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.

可以切成三角形的甜点形表面，可能没有一个以上的孔 - 或根本没有孔，如果我们记录切割的完成方式，我们可以通过粘合而将表面放回原处。另外，我们可以创建一个双面表面，在其中为每个原始的三角形制作新三角形，而侧面则与原始三角形的触摸数据相对应。 Poincaré从1895年开始的惊人定理告诉我们，双重形状可与原始的形状变形，而不管削减的方式如何。自从发现以来，这种二元性结果已在许多不同的数学领域进行了改进和概括。在同型理论中，不仅要研究对象的变形，而且还以简化的方式跟踪它们之间的变形以及相干数据。随着其最近扩展到派生的代数几何形状，同义理论已成为许多数学概念（包括二重性）的统一理论。 PI将与她的合作者合作探索两个双重性环境，其中同源和算术信息交织在一起。其中之一涉及建立Poitou和Tate关于二元性的经典结果的同源性扩展，并在数字领域的共同体中进行了二元性，并研究了这种扩展对算术问题的含义。另一个涉及理解所谓的色素同义理论中某些基本对象的二元性，从而根据周期性属性组织了同型的结构和计算信息。这将是称为Morava稳定剂组的色素群的共同体双重性特性的同质性增强，并且基于Gross和Hopkins的第二次工作。这两个项目的目标都可能涉及进一步开发针对涉及对象的小组行动的基础。该奖项反映了NSF的法定任务，并且我们使用基金会的知识分子优点和更广泛的影响审查标准，认为我们被认为是通过评估来诚实的支持。