Higher Representation Theory and Heegaard Floer Homology

Modern research at the interface of mathematics and physics has revealed startling connections between geometric properties of shapes, theories in high-energy physics, deep concepts of symmetry, and rapidly developing areas like materials science and quantum computation. A set of constructions known as "Heegaard Floer homology" brings together many of these perspectives and forms a key nexus in this web of ideas.

As a result, there is much interest in Heegaard Floer homology as a guiding light for understanding this web more broadly and pushing out toward the applications of most mathematical and practical significance. This project aims to exploit the locality inherent in Heegaard Floer homology to understand a long-sought-after algebraic operation based on principles of higher symmetry, and to use this new structure to advance adjacent fields of research.

This project will support the professional development of students and other early-career mathematicians through research collaborations, mentoring activities, and outreach, including organization of conferences and seminars.

In more detail, this project builds on recent work which uncovered a relationship between the lower-dimensional or more-local aspects of Heegaard Floer homology and tensor products for higher representations of categorified quantum groups. Such tensor products have long been recognized as a crucial element needed for further progress in this area, but only in certain cases have they been defined.

This project will expand our knowledge of the structure of Heegaard Floer homology using higher tensor products as an organizing principle, while also using insights from the extensive Heegaard Floer literature to understand more about higher tensor products in general and to develop new links with areas ranging from geometric representation theory to amplituhedra in physics.

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.