Spheres are the basic building blocks of geometry, which can fit together to build more complicated geometric objects. Enumerating combinations of spheres is one of the central questions of homotopy theory, and this problem is known as the computation of the homotopy groups of spheres. This project is a direct attack on that central problem. The main objective is to identify structure in the homotopy groups of spheres, and then to apply that structure to carry out large-scale explicit computations. The ultimate scientific goal is to actually carry out the computation at a practical scale, fully utilizing recent progress in computational power and artificial intelligence to achieve a higher level of understanding of this central question. For the broader community, the project capitalizes on the fact that machines are not only powerful for computation, but also powerful for communication. As the most important global culture conversations are occurring online, the rising generations of mathematicians are shifting the Mathematical Conversation in a similar direction. The electronic Computational Homotopy Theory (eCHT) online research community supports that shift and is an integral part of this project, aimed at fostering a community of software engineers within the culture of homotopy theory. It will also sponsor structured activities that encourage collaboration and networking.
The Adams spectral sequence is the primary tool for studying homotopy groups of spheres. This main tool is amenable to machine computation in at least two ways. First, machines can compute the Adams E2-page exhaustively in a range, much further than humans could possibly go without assistance. The project will improve the practical implementation of Adams E2-page computations, both via better use of existing hardware and also via faster algorithms. Second, machines can be used to study Adams differentials and hidden extensions and can indirectly deduce Adams differentials from structure such as the Leibniz rule and naturality. Machines repeat at scale the kinds of deductions with the Adams spectral sequence that were previously studied only manually. Computations with the unstable Adams spectral sequence are woefully underdeveloped. The most recent machine-based efforts are several decades out of date. The project will use modern hardware to revisit and extend computations with the unstable Adams spectral sequence. The project will also study unstable homotopy groups from the perspective of the EHP spectral sequence. The Adams and EHP spectral sequences are highly complementary, in the sense that they see the same structure from two very different perspectives. In combination, the spectral sequences are far more powerful than in isolation.
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.