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Chromatic and Arithmetic Duality

色彩和算术二元性

基本信息

批准号：
1812122
负责人：
Vesna Stojanoska
金额：
$ 23.44万
依托单位：
University of Illinois at Urbana-Champaign
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：
2018-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812122&HistoricalAwards=false
关键词：
Chromatic Arithmetic Duality

项目摘要

A doughnut-shaped surface, perhaps with more than one hole - or none at all, can be cut into triangles, and if we record how the cuts were done, we can put the surface back together by gluing. Alternatively, we can create a dual surface, where we make new triangles with a vertex for each of the original ones, and sides corresponding to the touching data of the original triangles. Poincaré's amazing theorem from 1895 tells us that the dual shape is deformable to the original, regardless of the way cuts were done. Since its discovery, this duality result has been improved on and generalized in many different areas of mathematics. In homotopy theory one not only studies objects up to deformations, but also keeps track of deformations between them as well as coherence data, all in a streamlined way. Along with its recent augmentation into derived algebraic geometry, homotopy theory has become a unifying ground for numerous mathematical concepts, including duality. The PI will work with her collaborators to explore two duality contexts in which homotopical and arithmetic information are intertwined. One of those involves establishing a homotopical extension of a classical result of Poitou and Tate about duality in the cohomology of number fields, as well as investigating the implications of such an extension to questions in arithmetic. The other involves understanding duality for some of the basic objects in so-called chromatic homotopy theory, whereby one organizes structural and computational information in homotopy according to periodicity properties. This would be a homotopical enhancement of a cohomological duality property of the chromatic Galois groups known as Morava stabilizer groups, and is based on seminal work of Gross and Hopkins. Both of these project goals may involve further developing the foundations for profinite group actions on profinite objects.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.

一个甜甜圈形状的表面，也许有一个以上的孔-或者根本没有，可以被切割成三角形，如果我们记录下切割是如何完成的，我们可以通过粘合将表面重新组合在一起。或者，我们可以创建一个对偶曲面，在这里我们创建一个新的三角形，每个三角形的顶点都对应于原始三角形的接触数据。庞加莱在1895年提出的惊人定理告诉我们，无论切割方式如何，对偶形状都可以变形为原始形状。自发现以来，这一对偶结果在数学的许多不同领域得到了改进和推广。在同伦理论中，人们不仅研究物体的变形，而且还跟踪它们之间的变形以及相干数据，所有这些都是以流线型的方式进行的。沿着它最近扩展到派生代数几何，同伦理论已成为许多数学概念的统一基础，包括对偶性。 PI将与她的合作者一起探索同伦和算术信息交织在一起的两个二元背景。其中之一涉及建立一个同伦扩展的经典结果普瓦图和泰特对偶的上同调数域，以及调查的影响，这样的扩展问题的算术。另一个涉及到在所谓的色同伦理论中理解一些基本对象的对偶性，其中一个根据周期性特性在同伦中组织结构和计算信息。这将是对色伽罗瓦群（称为摩拉瓦稳定群）的上同调对偶性质的同伦增强，并且基于格罗斯和霍普金斯的开创性工作。这两个项目的目标可能涉及进一步发展profinite集团对profinite对象的行动的基础。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。