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，可以随时可以研究我们有乘法的概念，尽管它取决于设置。在自然数量的世界中，只有1和-1是可逆的，但是所有非零合理数字都是逆的。在稳定的同型理论中，数字在普通算术中扮演的作用是由称为Spectra的对象扮演的，这是拓扑空间的代数不变性的，而拓扑空间不受连续变形的影响。没有太多可逆光谱，只有各个维度的球体。在这里，我们再次观察到一种现象，即当我们传递到类似于从整数到理由的某些本地化时，情况会发生变化。色彩同义理论研究了所讨论的本地化，该理论旨在将复杂的同位信息拆分为具有更常规行为的更易于理解的构件。该提案旨在对对称性的作用进行系统的研究，以掌握色调可逆物体在色素同拷贝中的作用，以期理解可以将这种外来物体理解为球体扭曲版本的程度。 PIS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，讨论，以及在数学研究所的研究计划，并特别强调批准的集团的批准能力，将其倾向于授予，这是一项更好的理解，这在数学机构上都在数学研究所中进行了更大的理解，从而更好地理解了这一点，因此，通过组织工作室，会议，研讨会，讨论，以及对授权的努力，朝着授予的努力方面的努力，朝着授予的努力方面的努力，将授予授予的努力，这是授予的努力。色素同义理论，即外来的K（n） - 局部PICARD组，试图将各种临时计算方法组织到使用Equivariant和表示理论方法的系统研究中。至少将采用两种不同的途径：一个是从亚组转变为摩拉瓦稳定剂组的商品，在PIS先前的工作中已经巧妙地存在（与Barthel，Beaudry，Bobkova，Bobkova，Goerss，Henn和Pham）在确定的Spher中，以及K（2）-local exotic exototial exototial exototial 2。 Mazurs经典形式表示的经典变形理论的全新使用量要少得多，该理论的探索程度要少得多，但有希望的新联系和对色素同型中可逆物体的更深入的理解。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力和更广泛影响的评估来通过评估来支持的，这是值得的。