Invertibility and deformations in chromatic homotopy theory

可逆性是数学中的一个概念，只要我们有乘法的概念，就可以研究它，尽管它依赖于设置。在自然数的世界里，只有1和-1是可逆的，而所有的非零有理数都有一个逆。在稳定同伦理论中，数字在普通算法中扮演的角色被称为谱的对象所取代，谱作为不受连续变形影响的拓扑空间的代数不变量出现。不可逆光谱并不多，只有不同维度的球体。在这里，我们再次观察到一种现象，当我们传递到某些局部时，情况发生了变化，类似于从整数传递到有理数。利用色同伦理论研究了局部化问题，该理论旨在将复杂的同伦信息分解成更容易理解、行为更规则的构建块。本提案旨在系统研究对称性的作用，以掌握色同伦中奇异的可逆物体，并希望了解这些奇异物体在多大程度上可以被理解为球体的扭曲版本。该倡议更广泛影响的主要重点是在冠状病毒大流行后重建和加强数学界，通过组织讲习班、会议、研讨会、讨论以及数学研究所的研究计划等努力，所有这些努力都特别强调包容性和对服务不足群体的支持。该基金支持的研究将致力于更好地理解色同伦理论中的大规模可逆性现象，即奇异的K(n)-局部皮卡德群，通过尝试使用等变和表示理论方法组织各种特设计算方法进行系统研究。至少有两种不同的途径将被追求：一种是将焦点从子群转移到Morava稳定群的商，这已经在PIs之前的工作（与Barthel， Beaudry, Bobkova, Goerss， Henn和Pham）中微妙地出现在决定球上，以及K(2)-局域奇异皮卡德群在质数2上。另一个想法，在与Dicks的合作中，是对Mazurs的模表示的经典变形理论的全新应用，这个理论很少被探索，但有希望在色同伦中建立新的联系和更深入地理解可逆对象。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。

Invertibility is a concept in mathematics which can be studied any time we have a notion of multiplication, though it is dependent on the setting. Within the world of natural numbers, only 1 and -1 are invertible, yet all non-zero rational numbers have an inverse. In stable homotopy theory, the role that numbers play in ordinary arithmetic is taken by objects called spectra, which arise as algebraic invariants of topological spaces that are unaffected by continuous deformations. There are not very many invertible spectra, only spheres of various dimensions. Here again we observe a phenomenon that the situation changes when we pass to certain localizations analogous to passing from the integers to the rationals. The localizations in question are studied by chromatic homotopy theory, which aims to disassemble complicated homotopical information into building blocks with more understandable with more regular behaviors. This proposal aims to undertake a systematic study of the role of symmetries to get a grasp of the exotic invertible objects in chromatic homotopy, with the hope of understanding the extent to which such exotic objects can be understood as twisted versions of spheres. The main focus of the PIs broader impacts is the rebuilding and strengthening the mathematical community in the post-coronavirus pandemic, through efforts such as organizing workshops, conferences, seminars, discussions, as well as a research program at a mathematical institute, all with a particular emphasis on inclusivity and support for underserved groups.The research supported by this grant will work towards a better understanding of large-scale invertibility phenomena in chromatic homotopy theory, namely the exotic K(n)-local Picard groups, by attempting to organize a variety of ad-hoc computational methods into a systematic investigation using equivariant and representation-theoretic methods. At least two different avenues will be pursued: one is a shift of focus from subgroups to quotients of the Morava stabilizer group, already subtly present in the PIs previous work (with Barthel, Beaudry, Bobkova, Goerss, Henn, and Pham) on determinant spheres, and on the K(2)-local exotic Picard group at the prime 2. The other idea, pursued in a collaboration with Dicks, is a completely new use of Mazurs classical deformation theory of modular representations, which is much less explored but is promising new connections and deeper understanding of invertible objects in chromatic homotopy.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.