Chromatic homotopy - stable and unstable

项目编号：DMS 1611786，项目负责人：Mark J. behrens本项目旨在解决代数拓扑领域的主要问题。拓扑学是对几何（在任何维度上）的研究，如果一个几何对象可以变形成另一个，那么您可以将其识别为另一个几何对象。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。这样的代数计算是可取的，因为它们可以由计算机来处理。代数拓扑学的早期成功是通过欧拉特征（欧拉在18世纪定义的一个数字）和可定向性（例如，莫比乌斯带是不可定向的）对所有可能的表面（二维物体）进行分类。相比之下，高维的情况要棘手得多，并且是积极研究的主题。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。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（如果算上时间的话是四维的）。这个宇宙的形状是什么？解决这个问题需要对三维或四维可能的形状有一定的了解。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。这些理论的配分函数的全局行为涉及到拓扑方面的考虑。在弦理论的背景下，这样的考虑是不可能避免的，因为运动的弦沿着一个表面运动。拓扑计算最近被应用于解决固体物理中的问题。此外，涉及大量变量相互关系的数据自然会在高维空间中描绘出高维几何对象。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。提出的具体项目旨在阐明围绕球体同伦群，色同伦理论和拓扑模形式（tmf）的代数拓扑中的许多问题。首席研究员（PI）和他的合作者将参与一个项目，开发基于tmf的亚当斯光谱序列，使其可以实际用于计算。经典的亚当斯谱序列成功地计算了前60个稳定的谱。我们期望，由于tmf是一个更敏感的上同调理论，当适当发展时，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126维的Kervaire不变量问题的唯一剩余情况。PI还计划使用基于tmf的亚当斯光谱序列在色阶2上研究望远镜猜想。利用基于拓扑Andre-Quillen上同调的Quillen-Sullivan理性同伦理论的推广，通过色透镜研究不稳定同伦。利用超滤子研究了稳定同伦理论在一般素数上的计算。利用古德威利微积分研究色分裂猜想。PI还将研究Ochanine属、拓扑模形式和球体环空间上的光滑结构之间的推测关系。

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.