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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.

在理解拓扑空间时，对称性和形变可以看作是两种对立的力量。一方面，对称代表了空间的刚性结构--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。这些特征的例子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在等变代数拓扑中结合在一起。代数拓扑是通过代数不变量来研究复杂且频繁的高维空间的一个数学领域，而等变代数拓扑将空间的对称性以一种稳健的方式结合到不变量中。这个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之间的关系，从而推动了这一领域的最先进知识的发展。特别感兴趣的内容还包括与代数和拓扑K理论有关的问题，这些不变量是现代拓扑学、代数和数论方法的核心。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代表不足的少数群体在数学领域的推广活动，包括她与数学妇女协会新成立的范德比尔特学生分会和范德比尔特定向阅读计划的工作。这些课程将扩大数学思维的范围，让更多的学生有机会成为数学项目的一部分，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突显了等方差的重要性，以及在许多方面对等方差仍然知之甚少。等变同伦理论是代数K-理论中现代计算的关键，在p-进Hodge理论中有着深刻的影响。等变同伦论也是导致非等变同伦论的核心，例如最近对Kervaire不变量一问题的解。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。她的程序关注于不同对称组之间的相互作用，包括在拓扑和代数水平上。在代数水平上，这一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。在拓扑学层面上，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，国际数学联合会计划继续开展目前旨在促进妇女和在数学领域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的新成果的机会，她将支持他们的职业生涯，并进一步提高从事数学工作的各种人的能见度。国际数学中心的拟议活动还将扩大对数学的参与，为具有广泛背景的学生提供参与数学研究经验的机会。该计划将扩大数学思维的范围，让更多的学生有机会成为数学项目的一部分，为数学社区和更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，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。