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.

Abstractaward：DMS 1611786，首席研究员：Mark J. Behrensthis Thisthis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（在任何数量的维度中），其中您可以将一个几何对象与另一个几何对象识别为另一个几何对象。 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。 例如，这种代数计算是可取的，因为它们可以通过计算机处理。 代数拓扑的早期成功是通过Euler特征（由Euler在18世纪定义的数字）和方向性（例如，Mobius带是不可方向的），对所有可能的表面（二维对象）进行了分类。相比之下，较高维度的情况更加棘手，并且是积极研究的主题。 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。 但是，代数拓扑也有重要的应用。我们生活在一个三维的宇宙中（如果包括时间，则为4维）。 这个宇宙的形状是什么？ 解决这个问题需要对3或4维中可能形状的工作知识。物质物理学中物质和力的基本相互作用受量子场理论的控制。 这些理论的分区功能的全球行为涉及拓扑考虑。 在字符串理论的上下文中，无法避免这样的考虑，因为一个移动的字符串将轨道轨迹射出。 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较高的维空间中找到高维几何对象。 使用代数拓扑结构对此类数据集的研究是拓扑数据分析的新的和主动领域的主题。拟议的特定项目旨在阐明围绕球体同质拓扑的代数拓扑中的许多问题，色素同拷贝理​​论和拓扑模块化形式（TMF）。 首席研究员（PI）及其合作者将参与一个项目，以开发基于TMF的ADAMS光谱序列，以便实际上可以用于计算。 经典的Adams光谱序列成功地计算了前60个稳定的茎。 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通过基于拓扑结构Andre-Quillen的共同体学的Quillen-Sullivan Ronication Holication Theory的概括，通过色镜。 将使用Drinfeld模块研究稳定的同型在通用素质上的计算。将使用Goodwillie演算研究色彩分裂的猜想。 PI还将研究球体环空间上绿烷属，拓扑模块形式和平滑结构之间的猜想关系。