Invertibility and deformations in chromatic homotopy theory

色同伦理论中的可逆性和变形

• 批准号: 2304797
• 负责人: Vesna Stojanoska
• 金额: $ 33.11万
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2304797&HistoricalAwards=false
• 关键词: Invertibility; deformations; chromatic; homotopy; theory

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.

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，尽管它取决于设置。在自然数的世界中，只有 1 和 -1 是可逆的，但所有非零有理数都有逆元。在稳定同伦理论中，数字在普通算术中所扮演的角色由称为谱的对象承担，谱是作为不受连续变形影响的拓扑空间的代数不变量而出现的。可逆光谱并不多，只有各种维度的球体。在这里，我们再次观察到一个现象，即当我们传递到某些局部化时，情况会发生变化，类似于从整数传递到有理数。所讨论的定位是通过色同伦理论来研究的，该理论旨在将复杂的同伦信息分解为更容易理解、行为更规则的构建块。该提案旨在对对称性的作用进行系统研究，以掌握色同伦中奇异的可逆物体，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。 PI 更广泛影响的主要重点是通过组织研讨会、会议、研讨会、讨论以及数学研究所的研究项目等努力，在后冠状病毒大流行中重建和加强数学界，所有这些都具有特别强调包容性和对服务不足群体的支持。这笔赠款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将各种临时计算方法组织成系统研究。至少有两种不同的途径：一种是将焦点从子群转移到 Morava 稳定群的商，这已经巧妙地出现在 PI 之前关于行列式的工作中（与 Barthel、Beaudry、Bobkova、Goerss、Henn 和 Pham 合作）球体，以及素数 2 处的 K(2) 本地奇异皮卡德群。与 Dicks 合作追求的另一个想法是一种全新的用途马祖尔的模表示经典变形理论，该理论的探索较少，但有望为色同伦中的可逆对象提供新的联系和更深入的理解。该奖项反映了 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。