Homotopical Coalgebras, Algebraic Models, and Realizing Derived Equivalences

同伦余代数、代数模型和实现导出等价

基本信息

批准号：
1811278
负责人：
Brooke Shipley
金额：
$ 24.17万
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：
2018
资助国家：
美国
起止时间：
2018-06-01 至 2022-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811278&HistoricalAwards=false
关键词：
Homotopical Coalgebras Algebraic Models Realizing

项目摘要

The 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, with co-authors, the PI will develop algebraic models for several different topological settings in spectra. The PI continues to train graduate students and disseminate research results. In addition, the PI is involved with several organizations that promote the participation of women and underrepresented minorities in mathematics and science. In the first broad project the PI will continue her work with various coauthors on developing homotopical settings for comodules and coalgebras. The motivations for this project include applications to algebraic K-theory, connections with chromatic homotopy theory and string theory, and development of computational tools. In a second broad project, the PI continues to study when derived equivalences can be realized by underlying richly structured equivalences. This includes recognizing some homotopy categories as rigid, or having a unique (up to equivalence) underlying homotopy theory (as has been shown, for example, for rational circle-equivariant homotopy theory) and also recognizing some homotopy categories as being modeled by non-equivalent homotopy theories.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.

该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，与合着者一起，PI将开发用于光谱中几种不同拓扑设置的代数模型。 PI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参与和代表性不足的数学和科学。在第一个宽广的项目中，PI将继续与各种合着者一起开发综合和山地的同型环境。该项目的动机包括对代数K理论的应用，与色质理论和弦理论的联系以及计算工具的开发。在第二个广泛的项目中，PI继续研究何时可以通过基础结构化等价来实现衍生的等价。这包括将某些同质类别识别为刚性，或具有独特的（等效性）基础理论（例如，如图所示，例如，对于理性的循环 - 均衡同质理论），并认识到某些同质类别，以识别非等量的同质性理论来建立不等的同类奖，以反映NSF的Infortional of Suttion te Indertial of Nection te Inderit of deem eyt of dee eyt eym eyre the ey eym eyt eym eyr ey eym awardy ey eym ever ey eym ever the nect ofer the ne eym ship the ne eym award awtion awmity。影响审查标准。