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Collaborative Research: Algebraic K-Theory, Arithmetic, and Equivariant Stable Homotopy Theory

合作研究：代数K理论、算术和等变稳定同伦理论

基本信息

批准号：
2104348
负责人：
Michael Mandell
金额：
$ 20.3万
依托单位：
Indiana University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-08-01 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104348&HistoricalAwards=false
关键词：
Collaborative Research Algebraic Theory Arithmetic

项目摘要

Algebraic topology began as the study of those algebraic invariants of geometric objects that are preserved under certain smooth deformations. Gradually, it was realized that the 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 that are suitable for performing constructions analogous to those of classical algebra. This has proved fruitful by providing invariants, which shed new light on old questions. In addition, it has raised new questions that have unexpected connections to other areas of mathematics and physics. This project works in the setting of an invariant called algebraic K-theory and related theories. 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. The award provides support for students who will be engaged in parts of this research.This grant funds a broad research program aimed at applying recent work of the PIs on algebraic K-theory, trace methods, and equivariant stable homotopy theory to study a wide variety of problems in homotopy theory. Prior work of the PIs studied the algebraic K-theory of the sphere spectrum and the fiber of the cyclotomic trace for algebraic number rings. The current project expands the investigation to the fiber of the cyclotomic trace on more general rings over algebraic p-integers in terms of Tate-Poitou duality and a related K-theory question more generally for other kinds of Artin duality. The project explores a connection between the geometric Soule embedding and the Kummer-Vandiver conjecture discovered in the PIs' prior work. The PIs' prior work also gives a splitting that is consistent with and gives evidence for the existence of p-adically interpolated Adams operations on the algebraic K-theory at least in the context of regular rings. The project investigates specific conjectures and approaches to this problem. The project advances a new approach to the the Hatcher-Waldhausen map that would have implications in geometric, differential, and symplectic topology. The project proposes a construction of multiplicative norm maps in equivariant stable homotopy theory for positive dimensional compact Lie groups. This requires a new foundation for non-equivariant factorization homology and holds the promise of constructing a genuine equivariant factorization homology theory for positive dimensional compact Lie groups. The project includes a collaboration of the PIs with Basterra, Hill, and Lawson to 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在某些光滑变形下保持不变的代数不变量。渐渐地，人们意识到代数不变量，称为上同调理论，可以用称为谱的几何对象来表示。现代同伦理论的一个中心胜利是构造了环谱范畴，这些范畴适合于进行类似于经典代数的构造。这已经证明是富有成效的，提供了不变量，揭示了新的老问题。此外，它还提出了一些新的问题，这些问题与数学和物理的其他领域有着意想不到的联系。这个项目的工作原理是在一个不变量的设置称为代数K理论和相关理论。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中的广泛应用。该奖项为将参与本研究的学生提供支持。该补助金资助了一个广泛的研究计划，旨在应用PI在代数K理论，跟踪方法和等变稳定同伦理论方面的最新工作来研究同伦理论中的各种问题。 PI的先前工作研究了代数数环的球谱和分圆迹的纤维的代数K-理论。目前的项目扩大调查的纤维割圆迹更一般的环代数p-整数的Tate-Poitou对偶和相关的K-理论问题更普遍的其他种类的Artin对偶。该项目探索了几何Soule嵌入和在PI之前的工作中发现的Kummer-Vandiver猜想之间的联系。 PI之前的工作也给出了一个分裂，这与代数K理论上的p-基插值亚当斯运算的存在是一致的，并且至少在正则环的上下文中给出了证据。该项目研究了解决这一问题的具体方法和途径。该项目提出了一种新的方法来Hatcher-Waldhausen映射，这将在几何，微分和辛拓扑中产生影响。该项目提出了一个建设的乘法范数映射在等变稳定同伦理论的正维紧李群。这需要一个新的基础，非等变分解同调，并持有承诺，建设一个真正的等变分解同调理论的正维紧李群。该项目包括与Basterra，Hill和Lawson合作的PI研究等变TAQ作为更广泛的计划的一部分，以开发等变导出代数几何的基础。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。