Modern Homotopical Obstruction Theory

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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.