Algebraic Structures in Equivariant Homotopy Theory

代数拓扑学是纯数学的一个领域，它通过使用代数不变量来研究复杂且通常是高维的空间。 它与从数据分析到物理学的各个领域都有联系。 PI的研究特别关注用于研究具有有趣对称性的空间的代数。 这种类型的结构在代数拓扑学的最近发展中变得越来越重要，但仍然相对缺乏了解。PI的工作将推进纯数学这一领域的最先进的知识。她的研究计划集中在开发丰富的代数结构，出现在这种情况下，并使用这些来构建和分析经典和新的代数拓扑变形不变量的对称空间。 此外，这笔赠款的资金将帮助PI开展旨在促进妇女和数学代表性不足的少数民族的外联活动，包括她与新成立的范德比尔特妇女数学小组的工作。PI计划进行等变稳定同伦理论的研究。这个领域主要涉及提取在变形和对称下不变的空间特征。同伦理论的最新发展突出了等方差的重要性，以及等方差的许多方面仍然知之甚少。等变同伦理论是代数K-理论中现代计算的关键，也是非等变同伦理论的结果，例如最近对Kervaire不变一问题的解决。PI的研究计划将开发新的工具来扩展这些计算，并加深我们对对称性在同伦考虑中表现出来的令人惊讶的方式的整体了解。她的工作的一个主要组成部分将是建立方法的建设上同调理论与组行动从代数数据，以提供新的和改进的工具，进一步研究该领域。她的工作也将集中在发展代数结构固有的等变上同调理论与交换环结构。这项工作将为同伦理论中的各种问题提供计算工具。 此外，PI将研究一个新的上同调理论的余代数，包括其连接到等变上同调理论所产生的代数。这些研究途径的总体目标是推进同伦理论的最先进的知识，更具体地说，在同伦理论背景下的群体行动。

Algebraic topology is an area of pure mathematics that approaches the study of complicated and usually high dimensional spaces via the use of algebraic invariants. It has connections to fields ranging from data analysis to physics. The PI's research specifically focuses on the algebra used to study spaces that have interesting symmetries. This type of structure has become increasingly important in recent developments in algebraic topology but is still comparatively poorly understood. The PI's work will advance state-of-the-art knowledge in this area of pure mathematics. Her research program is centered on developing the rich algebraic structures that arise in this context and using these to construct and analyze both classical and new algebro-topological deformation invariants of spaces with symmetries. Additionally, the funds from this grant will assist the PI in her outreach activities designed to promote women and underrepresented minorities in mathematics, including her work with the newly formed Vanderbilt Women in Math group.The PI plans to conduct research in equivariant stable homotopy theory. This area is primarily concerned with extracting features of spaces that are invariant under deformations and symmetries. Recent developments in homotopy theory have highlighted the importance of equivariance, as well as the many ways in which equivariance is remains poorly understood. Equivariant homotopy theory is key to modern computations in algebraic K-theory and in results in nonequivariant homotopy theory, such as the recent solution to the Kervaire invariant one problem. The PI's research program will develop new tools for extending these kinds of calculations as well as deepening our overall picture of the surprising ways in which symmetry manifests itself in homotopical considerations. A primary component of her work will be establishing methods of constructing cohomology theories with group actions from algebraic data in order to provide new and improved tools for further research in the field. Her work will also focus on developing algebraic structures inherent in equivariant cohomology theories with commutative ring structures. This work will provide calculational tools for use in a variety questions in homotopy theory. Additionally, the PI will investigate a new cohomology theory for coalgebras, including its connections to equivariant cohomology theories arising from algebras. The overall goals of these avenues of research are to advance state-of-the-art knowledge in homotopy theory and more concretely in group actions in a homotopy theoretic context.