring applications of algebraic K-theory and TP to symplectic topology via the wrapped Fukaya category. The PIs describe a series of projects that leverage their expertise and prior results to study fundamental questions in this area. PI Blumberg has previously worked with Mike Hill to develop the foundations of the theory of equivariant commutative ring spectra. PI Mandell is one of the foremost experts on topological Andre-Quillen homology (TAQ). In collaboration with Basterra, Hill, and Lawson, the PIs study equivariant TAQ as part of a broader program to develop the foundations for equivariant derived algebraic geometry. If successful, this program will provide an organizing principle for phenomenological data coming from work on topological modular forms.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.

代数拓扑开始于研究在某些平滑变形下保持的几何对象的代数不变量。逐渐地，人们意识到这些代数不变量（称为上同调理论）本身可以用几何对象（称为谱）来表示。现代同伦理论的一个主要胜利是环谱范畴的构造（代表乘法上同调理论的对象），它适合于执行与经典代数直接类似的构造。事实证明，这一举措取得了令人难以置信的成果，既提供了为旧问题提供新思路的不变量，又提出了与数学和物理其他领域有着意想不到的联系的新问题。由这笔赠款资助的项目在称为代数 K 理论的丰富不变量和称为拓扑 Hochschild、循环和周期同调的相关理论的背景下执行该计划。该项目研究这些理论在数论、代数几何和几何拓扑以及代数拓扑本身的广泛问题中的应用。这项研究继续了一项广泛的研究计划，旨在将 PI 的最新工作应用于代数 K-理论和迹方法来研究数论、非交换代数几何和辛拓扑中的各种基本问题。它还包括一个开发等变派生代数几何基础的项目，该项目可应用于组织拓扑模形式研究中观察到的计算现象。 PI 最近的工作对 K(S) 的同伦群（根据其他已知光谱）进行了完整的描述，并通过 Tate-Poitou 对偶性的光谱提升对分圆迹线的纤维进行了规范的识别。 PI 有一个计划来应用这项工作，为 Kummer-Vandiver 猜想提供新的证据。如果成功，这将提供代数拓扑输入解决数论问题的另一个例子。 PI 之前将他们的工作应用于分圆迹的纤维，以解决 p 进朗兰兹纲领中关于稳定同余子群的（共）同源性的猜想。 PI 描述了一系列项目，这些项目将在 p 进朗兰兹纲领的研究中使用有关光纤的同伦理论数据。 PI 最近的其他工作为可对偶 dg 类别的拓扑周期循环同调 (TP) 建立了 Kunneth 定理。由于将 TP 视为一种非交换 Weil 上同调理论，这一结果已经在非交换代数几何中得到了有趣的应用。该赠款包括一个建立这一观点并将 TP 应用到非交换代数几何中的项目。根据与 Abouzaid 和 Kragh 的对话，PI 已开始通过包装的 Fukaya 范畴探索代数 K 理论和 TP 在辛拓扑中的应用。 PI 描述了一系列项目，这些项目利用他们的专业知识和先前的成果来研究该领域的基本问题。 PI Blumberg 此前曾与 Mike Hill 合作开发了等变交换环谱理论的基础。 PI Mandell 是拓扑 Andre-Quillen 同调 (TAQ) 方面最重要的专家之一。 PI 与 Basterra、Hill 和 Lawson 合作，研究等变 TAQ，作为更广泛计划的一部分，为等变派生代数几何奠定基础。如果成功，该计划将为来自拓扑模块形式工作的现象学数据提供组织原则。该奖项反映了 NSF 的法定使命，并通过使用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。