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

• 批准号: 2104420
• 负责人: Andrew Blumberg
• 金额: $ 20万
• 依托单位: Columbia 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 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104420&HistoricalAwards=false
• 关键词: Collaborative; Research; Algebraic; Theory; Arithmetic

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.

代数拓扑开始是对在某些平滑变形下保存的几何物体代数不变的研究。 逐渐地，人们意识到，代数不变的人，称为同胞理论，本身可以用称为光谱的几何对象表示。 现代同质理论的中心胜利是构造环光谱类别的构造，这些类别适合执行类似于经典代数类似的结构。 通过提供不变性，这证明了这是富有成果的，这为旧问题提供了新的启示。此外，它提出了与其他数学和物理学领域有意外联系的新问题。 该项目在一个不变的代数K理论和相关理论的情况下起作用。 这些理论在数字理论，代数几何学和几何拓扑以及代数拓扑本身中的广泛问题研究中的应用。该奖项为将参与本研究部分的学生提供了支持。这项赠款资助了一项广泛的研究计划，旨在应用PI的最新工作，对代数K理论，痕量方法和均等稳定同型理论来研究各种各样的多样性同质理论中的问题。 PIS的先前工作研究了球形光谱的代数K理论和代数数环的环形迹线的纤维。 目前的项目将调查扩展到在代数p-Integers tate-poitou二重性方面对更通用环的环形痕迹的纤维，而相关的K理论问题则更为普遍地说明其他类型的Artin二元性。 该项目探讨了几何soule嵌入与PIS先前工作中发现的Kummer-vandiver猜想之间的联系。 PIS的先前工作还提供了与代数K理论上P-AFAID插值ADAMS操作存在的分裂，并为至少在常规环的背景下存在。 该项目研究了该问题的特定猜想和方法。 该项目推动了一种新的方法，用于孵化器 - 瓦尔德豪森地图，该图将对几何，差异和符号拓扑具有影响。 该项目提出了对正尺寸紧凑型谎言组的模棱两可稳定同型理论中乘法规范图的构建。这需要一个新的基础，用于非等级分解同源性同源性，并承诺为正尺寸紧凑型谎言群构建真正的eproivariant分解同源理论。 该项目包括PIS与Basterra，Hill和Lawson的合作，以研究Equivariant Taq，这是开发Equivariant派生的代数几何基础的更广泛计划的一部分。 如果成功的话，该计划将为来自拓扑模块化形式的作品提供的现象学数据提供组织原则。该奖项反映了NSF的法定任务，并被认为是使用基金会的知识分子优点和更广泛的影响评估标准的评估值得支持的。