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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：
1811820
负责人：
Michael Mandell
金额：
$ 29.2万
依托单位：
Indiana University
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 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811820&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.

代数拓扑学开始于研究几何对象的代数不变量，这些几何对象在某些光滑变形下保持不变。渐渐地，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来表示，称为谱。现代同伦理论的一个中心胜利是构造了环谱范畴（代表乘性上同调理论的对象），这些范畴适合于进行直接类似于经典代数的构造。这一举措已被证明是令人难以置信的富有成效的，无论是通过提供不变量，揭示了新的光在旧的问题，以及通过提出新的问题，有意想不到的连接到其他领域的数学和物理。该项目由这笔赠款资助，在一个丰富的不变量称为代数K-理论和相关理论称为拓扑Hochschild，循环和周期同源性的设置中进行该计划。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。该研究继续了一个广泛的研究计划，旨在将PI在代数K理论和迹方法方面的最新工作应用于研究数论、非交换代数几何和辛拓扑中的各种基本问题。它还包括一个项目，以发展等变派生代数几何的基础，这有组织的拓扑模形式的研究中观察到的计算现象的应用。 PI最近的工作导致了K（S）的同伦群的完整描述（根据其他已知的谱）和通过Tate-Poitou对偶的谱提升对分圆迹的纤维的规范识别。 PI有一个程序来应用这项工作，为Kummer-Vandiver猜想提供新的证据。如果成功的话，这将提供另一个从代数拓扑学输入解决数论问题的例子。PI以前应用他们的工作纤维的分圆迹解决progratures在p-adic朗兰兹计划的（上）同调稳定的同余子群。PI描述了一系列项目，这些项目将使用关于光纤的同伦理论数据来研究p-adic Langlands程序。 PI最近的其他工作建立了可对偶dg范畴的拓扑周期循环同调（TP）的Kunneth定理。这个结果已经有有趣的应用在非交换代数几何，作为一个后果，把TP作为一种非交换的Weil上同调理论。该补助金包括一个项目，以建立这种观点和应用TP在非交换代数几何。基于与Abouzaid和Kragh的对话，PI已经开始探索代数K理论和TP通过包裹福谷范畴在辛拓扑中的应用。 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结果来研究这一领域的基本问题。 PI Blumberg以前曾与Mike Hill合作开发等变交换环谱理论的基础。PI Mandell是拓扑Andre-Quillen同源性（TAQ）方面最重要的专家之一。在与巴斯特拉，希尔和劳森合作，PI研究等变TAQ作为一个更广泛的计划的一部分，以发展等变派生代数几何的基础。如果成功的话，这个计划将提供一个组织原则的现象学数据来自工作的拓扑modularforms.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。