Euler Products and Homological Densities via Factorization Homology

粒子在空间中移动是拓扑学中的一个基本研究对象，可以被视为汽车如何在道路上移动(希望不会发生碰撞！)或细胞如何在血液中相互移动的常见抽象模型。在之前的工作中，PI和他的合作者发现了令人惊讶的模式，将粒子在空间中的运动与数字集合如何构成素数联系在一起。这个由国家科学基金会资助的项目旨在为这些联系提供一个概念性的解释，这将有望阐明拓扑学中的问题以及代数和数论中的问题。PI将进一步研究同调密度，这是在PI与本森·法布和梅勒妮·伍德的联合工作中首次引入的。同调密度提供了一个新的拓扑不变量，这是Weil的数域/函数域词典的自然推广，并证明了流形的配置空间和0-圈空间(推广的配置空间)之间以前不被认识的关系。PI计划进行四个关键方面的研究：1)构造拓扑对象(即空间或有理同伦类型)，使得极限同伦密度是这些对象的内在不变量；2)提供一个负责PI、Farb和Wood观察到的重合的拓扑机制，这将从算术上解释启发式的有效性；3)使用从1)和2)获得的知识来构造Zeta函数的拓扑模拟，对于该拓扑模拟，上述密度是特定值；以及4)将上述重合从0圈空间扩展到粘性空间。PI建议因式分解同源应该为前三个问题提供一个统一的方法，而从这些问题中获得的证据应该为第四个问题的方法提供依据。PI还计划按照Ellenberg、Venkatesh和Westland在他们对函数域的Cohen-Lenstra启发式的证明中阐述的原则来调查除数空间。对于Farb，PI已经证明了他们的原理的强形式对于配置空间是成立的。他建议将这一点扩展到除数空间。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.