Equivariant and Motivic Deformations of Stable Homotopy Theory

这些项目旨在解决代数拓扑学领域的主要问题。拓扑学是对几何学的研究，如果一个几何对象可以变形为另一个几何对象，则可以将一个几何对象与另一个几何对象进行标识。代数拓扑学的目的是将离散的代数不变量赋予这些几何对象，以区分它们的拓扑类型。这样，区分几何对象就简化为代数计算。理解几何对象的拓扑类型是科学/数学研究的基本行为，可以与素数或构成物质和载力的基本粒子的分类相媲美。最近，拓扑计算已被应用于解决固体物理中的问题。此外，涉及大量变量的相互关系的数据自然地描绘出高维空间中的几何对象。利用代数拓扑学对这类数据集的研究是拓扑数据分析这一新的活跃领域的主题。这些项目的重点是经典代数拓扑、等变同伦理论和动机同伦理论的相互作用。等变同伦理论是研究对称的拓扑，而动机同伦理论是研究多项式方程组解的拓扑。近年来，通过引入等变方法和动机方法，在代数拓扑学领域取得了令人眼花缭乱的进展。这些项目的更广泛的影响包括与研究生的合作，暑期数学研究项目，定向阅读项目，以及进入研究生的桥梁项目。具体项目包括研究等变和动机稳定同伦理论中的新结构，并试图利用这些结构为经典稳定同伦理论中一些长期悬而未决的问题提供新的方法。望远镜猜想将使用出现在Kervaire不变一问题的Hill-Hopkins-Ravenel解中的一组光谱来研究。Pstragowski和Gheorghe-Isaksen-Krause-Ricka最近的工作给出了复动机稳定同伦理论的综合构造。主要研究人员计划使用等变同伦理论将其扩展到真实的动机语境。另一个项目是利用等变色同伦理论研究经典色同伦理论与Tate结构的相互作用，目的是改进色分裂猜想。Barthel-Schlank-Stapleton最近的工作给出了一种利用超滤子研究一般素数稳定同伦理论的方法。在这方面的计算将使用Drinfeld模块进行调查。这一裁决反映了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。

These projects aim to address major problems in the field of algebraic topology. Topology is the study of geometry where you identify one geometric object with another if one can be deformed into the other. The goal of algebraic topology is to ascribe discrete algebraic invariants to these geometric objects to distinguish their topological types. In this way, distinguishing geometric objects is reduced to algebraic computations. Understanding the topological type of geometric objects is a fundamental act of scientific/mathematical inquiry, comparable to the study of prime numbers, or the classification of the fundamental particles that constitute matter and carry forces. Topological computations have recently been applied to solve problems in solid state physics. Also, data involving the interrelation of a large number of variables naturally traces out a geometric object in a high dimensional space. The study of such data-sets using algebraic topology is the subject of the new and active field of topological data analysis. The focus of these projects is in the interaction of classical algebraic topology, equivariant homotopy theory, and motivic homotopy theory. Equivariant homotopy theory is the study of the topology of symmetry, whereas motivic homotopy theory is the study of the topology of solutions to systems of polynomial equations. Recent years have witnessed a dazzling array of progress in the field of algebraic topology through the importation of equivariant and motivic methods. Broader impacts of these projects include work with graduate students, a summer math research program, a directed reading program, and a bridge program for entering graduate students.Particular projects involve investigating novel structures in equivariant and motivic stable homotopy theory, and seek to leverage these structures to give new approaches to some long outstanding problems in classical stable homotopy theory. The telescope conjecture will be investigated using a tower of spectra which appeared in the Hill-Hopkins-Ravenel solution of the Kervaire Invariant One Problem. Recent work of Pstragowski and Gheorghe-Isaksen-Krause-Ricka gives a synthetic construction of complex motivic stable homotopy theory. The principal investigator plans to extend this to the real motivic context using equivariant homotopy theory. Another project uses equivariant chromatic homotopy theory to study the interaction of classical chromatic homotopy theory with the Tate construction, with the aim of making progress on the chromatic splitting conjecture. Recent work of Barthel-Schlank-Stapleton gives a means of studying stable homotopy theory at generic primes using ultra-filters. Computations in this context will be investigated using Drinfeld Modules.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.