Generalized Steenrod operations and equivariant geometry

广义 Steenrod 运算和等变几何

• 批准号: 2305016
• 负责人: Prasit Bhattacharya
• 金额: $ 31.9万
• 依托单位: New Mexico State University
• 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: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2305016&HistoricalAwards=false
• 关键词: Generalized; Steenrod; operations; equivariant; geometry

Topology is a subject that studies shapes. This subject was revolutionized in 1947 by Norman Steenrod when he introduced the Steenrod operations. These operations led to some of the most enigmatic results in geometry which popularized homotopy theory, a subbranch of topology, in the fifties and the sixties. This project will lead to advances in equivariant homotopy theory, an enhancement of homotopy theory which is sensitive to the symmetries of shapes. The resulting equivariant analogs of Steenrod operations will lead to new applications in equivariant geometry. The award will also be used to stimulate the research culture and support graduate students at New Mexico State University. The PI will establish a new method that generalizes the construction of classical Steenrod operations to equivariant homotopy theory and constructs equivariant Steenrod operations for all finite groups. Equivariant Steenrod operations will immediately lead to equivariant analogs of Stiefel-Whitney classes which can find many applications in the study of equivariant vector bundles and equivariant smooth manifolds. Atiyah Real vector bundles, which are important examples of equivariant vector bundles, will be studied. In particular, the James periodicity number of tautological Atiyah Real vector bundles will be determined using an equivariant analog of the Adams conjecture formulated using Atiyah Real K-theory. The PI will also continue to study periodic self-maps of chromatic homotopy theory and extend them to equivariant as well as motivic settings.This project is jointly funded by the Topology program and the Established Program to Stimulate Competitive Research (EPSCoR).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.

拓扑学是一门研究形状的学科。 1947 年，Norman Steenrod 引入了 Steenrod 操作，彻底改变了这一主题。这些运算导致了几何学中一些最神秘的结果，从而在五十年代和六十年代普及了同伦理论（拓扑学的一个分支）。该项目将推动等变同伦理论的进步，这是对形状对称性敏感的同伦理论的增强。 由此产生的 Steenrod 运算的等变类似物将导致等变几何中的新应用。 该奖项还将用于刺激新墨西哥州立大学的研究文化并支持研究生。 PI将建立一种新方法，将经典Steenrod运算的构造推广到等变同伦理论，并为所有有限群构造等变Steenrod运算。等变 Steenrod 运算将立即导致 Stiefel-Whitney 类的等变类似物，它可以在等变向量丛和等变光滑流形的研究中找到许多应用。将研究 Atiyah Real 向量丛，它是等变向量丛的重要例子。特别地，同义反复 Atiyah Real 向量束的 James 周期数将使用使用 Atiyah Real K 理论制定的 Adams 猜想的等变模拟来确定。 PI 还将继续研究色同伦理论的周期性自映射，并将其扩展到等变和动机设置。该项目由拓扑计划和刺激竞争研究既定计划 (EPSCoR) 联合资助。该奖项反映了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。