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，我们可以在任何时候研究乘法的概念，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数。在稳定同伦理论中，数字在普通算术中的作用被称为谱的对象所取代，谱是拓扑空间的代数不变量，不受连续变形的影响。不存在太多的可逆谱，只有不同维度的球面。这里我们再次观察到一个现象，当我们传递到某些局部化时，情况发生了变化，类似于从整数传递到有理数。我们用色同伦理论研究了这些局部化问题，其目的是将复杂的同伦信息分解成更容易理解、行为更规则的积木块。这个提议的目的是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，并希望了解这种奇异对象在多大程度上可以被理解为扭曲的球体。PI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，通过组织研讨会，会议，研讨会，讨论以及数学研究所的研究计划等努力，所有这些都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了解大型色同伦理论中的标度可逆现象，即奇异的K（n）-局部Picard群，通过尝试使用等变和表示论方法将各种特别计算方法组织成系统的研究。至少有两个不同的途径将被追求：一个是从子群的焦点转移到Morava稳定子群，已经巧妙地出现在PI以前的工作（与Barthel，Beaudry，Bobkova，Goerss，Henn和Pham）在行列式领域，以及在素数2的K（2）-局部奇异Picard群。另一个想法，追求在与迪克斯合作，是一个全新的使用马祖尔经典变形理论的模块化表示，这是少得多的探索，但有前途的新的连接和更深入的了解可逆对象的色同伦。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。