Motivic Symmetries

在过去的一个世纪里，代数拓扑学发展了许多复杂的方法和工具来研究几何“形状”。Motivic同伦理论提供了一个框架，用于将代数拓扑学中的工具应用于代数簇的研究以及重要的代数几何不变量，如向量丛、二次型、代数圈和有理点。这个项目将研究代数簇的同伦理论的结构和计算方面，重点是几何应用。该项目的更广泛影响包括与被监禁个人的合作，以及与本科生和研究生的合作。在这个项目中，PI将结合等变和更高级的范畴技术来研究动机同伦理论中最近发展的分支。在项目的第一部分，PI建议进一步开发工具来研究归一化的动机谱，重点是定向。在第二部分，PI建议将重点放在等变基元不变量和切片谱序列的计算上。在第三项中，PI建议发展等变框架对应理论，并将其应用于等变运动环空间的研究。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Over the past century, algebraic topology has developed many sophisticated methods and tools to study geometric "shapes". Motivic homotopy theory, provides a framework to apply tools from algebraic topology, to the study of algebraic varieties and to important algebro-geometric invariants such as vector bundles, quadratic forms, algebraic cycles, and rational points. This project will study structural and computational aspects of homotopy theory for algebraic varieties with an emphasis on geometric applications. Broader impacts of this project include work with incarcerated individuals, as well as work with undergraduate and graduate students. In this project, the PI will combine equivariant and higher categorical techniques to study ramifications of recent developments within motivic homotopy theory. In the first part of the project, the PI proposes to further develop tools to study normed motivic spectra, with a focus on orientations. In the second part, the PI proposes to focus on computations of equivariant motivic invariants and slice spectral sequences. In the third the PI proposes to develop a theory of equivariant framed correspondences and apply this to the study of equivariant motivic loop spaces.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.