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。 拓扑学是一门研究几何的学科，如果一个几何对象可以变形为另一个几何对象，那么你就可以将一个几何对象与另一个几何对象区分开来。代数拓扑的目标是赋予这些几何对象离散的代数不变量，以区分它们的拓扑类型。 通过这种方式，区分几何对象被简化为代数计算。 理解几何对象的拓扑类型是科学/数学探究的基本行为，与素数的研究或构成物质和承载力的基本粒子的分类相当。 拓扑计算最近已被应用于解决固体物理学中的问题。 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。 使用代数拓扑学研究这样的数据集是拓扑数据分析的新的和活跃的领域的主题。 这些项目的重点是在相互作用的经典代数拓扑，等变同伦理论和motivic同伦理论。 等变同伦理论是研究对称性的拓扑，而运动同伦理论是研究多项式方程组解的拓扑。 近年来，通过引进等变方法和motivic方法，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，暑期数学研究计划，定向阅读计划，以及进入研究生的桥梁计划。特别的项目涉及调查等变和motivic稳定同伦理论的新结构，并寻求利用这些结构来解决经典稳定同伦理论中一些长期悬而未决的问题。 望远镜猜想将使用出现在希尔-霍普金斯-拉文埃尔解决方案的Kervaire不变一问题的光谱塔进行调查。 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。 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。 另一个项目使用等变色同伦理论来研究经典色同伦理论与Tate构造的相互作用，目的是在色分裂猜想上取得进展。 Barthel-Schlank-斯台普顿最近的工作给出了一种利用超滤子研究类属素数上的稳定同伦理论的方法。 该奖项反映了NSF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.