ALGEBRAIC MODELS OF HOMOTOPY THEORIES AND HOMOTOPICAL MODELS OF ALGEBRA

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。这方面的例子出现在PI与John Greenlees的长期项目中，该项目是为紧李群g的有理g等变谱构建代数模型。PI和Greenlees计划将他们在自由情况下的连通群模型扩展到非连通群。PI和Greenlees还计划扩展他们的环面模型，以提供有理g等变交换和结合环谱的代数模型。作为向一般情况迈进的一步，我们也在考虑SO(3)的具体情况。PI和Kathryn Hess建议在不同的环境下为不同的公共空间上的代数开发模型类别。本文的主要目的是为研究环谱的Hopf-Galois扩展提供一个同调环境。提议的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和greenlee为某些类型的光谱开发了代数模型，允许进行完整的计算。PI还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群体参与科学的组织。

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. The PI and Greenlees plan to extend their model for connected groups in the free case to non-connected groups. The PI and Greenlees also plan to extend their model for tori to provide an algebraic model for rational G-equivariant commutative and associative ring spectra. As a step towards the general case, the specific case of SO(3) is also being considered. The PI and Kathryn Hess propose to develop model categories for coalgebras over various comonads in various settings. The main motivation here is to provide a homotopical setting for studying Hopf-Galois extensions of ring spectra.The proposed 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 which allow complete calculations. The PI also continues to train graduate students and disseminate research results. In addition, the PI is involved with several organizations which promote the participation of women and underrepresented minorities in science.