Equivariant and Motivic Deformations of Stable Homotopy Theory

稳定同伦理论的等变和动机变形

• 批准号: 2005476
• 负责人: Mark Behrens
• 金额: $ 33.88万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-15 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005476&HistoricalAwards=false
• 关键词: Equivariant; Motivic; Deformations; Stable; Homotopy

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.

这些项目旨在解决代数拓扑领域的主要问题。 拓扑是对几何形状的研究，您可以在其中识别一个几何对象，如果将一个对象变形为另一个几何对象。代数拓扑的目的是将离散代数不变性归因于这些几何对象，以区分它们的拓扑类型。 这样，区分几何对象就会简化为代数计算。 了解几何对象的拓扑类型是科学/数学探究的基本行为，与质数的研究相当，或者对构成物质和携带力的基本粒子的分类。 拓扑计算最近已用于解决固态物理学的问题。 同样，涉及大量变量相互关系的数据自然会在高维空间中找到几何对象。 使用代数拓扑来对此类数据集进行研究是拓扑数据分析的新现场的主题。 这些项目的重点是经典代数拓扑，均等同义理论和动机同型理论的相互作用。 模棱两可的同型理论是对对称拓扑的研究，而动机同义理论是对多项式方程系统解决方案拓扑的研究。 近年来，通过进口模棱两可的和动机的方法，在代数拓扑领域中见证了一系列令人眼花the乱的进步。这些项目的更广泛影响包括与研究生的合作，夏季数学研究计划，定向阅读计划以及用于进入研究生的桥梁计划。杂志项目涉及调查现象和动机稳定同型同苯二的新型结构，并试图利用这些结构来为经典稳定同型同质拷贝理论的一些杰出问题提供新的方法。 望远镜的猜想将使用光谱塔进行研究，该光谱塔出现在Kervaire不变的一个问题的山丘山脉 - 雷维内尔解决方案中。 Pstragowski和Gheorghe-Isaksen-Krause-Ricka的最新工作为复杂的动机稳定同型理论提供了合成的结构。 主要研究者计划使用均等同义理论将其扩展到真正的动机背景。 另一个项目使用地位色的同义理论来研究经典色度同义理论与泰特构建的相互作用，目的是在色度分裂的猜想上取得进展。 Barthel-Schlank-Stapleton的最新工作提供了一种使用Ultra-Filters在通用素质上研究稳定同型理论的方法。 在这种情况下，将使用德林菲尔德模块进行调查。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估值得支持的。

项目成果

The telescope conjecture at height 2 and the tmf resolution

• DOI: 10.1112/topo.12208
• 发表时间: 2019-09
• 期刊: Journal of Topology
• 影响因子: 1.1
• 作者: A. Beaudry;M. Behrens;P. Bhattacharya;D. Culver;Zhouli Xu