Equivariant and Motivic Deformations of Stable Homotopy Theory

这些项目旨在解决代数拓扑领域的主要问题。拓扑学是对几何学的研究，如果一个物体可以变形成另一个物体，你就可以把它和另一个物体区分开来。代数拓扑学的目标是赋予这些几何对象离散代数不变量，以区分它们的拓扑类型。通过这种方法，将几何对象的识别简化为代数计算。理解几何物体的拓扑类型是科学/数学探究的基本行为，类似于质数的研究，或构成物质和携带力的基本粒子的分类。拓扑计算最近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会在高维空间中描绘出几何对象。利用代数拓扑对这些数据集进行研究是拓扑数据分析这一新兴而活跃的领域的主题。这些项目的重点是在经典代数拓扑，等变同伦理论，和动力同伦理论的相互作用。等变同伦理论研究的是对称拓扑，而动力同伦理论研究的是多项式方程组解的拓扑。近年来，随着等变和动力方法的引入，代数拓扑学领域取得了令人眼花缭乱的进展。这些项目更广泛的影响包括与研究生的合作、夏季数学研究项目、定向阅读项目和进入研究生的桥梁项目。具体项目包括研究等变和动力稳定同伦理论中的新结构，并寻求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解中的光谱塔进行研究。Pstragowski和georghe - isaksen - krause - ricka最近的工作给出了复动力稳定同伦理论的综合构造。首席研究员计划使用等变同伦理论将其扩展到真实的动机背景。另一个项目利用等变色同伦理论研究经典色同伦理论与Tate构造的相互作用，以期在色分裂猜想上取得进展。Barthel-Schlank-Stapleton最近的工作给出了一种利用超滤研究一般素数上稳定同伦理论的方法。在这种情况下的计算将使用Drinfeld模块进行研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.