Euler Products and Homological Densities via Factorization Homology

在空间中移动的粒子是拓扑学的基本研究对象，可以被认为是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分解成质数联系起来。这个由美国国家科学基金会资助的项目旨在对这些联系给出一个概念性的解释，这将有希望阐明拓扑问题、代数问题和数论问题。PI将进一步研究同密度，这是PI与Benson Farb和Melanie Wood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，由Weil的“数域/函数域”字典的自然扩展提出，并展示了流形的构型空间和0环空间（广义构型空间）之间以前未被认识的关系。PI计划开展四个重点研究：1)构建拓扑对象（即空间或有理同伦类型），使这些对象的极限同伦密度是其固有不变量；2)为PI、Farb和Wood观察到的一致性提供了一种拓扑机制，这将解释算法启发式的有效性；3)利用从1)和2)中获得的知识，形成zeta函数的拓扑模拟，其中上述密度为特殊值；(4)将上述巧合从0圈空间推广到除数空间。PI建议因式分解同源性应该为前三个问题提供一个统一的方法，并且从这些问题中获得的证据应该为第四个问题的方法提供信息。PI还计划根据Ellenberg、Venkatesh和Westerland在证明函数场的Cohen—Lenstra启发式中所阐述的原则来研究除数空间。通过Farb， PI证明了其原理的强形式适用于位形空间。他建议将其推广到除数空间。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.