HOMOTOPICAL ALGEBRA: COALGEBRAS, DGAS, AND RATIONAL EQUIVARIANT SPECTRA

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，PI和Greenlees为某些类型的对称光谱开发了代数模型，允许完整的计算。 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性不足的少数民族参与数学和科学的组织。PI继续研究何时可以通过基础的丰富结构的等同关系实现衍生的等同关系。这方面的例子出现在PI的长期项目与约翰Greenlees构建代数模型的合理G-等变谱紧凑李群G。更具体地说，PI和Greenlees计划扩展他们的环面模型，为有理G-等变交换和结合环谱提供一个代数模型。 PI和Kathryn Hess继续发展余代数的同伦设置，其动机来自罗涅的环谱的Hopf-Galois扩展和Hess的下降同伦框架。 PI和Birgit Richter正在为同伦交换微分分次代数开发一个简单的代数模型。

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, the PI and Greenlees develop algebraic models for certain types of spectra with symmetries that allow complete calculations. 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 math and science. The PI continues to study when derived equivalences can be realized by underlying richly structured equivalences. Examples of this arise in the PI's long term project with John Greenlees of constructing algebraic models for rational G-equivariant spectra for compact Lie groups G. More specifically, the PI and Greenlees plan to extend their model for tori to provide an algebraic model for rational G-equivariant commutative and associative ring spectra. The PI and Kathryn Hess continue to develop the homotopical setting for coalgebras with motivations coming from Rognes' Hopf-Galois extensions of ring spectra and Hess' homotopical framework for descent. The PI and Birgit Richter are developing a simple algebraic model for homotopically commutative differential graded algebras.