Algebraic Structures in Equivariant Homotopy Theory and K Theory

在理解拓扑空间时，对称性和形变可以被看作是两种对立的力量。 一方面，对称性代表了空间的刚性结构，即在旋转或反射等操作下空间是相同的。另一方面，变形有意地省略了刚性结构，以便让我们了解空间的粗略特征。 这些特征的示例包括孔的数量或空间的分离件的数量。 这两种理解空间的方法联合收割机结合在等变代数拓扑中。 代数拓扑是一个通过代数不变量研究复杂且经常高维空间的数学领域，而等变代数拓扑以稳健的方式将空间的对称性纳入不变量中。 该领域与数学物理和数据分析等学科有关。 PI的工作通过开发计算和理论工具来分析等变代数拓扑中不变量的结构和关系，从而推进了这一领域的最新知识。 感兴趣的特定背景还包括与代数和拓扑K理论相关的问题，这些问题是现代拓扑，代数和数论方法的核心不变量。 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数学妇女协会学生分会和范德比尔特定向阅读计划的合作。 这些计划将扩大数学思维的范围，并为更广泛的学生提供机会成为数学项目的一部分，为数学界和社会上更多的数学素养成员创造更广泛的基础。 同伦理论的最新发展强调了等方差的重要性，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，并在p进霍奇理论中有深刻的分支。 等变同伦理论也是非等变同伦理论的核心成果，例如最近解决了Kervaire不变一问题。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。 她的计划侧重于不同群体的对称性的相互作用，无论是在拓扑和代数水平。在代数层面上，这些新的发展，在这方面将允许数学家充分利用代数工具，在理解拓扑空间与群作用。 在拓扑水平上，她的工作将提供一个基础，有关对偶性，色同伦理论和代数K理论的进展。 在进行这项研究的同时，PI计划继续开展旨在促进妇女和数学代表性不足的少数民族的现有活动。 通过为这些团体的数学家提供传播他们新成果的机会，她将支持他们的职业生涯，并增加数学工作者的知名度。 PI的拟议活动也将扩大数学的参与，为具有广泛背景的学生提供机会，成为数学研究经验的一部分。 该计划将扩大数学思维的影响范围，并为更广泛的学生提供参与数学项目的机会，为数学界和社会中更多的数学素养成员创造更广泛的基础。该奖项反映了NSF的法定使命，并被认为值得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来支持。

Symmetry and deformation may be viewed as two opposing forces in understanding topological spaces. On the one hand, symmetries represent rigid structure of a space-ways in which the space is the same under operations like rotation or reflection. Deformations, on the other hand, purposely elide rigid structures so as to allow us to understand rough features of spaces. Examples of these features include the number of holes or the number of separate pieces of the space. These two approaches to understanding spaces combine in equivariant algebraic topology. Algebraic topology is an area of mathematics that studies complicated and frequently high dimensional spaces via algebraic invariants, and equivariant algebraic topology incorporates the symmetries of the spaces into the invariants in a robust way. This field has connections to subjects such as mathematical physics and data analysis. The PI's work advances state-of-the-art knowledge in this area by developing both computational and theoretical tools to analyze the structures of and relationships between the invariants in equivariant algebraic topology. Particular contexts of interest also include questions related to algebraic and topological K-theory, invariants that are at the heart of modern approaches to topology, algebra and number theory. 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 student chapter of the Association for Women in Mathematics and the Vanderbilt Directed Reading Program. These programs will expand the reach of mathematical thinking and give a wider variety of students the opportunity to be part of the project of mathematics, creating both a broader base for the mathematical community and more mathematically literate members of society. Recent developments in homotopy theory have highlighted the importance of equivariance, as well as the many ways in which equivariance remains poorly understood. Equivariant homotopy theory is key to modern computations in algebraic K-theory and has deep ramifications in p-adic Hodge theory. Equivariant homotopy theory is also central to 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. Her program focuses on the interplay of different groups of symmetries, both at a topological and algebraic level. At the algebraic level, these new developments in this area will allow mathematicians to fully exploit algebraic tools in understanding topological spaces with group actions. At the topological level, her work will provide a basis for advances relating to duality, chromatic homotopy theory and algebraic K-theory. While undertaking this research, the PI plans to continue current activities designed to promote women and underrepresented minorities in mathematics. By providing mathematicians from these groups with the opportunity to disseminate their new results, she will support their careers and additionally increase the visibility of the diverse range of people doing mathematics. The PI's proposed activities will also broaden participation in mathematics by providing opportunities for students with a wide range of backgrounds to be part of the mathematical research experience. This program will expand the reach of mathematical thinking and give a wider variety of students the opportunity to be part of the project of mathematics, creating both a broader base for the mathematical community and more mathematically literate members of society.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.