Homotopical Coalgebras, Algebraic Models, and Realizing Derived Equivalences

研究项目涉及代数结构和拓扑研究之间的相互作用，形状或空间的研究。代数拓扑学家用代数结构来描述和简化拓扑现象。光谱代表上同调理论，是建立在拓扑空间上的代数结构，因此对于从一个场转换到另一个场是有用的。在一个项目中，PI将与合作者一起开发光谱中几种不同拓扑设置的代数模型。PI继续培养研究生和传播研究成果。此外，PI还参与了几个促进妇女和代表性不足的少数民族参与数学和科学的组织。在第一个广泛的项目中，PI将继续她的工作，与其他合作者一起开发模和余代数的同调设置。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，以及计算工具的发展。在第二个广泛的项目中，PI继续研究何时可以通过底层结构丰富的等价来实现派生等价。这包括承认一些同伦范畴是刚性的，或者有一个唯一的（直到等价的）底层同伦理论（例如，对于有理圆等变同伦理论），也承认一些同伦范畴是由非等价同伦理论建模的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.