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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.

在空间中运动的粒子是拓扑学中的一个基本研究对象，可以被认为是汽车如何在道路上移动的一个常见的抽象模型（希望没有碰撞！），或者细胞如何在血液中相互移动。在之前的工作中，PI和他的合作者发现了令人惊讶的模式，将空间中移动的粒子与数字集合如何分解成素数联系起来。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性的解释，这将有望阐明拓扑学问题以及代数和数论问题。PI将进一步研究同调密度，首次介绍了PI与Benson Farb和Melanie Wood的联合工作。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”字典的自然扩展，并证明了流形的配置空间与0-圈空间（广义配置空间）之间以前未被认识的关系。 PI计划进行四个关键的研究：1）构建拓扑对象（即空间或有理同伦类型），使得极限同调密度是这些对象的内在不变量; 2）提供一种拓扑机制，负责PI，Farb和Wood观察到的重合，这将解释算术的功效; 3）利用从1）和2）中得到的知识，构造了zeta函数的一个拓扑类似物，其中上述密度是特殊值; 4）将上述重合从0-圈空间推广到因子空间。PI提出，因子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还计划调查空间的除数以下的原则阐述了艾伦伯格，文卡特什，和韦斯特兰在他们的证明科恩-伦斯特拉questiistics的功能领域。通过Farb，PI已经证明了他们的原理的强形式对位形空间成立。该奖项反映了NSF的法定使命，并被认为值得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来支持。