Invertibility and deformations in chromatic homotopy theory

可逆性是数学中的一个概念，它可以在任何时候我们有乘法的概念时学习，尽管它取决于背景。在自然数的世界里，只有1和-1是可逆的，然而所有非零有理数都有一个逆数。在稳定同伦理论中，数字在普通算术中所起的作用是由称为谱的对象来承担的，这些对象是不受连续变形影响的拓扑空间的代数不变量。没有太多的可逆光谱，只有不同维度的球体。在这里，我们再次观察到一种现象，当我们转到某些局部化时，情况发生了变化，类似于从整数到有理数的转变。利用色同伦理论对局部化进行了研究，目的是将复杂的同伦信息分解成行为更规则、更易于理解的积木。这一建议旨在对对称性的作用进行系统的研究，以掌握色同伦中的奇异可逆对象，以期了解这种奇异对象在多大程度上可以被理解为球体的扭曲版本。PIS更广泛的影响的主要焦点是重建和加强后冠状病毒大流行中的数学界，通过组织研讨会、研讨会、讨论和数学研究所的研究计划，所有这些都特别强调包容性和对服务不足的群体的支持。由这笔赠款支持的研究将努力更好地理解色同伦理论中的大规模可逆现象，即奇异的K(N)局部Picard群，试图组织各种特别的计算方法，使用等变和表示理论方法进行系统调查。至少将寻求两种不同的方法：一种是将焦点从子群转移到Morava稳定子群的商，这在PI以前关于行列式球面的工作(与Barthel，Beaudry，Bobkova，Goerss，Henn和Pham)中已经巧妙地存在，以及关于K(2)-局部奇异Picard群在素数2上。另一种想法，与Dicks合作，是Mazur经典的模表示形变理论的全新使用，这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.