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，对称性和变形可能被视为两个相反力量。一方面，对称代表了在旋转或反射（例如旋转或反射）下空间相同的太空通道的刚性结构。另一方面，变形是故意使用的刚性结构，以使我们能够理解空间的粗糙特征。这些功能的示例包括孔的数量或空间的单独片段的数量。这两种理解空间的方法结合了模棱两可的代数拓扑。代数拓扑是数学领域，通过代数不变式研究复杂且经常高维空间，而均值代数拓扑结合了空间的对称性，以强大的方式将其纳入不变性。该领域与数学物理和数据分析等主题有联系。 PI的工作通过开发计算和理论工具来分析doivariant代数拓扑中不变性的结构和关系，从而促进了该领域的最新知识。感兴趣的特定背景还包括与代数和拓扑K理论有关的问题，不变性是现代拓扑，代数和数字理论的核心。此外，这笔赠款的资金将帮助PI进行她的外展活动，旨在促进数学领域的妇女和代表性不足的少数民族，包括她与新成立的数学妇女范德比尔特学生分会的合作以及范德比尔特的指导阅读计划。这些计划将扩大数学思维的范围，并使多种学生有机会成为数学项目的一部分，为数学社区创造更广泛的基础，并在数学上更素养社会。同型理论的最新发展强调了均衡性的重要性，以及肩variance的许多方式仍然鲜为人知。模棱两可的同质理论是代数K理论中现代计算的关键，并且在P-Adic Hodge理论中具有深刻的影响。模棱两可的同义理论也是非公平同型理论的结果，例如最近对kervaire不变的一个问题的解决方案。 PI的研究计划将开发新的工具，以扩展这些类型的计算，并加深我们对对称性在同质考虑方面表现出令人惊讶的方式的整体情况。她的计划着重于拓扑和代数级别的不同对称组的相互作用。在代数层面，这些领域的这些新发展将使数学家能够完全利用代数工具，以通过小组行动理解拓扑空间。在拓扑层面上，她的工作将为双重性，色度同义理论和代数K理论提供基础。在进行这项研究的同时，PI计划继续进行当前的活动，以促进妇女和代表性不足的数学活动。通过为这些团体的数学家提供机会传播新成果，她将支持他们的职业，并增加从事数学的人的各种各样的人的知名度。 PI的拟议活动还将通过为具有广泛背景的学生提供数学研究经验的一部分来扩大对数学的参与。该计划将扩大数学思维的影响力，并使多种学生有机会成为数学项目的一部分，为数学社区创造更广泛的基础，并在数学上更广泛地识字社会成员。该奖项反映了NSF的法定使命，并通过使用该基金会的知识优点和广泛的影响来评估NSF的法定任务，并被认为是值得的支持。