Euler Products and Homological Densities via Factorization Homology

通过分解同调的欧拉积和同调密度

基本信息

批准号：
1811846
负责人：
Jesse Wolfson
金额：
$ 15.3万
依托单位：
University of California-Irvine
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811846&HistoricalAwards=false
关键词：
Euler Products Homological Densities via

项目摘要

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.

在空间中移动的粒子是拓扑研究的基本对象，可以将其视为汽车如何在道路上移动的常见抽象模型（希望不会碰撞！），或者细胞如何在血液中相互移动。在先前的工作中，PI和他的合作者发现了令人惊讶的模式，这些模式将粒子在太空中移动的粒子与数字因素的收集因素如何为素数。这个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上的解释，希望该链接既阐明了拓扑问题和代数理论的问题和数字理论的问题。PI将进一步研究同源密度，首次在PI的联合工作中引入与Benson Farb和Melanie Wood在一起。同源密度提供了一种新的拓扑不变，这是由Weil的“数字字段/功能字段”字典的自然扩展所表明的，并证明了先前未识别的流形和0个循环空间之间的构型之间的关系（概括配置空间）。 PI计划进行研究的四个关键项：1）构建拓扑对象（即空间或合理同型类型），以便限制同源密度是这些对象的内在不变性； 2）提供了负责PI，Farb和Wood观察到的巧合的拓扑机制，这将解释算术中启发式方法的功效； 3）使用从1）和2）获得的知识来制定Zeta功能的拓扑类似物，上面的密度为特殊值； 4）将上述一致性从0循环的空间扩展到除数的空间。 PI提出，分解同源性应为前三个问题提供统一的方法，并且从这些问题中获得的证据应为第四种方法提供信息。 PI还计划根据Ellenberg，Venkatesh和Westerland所阐明的原则来调查除数空间，以证明其功能领域的Cohen-Lenstra启发式方法。借助Farb，PI表明其原理的强大形式适用于配置空间。他建议将其扩展到分隔线的空间。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响标准来评估值得支持。