Chromatic and Arithmetic Duality

一个甜甜圈形状的表面，也许有多个洞，或者根本没有，可以被切成三角形，如果我们记录下切割是如何完成的，我们可以用胶水把表面粘在一起。或者，我们可以创建一个对偶曲面，在那里我们为每个原始三角形创建一个顶点的新三角形，并与原始三角形的触摸数据对应边。1895年庞加莱的神奇定理告诉我们，无论切割的方式如何，对偶形状都可以变形为原始形状。自发现以来，这个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。随着其最近扩展到派生代数几何，同伦理论已经成为许多数学概念的统一基础，包括对偶。PI将与她的合作者一起探索两种对偶性背景，其中同调和算术信息交织在一起。其中之一是建立Poitou和Tate关于数域上同调对偶的经典结果的同调推广，并研究这种推广对算术问题的影响。另一个涉及到理解所谓的色同伦理论中一些基本对象的对偶性，根据周期性组织同伦中的结构和计算信息。这将是对被称为Morava稳定群的色伽罗瓦群的上同调对偶性的同位增强，并且是基于Gross和Hopkins的开创性工作。这两个项目目标都可能涉及到进一步开发在无限对象上的无限群体行动的基础。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。

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.