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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.

研究项目涉及代数结构和拓扑研究、形状或空间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 将与合著者一起开发针对光谱中几种不同拓扑设置的代数模型。 PI 继续培养研究生并传播研究成果。此外，PI 还参与了多个促进女性和代表性不足的少数群体参与数学和科学领域的组织。在第一个广泛的项目中，PI 将继续与各个合著者一起开发余模和余代数的同伦设置。该项目的动机包括代数 K 理论的应用、与色同伦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底层丰富的结构等价来实现派生等价。这包括承认一些同伦类别是刚性的，或者具有独特的（直至等价）基础同伦理论（例如，对于有理圆等变同伦理论所示），并且还承认一些同伦类别是由非等价同伦理论建模的。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的评估进行评估，被认为值得支持。影响审查标准。