Modern Homotopical Obstruction Theory

现代同伦阻碍理论

• 批准号: 2208062
• 负责人: Tyler Lawson
• 金额: $ 35万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2208062&HistoricalAwards=false
• 关键词: Modern; Homotopical; Obstruction; Theory

The mathematical field of algebraic topology begins by taking complicated geometric shapes and qualitatively distinguishing them using simpler data: algebraic invariants. One of the great successes of the subject is called obstruction theory, and it has largely made this procedure reversible. Starting with algebraic invariants associated to geometric shapes, obstruction theory gives information regarding objects that could or would have these invariants. Since its development, the basic techniques of obstruction theory have been applied in a wide variety of settings and been at the heart of several major developments in many areas of mathematics. The goal of this project is to develop the lessons learned in these separate branches into a modern, reusable library that is general enough to apply within the array of subjects where obstruction theory is found today, and more. Additionally, the PI is planning a series of workshops on mathematical writing, as well as regular "write-ins": shared, common writing time with peers and faculty hosts to provide Ph.D. students and early-career researchers tools for effective mathematical writing and sustainable practices for writing productively. The PI is also committed to establishing a supportive culture inside and outside the Ph.D. program and organizing discussion groups on mental health issues in the mathematical sciences.This project builds upon recent developments in multiplicative obstruction theory to address new and old questions in the subject. By applying these techniques to adjacent fields, such as algebraic geometry and geometric topology, the PI aims to provide new solutions to interesting problems in those areas and also develop interface points for outside researchers hoping to make use of techniques and tools in higher algebra. Specific goals include: the study of multiplication for ring spectra and ring spaces; commutativity in equivariant homotopy theory; monoidality in the May spectral sequence; "tame" versions of the commutative algebras in K(n)-local homotopy theory; and construction of homotopy-theoretic topological quantum field theories for application to knot invariants. Tying these individual problems together would facilitate the development of a modern and flexible framework for obstruction theory that unites the developments and lessons from several of its branches. In particular, the PI aims to combine the synthetic approach of Hopkins-Lurie and Pstragowski with more classical resolution-theoretic calculational techniques due to Robinson and developed further by Goerss-Hopkins.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.

代数拓扑学的数学领域开始于采用复杂的几何形状，并使用更简单的数据（代数不变量）定性地区分它们。这门学科的一个伟大成就被称为阻碍理论，它在很大程度上使这一过程可逆。从与几何形状相关的代数不变量开始，阻碍理论给出了关于物体可能或将具有这些不变量的信息。自其发展以来，阻碍理论的基本技术已被广泛应用于各种各样的设置，并在许多数学领域的几个主要发展的核心。这个项目的目标是将这些独立分支的经验教训发展成一个现代的、可重用的库，这个库足够通用，可以应用于今天发现阻碍理论的一系列主题，甚至更多。此外，PI正在计划一系列关于数学写作的研讨会，以及定期的“写作”：与同行和教师主持共享，共同的写作时间，为博士生和早期职业研究人员提供有效的数学写作工具和有效写作的可持续实践。PI还致力于在博士项目内外建立一种支持性文化，并组织关于数学科学心理健康问题的讨论小组。这个项目建立在乘法阻碍理论的最新发展上，以解决这个主题中的新问题和老问题。通过将这些技术应用于相邻的领域，如代数几何和几何拓扑，PI旨在为这些领域的有趣问题提供新的解决方案，并为希望利用高等代数技术和工具的外部研究人员开发接口点。具体目标包括：研究环谱和环空间的乘法；等变同伦理论中的交换性五月谱序列的单单调性；K(n)-局部同伦理论中交换代数的“驯服”版本并构造了应用于结不变量的同伦拓扑量子场论。将这些单独的问题联系在一起，将有助于形成一个现代而灵活的障碍理论框架，将其几个分支的发展和经验教训结合起来。特别是，PI的目标是将Hopkins-Lurie和Pstragowski的综合方法与Robinson提出的更经典的分辨率理论计算技术结合起来，并由goers - hopkins进一步发展。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。