Homotopical Coalgebras, Algebraic Models, and Realizing Derived Equivalences

这些研究项目涉及研究代数结构和拓扑学、研究形状或空间之间的相互作用。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，是由拓扑空间建立的代数结构，因此对于从一个域到另一个域的转换是有用的。在一个项目中，PI将与合著者一起为光谱中的几个不同的拓扑设置开发代数模型。国际和平研究所继续培训研究生并传播研究成果。此外，促进妇女和代表不足的少数群体参与数学和科学的几个组织也参与了该协会的工作。在第一个广泛的项目中，PI将继续与不同的合著者一起开发余模和余代数的同伦设置。这个项目的动机包括在代数K理论中的应用，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继续研究什么时候可以通过基础的丰富结构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，或者具有唯一的(直到等价的)同伦理论(例如，对于有理圆等变同伦理论已经表明)，以及承认一些同伦范畴是由非等价同伦理论建模的。这一奖项反映了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.