Algebraic Structures in Equivariant Homotopy Theory

代数拓扑学是一个纯数学领域，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的各个领域都有联系。PI的研究主要集中在用于研究具有有趣对称性的空间的代数上。这种类型的结构在代数拓扑学最近的发展中变得越来越重要，但人们对它的了解仍然相对较少。PI的工作将推进这一纯数学领域的最先进知识。她的研究计划集中于发展在这种背景下产生的丰富的代数结构，并利用这些结构来构造和分析具有对称性的空间的经典和新的代数拓扑变形不变量。此外，这笔赠款的资金将帮助国际数学协会开展旨在促进妇女和数学领域代表性不足的少数群体的推广活动，包括她与新成立的范德比尔特女性数学小组的工作。国际数学协会计划开展等变稳定同伦理论研究。该领域主要涉及提取在变形和对称下不变的空间的特征。同伦理论的最新发展突显了等差的重要性，以及对等差的许多理解仍然很少。等变同伦论是代数K-理论中现代计算的关键，也是非等变同伦论的结果，例如最近Kervaire不变一问题的解。PI的研究计划将开发新的工具来扩展这些类型的计算，并加深我们对对称性在同伦考虑中表现出的令人惊讶的方式的总体了解。她工作的一个主要组成部分将是根据代数数据建立具有群作用的上同调理论的方法，以便为该领域的进一步研究提供新的和改进的工具。她的工作还将集中于发展具有交换环结构的等变上同调理论中固有的代数结构。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将研究一种新的余代数上同调理论，包括它与由代数产生的等变上同调理论的联系。这些研究途径的总体目标是在同伦理论中发展最先进的知识，更具体地说，在同伦理论背景下的群体行动中。

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.