Collaborative Research: Factorization Homology, Deformation Theory, and Duality

Project: Research project

Project Details

Description

Overview
This project involves the development of the theory of factorization homology (FH) for variants of (∞, n)-categories, and applications thereof to mathematical physics and differential topology. First, the PIs propose to use FH with adjoints to complete their proof of the cobordism hypothesis of Baez–Dolan and Lurie, which asserts that topological quantum field theories (TQFTs) in the sense of Atiyah’s axioms are uniquely determined by their value on a point. They propose to do this using a purely combinatorial description of GL(Rn), amplifying the Bruhat decomposition, crafted to be commensurate with higher adjunction data. Second, the PIs propose to use linear FH to conceptually establish the state-operator correspondence, as well as instances of the holographic principle, of quantum physics. They propose to establish these results through Poincare/Koszul duality.
Intellectual Merit
The cobordism hypothesis is the structural crux in the current mathematical view of TQFT. It informs the work of every mathematician in this field, but a complete proof has not, to this point, been written. The PIs propose to complete their proof of the cobordism hypothesis. Their approach is conceptual, and based on the theory of stratifications and Bruhat cells, rather than on framed Morse theory and handlebody presentations which is the basis for locality since Atiyah’s axioms. This new basis for locality in TQFT allows for new manifold invariants which do not conform to the cobordism hypothesis.
The holographic principle is an expected duality in QFT that exchanges bulk and boundary theories, and stands as one of the central topics of study in contemporary particle physics. A full understanding of the holographic principle requires nonperturbative mathematical models, to advance past the BV formalism. The PIs’ theory of FH and the moduli of stratifications, is designed for this purpose. They propose to realize instances of the holographic principle and other dualities as a form of Poincar´e/Koszul duality within FH; this uses a proposed characterization of parametrized deformation theory in terms of higher categories, together with a homotopical Verdier duality on the moduli space of stratifications.
Broader Impacts
The PIs have a number of graduate students, and the scope and design of their agenda is well-suited toward incorporating young researchers and graduate students into the work. They have previously collaborated with and mentored graduate students. They have organized conferences and workshops tailored for graduate students.
StatusActive
Effective start/end date6/1/185/31/21

Funding

  • National Science Foundation (DMS-1812057-002)

Fingerprint Explore the research topics touched on by this project. These labels are generated based on the underlying awards/grants. Together they form a unique fingerprint.