ALGEBRAIC MODELS OF HOMOTOPY THEORIES AND HOMOTOPICAL MODELS OF ALGEBRA

同伦理论的代数模型和代数的同伦模型

• 批准号: 1104396
• 负责人: Brooke Shipley
• 金额: $ 17.75万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104396&HistoricalAwards=false
• 关键词: ALGEBRAIC; MODELS; HOMOTOPY; THEORIES; HOMOTOPICAL

The PI continues to study when derived equivalences can be realized by underlying richly structured equivalences. Examples of this arise in the PI's long term project with John Greenlees of constructing algebraic models for rational G-equivariant spectra for compact Lie groups G. The PI and Greenlees plan to extend their model for connected groups in the free case to non-connected groups. The PI and Greenlees also plan to extend their model for tori to provide an algebraic model for rational G-equivariant commutative and associative ring spectra. As a step towards the general case, the specific case of SO(3) is also being considered. The PI and Kathryn Hess propose to develop model categories for coalgebras over various comonads in various settings. The main motivation here is to provide a homotopical setting for studying Hopf-Galois extensions of ring spectra.The proposed research projects involve the interplay between the study of algebraic structures and topology, the study of shapes or spaces. Algebraic topologists use algebraic structures to describe and simplify topological phenomena. Spectra, which represent cohomology theories, are algebraic structures built out of topological spaces and hence are useful for translating from one field to the other. In one project, the PI and Greenlees develop algebraic models for certain types of spectra which allow complete calculations. The PI also continues to train graduate students and disseminate research results. In addition, the PI is involved with several organizations which promote the participation of women and underrepresented minorities in science.

PI 继续研究何时可以通过底层丰富的结构等价来实现派生等价。这方面的例子出现在 PI 与 John Greenlees 的长期项目中，该项目为紧李群 G 的有理 G 等变谱构建代数模型。PI 和 Greenlees 计划将其自由情况下连通群的模型扩展到非连通群。 PI 和 Greenlees 还计划扩展他们的 tori 模型，为有理 G 等变交换和缔合环谱提供代数模型。 作为迈向一般情况的一步，SO(3) 的具体情况也正在考虑中。 PI 和 Kathryn Hess 建议开发各种环境中各种共代数的模型类别。这里的主要动机是为研究环谱的 Hopf-Galois 扩展提供同伦环境。拟议的研究项目涉及代数结构和拓扑的研究以及形状或空间的研究之间的相互作用。代数拓扑学家使用代数结构来描述和简化拓扑现象。 谱代表上同调理论，是由拓扑空间构建的代数结构，因此可用于从一个领域转换到另一个领域。在一个项目中，PI 和 Greenlees 为某些类型的光谱开发了代数模型，可以进行完整的计算。 PI 还继续培训研究生并传播研究成果。 此外，PI 还参与了多个促进妇女和代表性不足的少数群体参与科学的组织。