Algebraic Structures in Equivariant Homotopy Theory and K Theory

等变同伦理论和K理论中的代数结构

基本信息

批准号：
2104300
负责人：
AnnaMarie Bohmann
金额：
$ 27.01万
依托单位：
Vanderbilt University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-08-15 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104300&HistoricalAwards=false
关键词：
Algebraic Structures Equivariant Homotopy Theory

项目摘要

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.

对称和变形可以被视为理解拓扑空间的两种相反的力量。一方面，对称性代表了空间通道的刚性结构，其中空间在旋转或反射等操作下是相同的。另一方面，变形故意消除刚性结构，以便让我们了解空间的粗糙特征。这些特征的示例包括孔的数量或空间的独立部分的数量。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且频繁的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合并到不变量中。该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之间的关系，从而推进了该领域的最先进知识。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，这些不变量是现代拓扑、代数和数论方法的核心。此外，这笔赠款的资金将协助 PI 开展旨在促进数学领域女性和代表性不足的少数群体的外展活动，包括她与新成立的女性数学协会范德比尔特学生分会和范德比尔特定向阅读计划的合作。这些项目将扩大数学思维的范围，让更多种类的学生有机会参与数学项目，为数学界和更多具有数学素养的社会成员创造更广泛的基础。同伦理论的最新发展凸显了等变性的重要性，以及人们对等变性仍然知之甚少的许多方面。等变同伦理论是代数 K 理论现代计算的关键，并且在 p 进 Hodge 理论中具有深远的影响。等变同伦理论也是非等变同伦理论结果的核心，例如最近对 Kervaire 不变一问题的解决方案。 PI 的研究计划将开发新的工具来扩展此类计算，并加深我们对对称性在同伦考虑中表现出令人惊讶的方式的整体了解。她的课程重点关注拓扑和代数层面上不同对称群的相互作用。在代数层面，该领域的这些新发展将使数学家能够充分利用代数工具来理解具有群作用的拓扑空间。在拓扑层面，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。在开展这项研究的同时，PI 计划继续当前旨在促进数学领域女性和代表性不足的少数群体的活动。通过为这些群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，并提高不同领域从事数学研究的人的知名度。 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对数学的参与。该计划将扩大数学思维的范围，并为更多种类的学生提供参与数学项目的机会，为数学界和更多具有数学素养的社会成员创造更广泛的基础。该奖项反映了 NSF 的法定使命通过使用基金会的智力优点和更广泛的影响审查标准进行评估，并被认为值得支持。