Chromatic Phenomena with an Equivariant Perspective

该奖项支持的项目是理解球面同伦群的大规模努力的一部分，球面同伦群是拓扑学中的基本问题之一。球体是最简单的几何对象之一，是更复杂实体的构建块。球面的同伦群是考虑到变形的球面之间的连续函数的集合。它们被用来研究这些基本组成部分之间的联系。这些小组和用于研究它们的工具与其他领域有着深厚的联系。例如，当有趣的函数和序列作为模式和结构出现在球面的同伦群中时，就与数论有了联系。与微分几何有联系，因为这些组用于几何对象的分类。最近，用于研究球面同伦群的工具也被用于物理中物质相的分类。虽然球面同伦群的研究是出了名的困难，但在代数几何和代数拓扑之间架起了一座桥梁，它允许我们使用复杂的代数理论来计算球面的同伦群。这座桥被称为色同伦理论，是这个项目的中心主题。对偶性和可逆性是数学中普遍存在的概念，它们是理解数学对象之间关系的核心。该奖项支持的项目在色同伦理论中研究这些现象。该项目还提出了理论和计算，将提供数据来帮助研究这些基本现象。色同伦理论的核心是研究关于mod p K-理论的局部谱的同伦范畴的一个更高的类似：K(N)-局部稳定同伦范畴。它等价于G-等变谱中K(N)-局部E-模的同伦范畴，其中Morava E-理论E是p-完备K理论的高阶模拟，稳定子群G是Adams运算群的推广。该建议的策略是将问题限制在G的有限子群F上，并利用有限归结理论将从有限子群获得的信息传递到关于K(N)-局部范畴的信息。该提案中的一个重要思想圈涉及二元性和可逆性。PI及其合作者利用等变同伦理论中的技巧研究了K(N)-局部西班牙-怀特黑德对偶。他们利用J-同态的模拟来计算G的某些有限子群F的作用在E上的同伦不动点谱的对偶。这种同态也被用来研究K(N)-局部E-模F-谱范畴的Picard群。PI是p=2的色度高度为2的计算方面的专家，他提出了一些项目来加深我们对这一困难情况的理解。其中包括K(2)-局部范畴的Picard群的计算、K(2)-局部Brown-Comenetz对偶的研究和有限谱K(2)-局部同伦群的计算。高度为2的色同伦理论的进展加深了我们对K(N)局域现象的理解。PI提出了各种项目，以在更高的高度概括这一见解。最后，该项目还研究了色同伦理论中最重要的问题之一，色分裂猜想，它在色同伦理论和形式群的代数几何之间的关系中起着核心作用。这个奖项反映了NSF的法定使命，并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，被认为值得支持。

The project supported by this award is part of a large scale effort to understand the homotopy groups of spheres, one of the fundamental problems in topology. The spheres, which are among the simplest geometric objects, are building blocks for more complex entities. The homotopy groups of spheres are collections of continuous functions between spheres considered up to deformations. They are used to study the connections between these basic components. These groups and the tools used to study them have deep connections with other fields. For example, there are connections with number theory as interesting functions and sequences appear as patterns and structure in the homotopy groups of spheres. There are connections with differential geometry, as these groups are used in the classification of geometric objects. Recently, the tools used to study the homotopy groups groups of spheres have also been used in physics in the classification of phases of matter. Although the study of the homotopy groups of spheres is notoriously difficult, a bridge has been built between algebraic geometry and algebraic topology that allows us to use sophisticated algebraic theory to enable calculations of the homotopy groups of spheres. This bridge is known as chromatic homotopy theory and is the central theme of this project. Duality and invertibility are ubiquitous concepts in mathematics which are central to understanding relationships between mathematical objects. The project supported by this award studies these phenomena within chromatic homotopy theory. The project also proposes theory and computations that will provide data to help study these fundamental phenomena.At the heart of chromatic homotopy theory is the study of a higher analogue of the homotopy category of spectra local with respect to mod p K-theory: The K(n)-local stable homotopy category. It is equivalent to the homotopy category of K(n)-local E-modules in G-equivariant spectra, where Morava E-theory E is a higher analogue of p-complete K theory and the stabilizer group G is a generalization of the group of Adams operations. The strategy of the proposal is to restrict problems to finite subgroups F of G and use the theory of finite resolutions to pass from information obtained from finite subgroups to information about the K(n)-local category. An important circle of ideas in the proposal concerns duality and invertibility. The PI and collaborators study K(n)-local Spanier-Whitehead duality using techniques from equivariant homotopy theory. They use an analogue of the J-homomorphism to compute the duals of homotopy fixed point spectra for the action on E of some finite subgroups F of G. This homomorphism is also used to study Picard groups of categories of K(n)-local E-module F-spectra. The PI is an expert in computations at chromatic height 2 for p = 2 and proposes projects to further our understanding in this difficult case. These include the computation of the Picard group of the K(2)-local category, the study of K(2)-local Brown-Comenetz duality and computations of K(2)-local homotopy groups for finite spectra. Advancements in chromatic homotopy theory at height 2 have informed our understanding of K(n)-local phenomena. The PI proposes various projects to generalize this insight at higher heights. Finally, the project also examines one of the most important problems in chromatic homotopy theory, the chromatic splitting conjecture, which plays a central role in the relationship between chromatic homotopy theory and the algebraic geometry of formal groups.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.