Chromatic homotopy - stable and unstable

色同伦 - 稳定和不稳定

• 批准号: 1611786
• 负责人: Mark Behrens
• 金额: $ 32.71万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-15 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1611786&HistoricalAwards=false
• 关键词: Chromatic; homotopy; stable; unstable

AbstractAward: DMS 1611786, Principal Investigator: Mark J. BehrensThis project aims to address major problems in the field of algebraic topology. Topology is the study of geometry (in any number of dimensions) 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. Such algebraic computations are desirable, because they can be handled by a computer, for example. An early success of algebraic topology was the classification of all possible surfaces (2-dimensional objects) by means of Euler characteristic (a number, defined by Euler in the 18th century) and orientability (e.g., a Mobius strip is nonorientable). By contrast, the situation in higher dimensions is much more intractable, and is the subject of active research. 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. However, there are also important applications of algebraic topology. We live in a 3-dimensional universe (4-dimensions if you include time). What is the shape of this universe? Addressing this question requires a working knowledge of the possible shapes in 3 or 4 dimensions. The fundamental interactions of matter and forces in particle physics is governed by quantum field theory. The global behavior of the partition functions of such theories involves topological considerations. Such considerations are impossible to avoid in the context of string theory, as a moving string traces out a surface. 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 high dimensional geometric object in a higher 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 specific projects proposed are aimed at shedding light on many problems in algebraic topology surrounding the homotopy groups of spheres, chromatic homotopy theory, and topological modular forms (tmf). The principal investigator (PI) and his collaborators will engage in a project to develop the tmf-based Adams Spectral Sequence to the point where it can be actually used for calculations. The classical Adams spectral sequence has succeeded in computing the first 60 stable stems. We expect that since tmf is a much more sensitive cohomology theory, when properly developed, its associated Adams spectral sequence could push these computations into the 90s, which would shed light on the only remaining case of the Kervaire Invariant Problem, in dimension 126. The PI also plans on using the tmf-based Adams spectral sequence to investigate the Telescope Conjecture at chromatic level 2. Unstable homotopy will also be studied through the chromatic lens, using a generalization of Quillen-Sullivan rational homotopy theory based on topological Andre-Quillen cohomology. Computations in stable homotopy theory at generic primes using ultra-filters will be investigated using Drinfeld Modules. The Chromatic Splitting Conjecture will be investigated using Goodwillie calculus. The PI will also study the conjectural relationship between the Ochanine genus, topological modular forms, and smooth structures on loop spaces of spheres.

摘要奖：DMS 1611786，主要研究者：Mark J. Behrs该项目旨在解决代数拓扑学领域的主要问题。 拓扑学是对几何的研究（在任何数量的维度上），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识别。 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。 这样的代数计算是可取的，因为它们可以由例如计算机处理。 代数拓扑学的早期成功是通过欧拉特征（一个数，由欧拉在世纪定义）和可定向性（例如，莫比乌斯带是不可定向的）。相比之下，更高维度的情况要棘手得多，并且是积极研究的主题。 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。 然而，代数拓扑也有重要的应用。我们生活在一个三维的宇宙中（如果你包括时间，那就是四维）。 宇宙的形状是什么？ 解决这个问题需要对三维或四维的可能形状的工作知识。粒子物理学中物质和力的基本相互作用是由量子场论支配的。 这种理论的配分函数的全局行为涉及拓扑考虑。 在弦理论的背景下，这样的考虑是不可能避免的，因为运动的弦描绘出一个表面。 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在更高维空间中描绘出高维几何对象。 用代数拓扑学研究这类数据集是拓扑数据分析这一新的活跃领域的主题，提出的具体项目旨在阐明围绕球面同伦群、色同伦理论和拓扑模形式（tmf）的代数拓扑学中的许多问题。 主要研究者（PI）和他的合作者将参与一个项目，以开发基于tmf的亚当斯光谱序列，使其能够实际用于计算。 经典的亚当斯谱序列已成功地计算了前60个稳定的茎。 我们期望，由于tmf是一个更敏感的上同调理论，当适当地发展时，其相关的亚当斯谱序列可以将这些计算推进到90年代，这将揭示唯一剩下的情况下，在126维的科维尔不变问题。PI还计划使用基于tmf的亚当斯光谱序列来研究色级2的望远镜猜想。 不稳定同伦也将通过色透镜研究，使用基于拓扑Andre-Quillen上同调的Quillen-Sullivan有理同伦理论的推广。 在一般素数使用超过滤器的稳定同伦理论的计算将使用Drinfeld模块进行研究。色分裂猜想将使用Goodwillie演算进行研究。 PI还将研究Ochanine亏格，拓扑模形式和球面循环空间上的光滑结构之间的关系。