Algebraic Structures in Equivariant Homotopy Theory and K Theory

对称和变形在理解拓扑空间时可以看作是两种相反的力量。一方面，对称性代表了空间的刚性结构——在旋转或反射等操作下，空间是相同的。变形，另一方面，故意省略刚性结构，使我们能够了解空间的大致特征。这些特征的示例包括孔的数量或空间的独立块的数量。这两种理解空间的方法结合在等变代数拓扑中。代数拓扑是通过代数不变量研究复杂高维空间的一个数学领域，而等变代数拓扑以鲁棒的方式将空间的对称性纳入不变量中。这个领域与数学物理和数据分析等学科有联系。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。特别感兴趣的上下文还包括与代数和拓扑k理论相关的问题，不变量是现代拓扑，代数和数论方法的核心。此外，这笔拨款将帮助PI开展旨在促进女性和少数族裔在数学领域的推广活动，包括她与新成立的范德比尔特大学数学女性协会学生分会和范德比尔特大学定向阅读计划的合作。这些项目将扩大数学思维的范围，并为更多的学生提供参与数学项目的机会，为数学界和更有数学素养的社会成员创造更广泛的基础。最近同伦理论的发展突出了等方差的重要性，以及许多方面对等方差的理解仍然很差。等变同伦理论是代数k理论中现代计算的关键，在p进霍奇理论中有着深刻的分支。等变同伦理论也是非等变同伦理论的核心成果，例如最近Kervaire不变量1问题的解。PI的研究计划将开发新的工具来扩展这类计算，并加深我们对对称在同调考虑中表现出来的令人惊讶的方式的全面了解。她的项目侧重于不同组的对称的相互作用，无论是在拓扑和代数水平。在代数层面上，这一领域的新发展将使数学家能够充分利用代数工具来理解具有群行为的拓扑空间。在拓扑层面上，她的工作将为有关对偶性、色同伦理论和代数k理论的进展提供基础。在进行这项研究的同时，PI计划继续目前旨在促进妇女和代表性不足的少数民族在数学方面的活动。通过为这些团体的数学家提供机会来传播他们的新成果，她将支持他们的职业生涯，并进一步提高数学工作者的知名度。PI提议的活动还将通过为具有广泛背景的学生提供参与数学研究经验的机会来扩大数学参与。该计划将扩大数学思维的范围，并为更多的学生提供参与数学项目的机会，为数学界和社会上更多具有数学素养的成员创造更广泛的基础。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.