RUI: Higher Structures in Stable, Equivariant, and Motivic Homotopy Theory

RUI：稳定、等变和动机同伦理论中的高级结构

• 批准号: 1709302
• 负责人: Kyle Ormsby
• 金额: $ 36.76万
• 依托单位: Reed College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-09-01 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1709302&HistoricalAwards=false
• 关键词: RUI; Higher; Structures; Stable; Equivariant

This research investigates the interface between three branches of contemporary mathematics: higher categories, homotopy theory, and algebraic geometry. Loosely speaking, these subjects study structure, deformations, and polynomials, respectively. The projects undertaken via this award link these concepts in novel ways, seeking to expose new theorems. The Principal Investigors will also support the participation of a diverse group of undergraduate students in their research by organizing summer research groups and leading senior theses related to these projects.The PIs' approach to their research program is five-fold. First, they will jointly study the fashion in which spectral Mackey functors of an absolute Galois group map to "spectral Gysin functors" (constructed from Garkusha-Panin-Voevodksy's theory of framed correspondences) which model motivic spectra. Second, they will study the eta-periodic motivic sphere spectrum via connective Witt K-theory. Third, they will explicate further structure in the Balmer spectrum of the stable motivic homotopy category via comodules over the Lazard Hopf algebroid. Fourth, they will construct equivariant infinite loop space machines which handle pairings of genuine G-permutative categories. And finally, they will study the homotopy theory of Picard 2-groupoids and stable 2-types, and the interaction between the two.

本研究探讨当代数学的三个分支：高等范畴、同伦理论和代数几何之间的接口。粗略地说，这些学科分别研究结构、变形和多项式。通过该奖项开展的项目以新颖的方式将这些概念联系起来，寻求揭示新的定理。首席研究员还将通过组织夏季研究小组和撰写与这些项目相关的高级论文来支持不同群体的本科生参与他们的研究。私人学院的研究项目有五个方面。首先，他们将共同研究绝对伽罗瓦群的谱Mackey函子映射到模拟动力谱的“谱Gysin函子”（由Garkusha-Panin-Voevodksy的框架对应理论构建）的方式。其次，他们将通过连接Witt - k理论研究eta周期动力球谱。第三，他们将通过Lazard Hopf代数群上的模进一步解释稳定动力同伦范畴的Balmer谱中的结构。第四，他们将构造处理真g置换范畴配对的等变无限循环空间机。最后，他们将研究Picard 2-群和稳定2型的同伦理论，以及两者之间的相互作用。

项目成果

A multiplicative comparison of Segal and Waldhausen K-Theory

Segal 和 Waldhausen K 理论的乘法比较

• DOI: 10.1007/s00209-019-02394-7
• 发表时间: 2019
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Bohmann, Anna Marie;Osorno, Angélica M.
• 通讯作者: Osorno, Angélica M.

Comparison of Waldhausen constructions

Waldhausen 结构比较

• DOI: 10.2140/akt.2021.6.97
• 发表时间: 2021
• 期刊: Annals of K-Theory
• 影响因子: 0.6
• 作者: Bergner, Julia E.;Osorno, Angélica M.;Ozornova, Viktoriya;Rovelli, Martina;Scheimbauer, Claudia I.
• 通讯作者: Scheimbauer, Claudia I.

The Tambara structure of the trace ideal for cyclic extensions

迹线的 Tambara 结构非常适合循环扩展

• DOI: 10.1016/j.jalgebra.2020.04.036
• 发表时间: 2020
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Calle, Maxine;Ginnett, Sam
• 通讯作者: Ginnett, Sam

VANISHING IN STABLE MOTIVIC HOMOTOPY SHEAVES

消失在稳定的基元同伦滑轮中

• DOI: 10.1017/fms.2018.3
• 发表时间: 2018
• 期刊: Sigma
• 作者: ORMSBY, KYLE;RÖNDIGS, OLIVER;ØSTVÆR, PAUL ARNE
• 通讯作者: ØSTVÆR, PAUL ARNE

2–Segal objects and the Waldhausenconstruction

2—Segal 物品和 Waldhausen 建筑

• DOI: 10.2140/agt.2021.21.1267
• 发表时间: 2021
• 期刊: Algebraic & Geometric Topology
• 影响因子: 0.7
• 作者: Bergner, Julia E;Osorno, Angélica M;Ozornova, Viktoriya;Rovelli, Martina;Scheimbauer, Claudia I
• 通讯作者: Scheimbauer, Claudia I