Euler Products and Homological Densities via Factorization Homology

University Of California-Irvine, Irvine CA

Investigators

Abstract

Particles moving around in a space are a basic object of study in topology, and can be thought of as a common abstract model for how cars move on roads (hopefully without colliding!), or how cells move around each other in the bloodstream. In prior work, the PI and his collaborators discovered surprising patterns linking particles moving in space with how collections of numbers factor into primes. This National Science Foundation funded project aims to give a conceptual explanation for these links, which will hopefully shed light both on problems in topology and problems in algebra and number theory. The PI will further study homological densities, first introduced in joint work of the PI with Benson Farb and Melanie Wood. Homological densities provide a new topological invariant, suggested by a natural extension of Weil's "number field/function field" dictionary, and demonstrating previously unrecognized relationships between configuration spaces of manifolds and spaces of 0-cycles (generalizing configuration spaces). The PI plans to carry out four key prongs of research: 1) construct topological objects (i.e. spaces or rational homotopy types) such that the limiting homological densities are intrinsic invariants of these objects; 2) provide a topological mechanism responsible for the coincidences observed by the PI, Farb and Wood, which would explain the efficacy of the heuristics from arithmetic; 3) use the knowledge gained from 1) and 2) to formulate a topological analogue of a zeta function, for which the densities above are special values; and 4) extend the above coincidences from spaces of 0-cycles to spaces of divisors. The PI proposes that factorization homology should provide a unified approach to the first three problems, and that the evidence gained from these should inform the approach to the fourth. The PI also plans to investigate spaces of divisors following the principle articulated by Ellenberg, Venkatesh, and Westerland in their proof of the Cohen--Lenstra heuristics for function fields. With Farb, the PI has shown that a strong form of their principle holds for configuration spaces. He proposes to extend this to spaces of divisors. 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.