RUI: Motivic, Operadic, and Combinatorial Homotopy Theory

RUI：动机、操作和组合同伦理论

• 批准号: 2204365
• 负责人: Kyle Ormsby
• 金额: $ 34.5万
• 依托单位: Reed College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204365&HistoricalAwards=false
• 关键词: RUI; Motivic; Operadic; Combinatorial; Homotopy

The scope of contemporary homotopy theory — the mathematics of shape and deformation — has expanded dramatically in recent years, and new tools and perspectives are needed to understand its features and potential. The PIs will explore homotopy theory through the lenses of combinatorics (enumerating structures), operads (parametrizing operations), and algebraic geometry (shapes described by polynomial equations). The PIs will also support the participation of a diverse group of undergraduate students in their research by organizing the Collaborative Mathematics Research Group (CMRG). CMRG participants will produce novel research results and also engage their local community in mathematics outreach efforts.The PIs will investigate operadic, combinatorial, and motivic aspects of homotopy theory with the aim of producing structural, enumerative, and computational results. Their projects include the following:(1) Further explore the combinatorics of model structures on finite categories, especially with an eye towards Catalan combinatorics of arbitrary type (in the sense of cluster algebras).(2) Uncover additional structure in the Balmer spectrum of the stable motivic homotopy category by leveraging a recollement with its étale variant.(3) Extend our computational understanding of stable motivic homotopy theory by computing the slice spectral sequence for the Bachmann-Hopkins connective Hermitian image-of-J spectrum.(4) Construct a rigid model for E_n-spaces as commutative monoids in a certain diagram category.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.

当代同伦理论（形状和变形的数学）的范围近年来急剧扩大，需要新的工具和视角来理解其特征和潜力，PI 将通过组合学（枚举结构）的视角来探索同伦理论。 、运算（参数化运算）和代数几何（由多项式方程描述的形状） PI 还将通过组织不同的本科生群体来支持他们的研究。数学协作研究小组 (CMRG) 的参与者将产生新颖的研究成果，并让当地社区参与数学推广工作。PI 将研究同伦理论的运算、组合和动机方面，旨在产生结构性、枚举性和可预测性。他们的项目包括以下内容：（1）进一步探索有限类别上的模型结构的组合学，特别是着眼于任意类型的加泰罗尼亚组合学（在集群的意义上）。 (2) 通过利用其étale变体的重新整理，揭示稳定动机同伦范畴的巴尔默谱中的附加结构。(3) 通过计算巴赫曼-的切片谱序列，扩展我们对稳定动机同伦理论的计算理解霍普金斯连通埃尔米特 J 谱像。(4) 为 E_n 空间构建一个作为特定图类别中交换幺半群的刚性模型。此奖通过使用基金会的智力价值和更广泛的影响审查标准进行评估，NSF 的法定使命被认为值得支持。

项目成果

Self-duality of the lattice of transfer systems via weak factorization systems

通过弱分解系统的传递系统格的自对偶性

• DOI: 10.4310/hha.2022.v24.n2.a6
• 发表时间: 2022
• 期刊: Homotopy and Applications
• 作者: Franchere, Evan E.;Ormsby, Kyle;Osorno, Angélica M.;Qin, Weihang;Waugh, Riley
• 通讯作者: Waugh, Riley

Model structures on finite total orders

有限总阶数的模型结构

• DOI: 10.1007/s00209-023-03287-6
• 发表时间: 2023
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Balchin, Scott;Ormsby, Kyle;Osorno, Angélica M.;Roitzheim, Constanze
• 通讯作者: Roitzheim, Constanze

Multiplicative equivariant K-theory and the Barratt-Priddy-Quillen theorem

乘法等变 K 理论和 Barratt-Priddy-Quillen 定理

• DOI: 10.1016/j.aim.2023.108865
• 发表时间: 2023
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Guillou, Bertrand J.;May, J. Peter;Merling, Mona;Osorno, Angélica M.
• 通讯作者: Osorno, Angélica M.

Saturated and linear isometric transfer systems for cyclic groups of order pq

pq 阶循环群的饱和线性等距传递系统

• DOI: 10.1016/j.topol.2022.108162
• 发表时间: 2022
• 期刊: Topology and its Applications
• 影响因子: 0.6
• 作者: Hafeez, Usman;Marcus, Peter;Ormsby, Kyle;Osorno, Angélica M.
• 通讯作者: Osorno, Angélica M.