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基本的结构化等价来实现衍生的等价。 PI的长期项目中出现了这一点的示例，约翰·格林勒（John Greenlees）构建了用于紧凑型谎言组的理性G-均衡光谱的代数模型。PI和Greenlees计划将自由案例中的连接组模型扩展到非连接群体。 PI和Greenlees还计划将其模型扩展为Tori，以提供一个代数模型，以用于理性的G-量相位和关联环光谱。 作为迈向一般情况的一步，也正在考虑SO（3）的具体情况。 PI和Kathryn Hess提议在各种环境中针对各种共产党开发模型类别。这里的主要动机是为研究环光谱的HOPF-GALOIS扩展提供一个同质环境。拟议的研究项目涉及代数结构和拓扑的研究，形状或空间的研究之间的相互作用。代数拓扑师使用代数结构来描述和简化拓扑现象。 代表共同体理论的光谱是由拓扑空间构建的代数结构，因此有助于从一个领域转换为另一个领域。在一个项目中，PI和Greenlees为某些类型的光谱开发了代数模型，这些模型允许完整计算。 PI还继续培训研究生并传播研究结果。 此外，PI还参与了几个组织，这些组织促进了妇女和代表性不足的少数群体的参与。