Modern Homotopical Obstruction Theory

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定性地区分它们：代数不变量。这门学科最大的成就之一就是所谓的阻碍理论，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，阻塞理论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应用于各种各样的设置，并在许多数学领域的几个重大发展的核心。该项目的目标是将这些独立分支中学到的经验教训开发成一个现代化的，可重复使用的库，该库足够通用，可以应用于今天发现的障碍理论的主题阵列中，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”：与同行和教师主机共享共同的写作时间，以提供博士学位。学生和早期职业研究人员有效的数学写作工具和可持续的写作实践。PI还致力于在博士内外建立支持性文化。计划和组织数学科学中心理健康问题的讨论小组。2该项目建立在乘法障碍理论的最新发展之上，以解决该学科中的新老问题。通过将这些技术应用于相邻领域，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：研究乘法环谱和环空间;交换性等变同伦理论; monoidality在五月谱序列;"驯服"版本的交换代数在K（n）-本地同伦理论;和建设同伦理论拓扑量子场论的应用结不变。将这些单独的问题联系在一起将有助于发展一个现代和灵活的阻碍理论框架，该框架将其几个分支的发展和经验教训结合起来。特别是，PI旨在将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相结合，并由Goerss-Hopkins进一步发展。该奖项反映了NSF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而被认为值得支持。

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.