Modern Homotopical Obstruction Theory

代数拓扑学的数学领域开始于获取复杂的几何形状，并使用更简单的数据来定性地区分它们：代数不变量。这门学科的最大成功之一被称为障碍理论，它在很大程度上使这一过程可逆。障碍理论从与几何形状相关的代数不变量开始，给出了关于可能或将具有这些不变量的对象的信息。自其发展以来，障碍理论的基本技术已被广泛应用于各种环境中，并成为许多数学领域中几个主要发展的核心。这个项目的目标是将在这些独立分支中学到的经验发展成一个现代的、可重复使用的库，它足够通用，可以应用于当今发现障碍理论的一系列学科中，等等。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”：与同行和教师东道主共享共同的写作时间，为博士生和职业生涯早期研究人员提供有效的数学写作工具和高效写作的可持续实践。该项目还致力于在博士项目内外建立一种支持性的文化，并组织关于数学科学中心理健康问题的讨论小组。这个项目建立在乘法障碍理论的最新发展基础上，以解决该学科中的新老问题。通过将这些技术应用于邻近的领域，如代数几何和几何拓扑，PI旨在为这些领域中有趣的问题提供新的解决方案，并为希望利用高等代数中的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；May谱序列中的单纯性；K(N)-局部同伦理论中交换代数的“驯服”形式；以及应用于纽结不变量的同伦论拓扑量子场论的构造。将这些单独的问题联系在一起将有助于发展一个现代的、灵活的障碍理论框架，将其几个分支的发展和教训统一起来。特别是，PI旨在将Hopkins-Lurie和Pstragowski的综合方法与由Robinson提出并由Goerss-Hopkins进一步发展的更经典的分辨率理论计算技术相结合。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。

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.