Homotopical Coalgebras, Algebraic Models, and Realizing Derived Equivalences

研究项目涉及代数结构和拓扑学的研究之间的相互作用，形状或空间的研究。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，是建立在拓扑空间之外的代数结构，因此对于从一个领域到另一个领域的转换是有用的。在一个项目中，与合作者，PI将开发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成果。 此外，PI还参与了几个促进妇女和代表人数不足的少数群体参与数学和科学的组织。在第一个广泛的项目中，PI将继续与各种合著者合作，为余模和余代数开发同伦设置。这个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基础的丰富结构的等价物来实现衍生等价物。这包括将某些同伦范畴识别为刚性的，或者具有唯一的（直到等价）基本同伦理论（如已经示出的，例如，对于有理圆等变同伦理论），也认识到一些同伦范畴是由非-该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准。

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.