Collaborative Research: Homotopy Theory: Applications and New Dimensions

合作研究：同伦理论：应用和新维度

• 批准号: 0906194
• 负责人: Michael Hopkins
• 金额: $ 107.52万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906194&HistoricalAwards=false
• 关键词: Collaborative; Research; Homotopy; Theory; Applications

The work of this proposal involves the collaborative efforts of three senior (Hopkins, Lurie and Miller) and two junior (Barwick and Behrens) investigators. During the last few years revolutionary new directions have opened for algebraic topology. At the center is the theory of higher categories, which appear in diverse ways. Hopkins and Lurie have been using the homotopy theory of infinity n-categories to classify topological quantum field theories. They have already done this in dimension less than or equal to 2, and propose to pursue a program outlined by Lurie to extend this to all dimensions. The terms of the classification represent a refinement of the Baez-Dolan cobordism hypothesis. Barwick and Lurie propose to develop new approaches to infinity n-categories, better suited to the demands placed on the subject by the many new directions. Lurie proposes a program using derived algebraic geometry to study the problem of lifting the affine algebraic group schemes to derived group schemes defined over the sphere spectrum. Behrens, Lurie and Miller propose to study the Goodwillie tower in this context, as giving a functor from the infinity 2-category of infinity 1-categories, to the infinity 2-category of stable multicategories. New directions in topology have also been created by significant computational advances. Hopkins, Mike Hill and Doug Ravenel have made important progress computing the homotopy groups of the Hopkins-Miller cohomology theories associated to orbifold families of formal group laws. These computations have very recently led to a solution of the longstanding "Kervaire invariant" problem. There are many new directions opened up by this work. The computations themselves are what mediates between classical and topological automorphic forms, and Behrens and Hopkins are planning on determining new rings of topological automorphic forms. Behrens and Hopkins are also working on the problem of determining the structures needed by a vector bundle in order that it be oriented in the theory of topological automorphic forms. These orientations are fundamental to any geometric interpretation of these theories, and represent yet another interface with the theory of infinity n-categories.In broad strokes, the work in this proposal represents deep progress and new directions on the oldest problem in algebraic topology: how to count the number of solutions to a system of equations. When the number of equations is equal to the number of unknowns, the answer to the problem is known as the "degree," and many of the triumphs of the subject in the 1920's and early 1930's result from a clear understanding of the degree. In the mid 1930's, Pontryagin introduced new topological methods in case the number of equations is smaller than the number of unknowns. This led to a remarkable interrelation between algebraic topology and geometry and over the next 50 years to dramatic progress in the fundamental problems of geometry. The important "Kervaire Invariant" problem dates from this work of Pontryagin and remained open until very recently, when it was solved by Mike Hill, Hopkins, and Doug Ravenel, using some of the ideas of this proposal. Part of the work proposed here is to carry this development further using these new ideas. In the late 1980's a different mechanism for counting the solutions to a system of equations was developed by Atiyah and Witten, in response to the demands of quantum field theory. They introduced the notion of a "topological field theory." Relating this notion to Pontryagins' work forced a reexamination of the most basic ideas about "space," and what emerged was a kind of hybrid object, an ``infinity $n$-category,''part of which is best probed by the traditional methods of algebraic topology, and part of which is best understood in the essentially combinatorial conceptual framework of category theory. Jacob Lurie is one of the worlds leading experts on the theory of infinity n-categories, and he and Clark Barwick have proposed to investigate new approaches to the theory. Working partly with Hopkins, Lurie has made dramatic progress on what one might call the "quantum counting"of the number of solutions to a system of equations. In more mathematical terms, he has articulated a clear framework for classifying topological field theories, and made made substantive progress on its realization. Once one has decided "how" to count the number of solutions to a system of equations, fundamental questions emerge about the mathematical nature of the "value" of such a count.About ten years ago, Hopkins and Miller defined the theory of "topological modular forms" designed to be a particularly useful receptacle for these values. Recently, Mark Behrens and Tyler Lawson introduced a generalization, the theory of "topological automorphic forms."Behrens proposes work on several projects with Lurie and Hopkins which will further our understanding of these topological automorphic forms.

这项提案的工作涉及三名高级研究人员（霍普金斯、卢里和米勒）和两名初级研究人员（巴维克和贝伦斯）的合作努力。 在过去的几年里革命性的新方向已经开放的代数拓扑结构。 处于中心地位的是以各种方式出现的高级范畴理论。 霍普金斯和卢里一直在使用无穷n-范畴同伦理论来分类拓扑量子场论。 他们已经在小于或等于2的维度上做到了这一点，并建议追求Lurie概述的计划，将其扩展到所有维度。 分类的术语代表了Baez-Dolan协边假说的改进。 巴维克和卢里建议发展新的方法无穷n-范畴，更好地适应的要求，对这个问题的许多新的方向。 Lurie提出了一个程序，使用派生代数几何研究的问题，提升仿射代数群计划的派生群计划定义在球谱。 Behrens，Lurie和米勒提出在这种背景下研究Goodwillie塔，给出一个从无穷大1-范畴的无穷大2-范畴到稳定多范畴的无穷大2-范畴的函子。 拓扑学的新方向也被重要的计算进步所创造。 霍普金斯、麦克·希尔和道格·拉文埃尔在计算与正规群律的轨道族相关的霍普金斯-米勒上同调理论的同伦群方面取得了重要进展。 这些计算最近导致了长期存在的“Kervaire不变量”问题的解决方案。 这项工作开辟了许多新的方向。 计算本身是什么调解之间的经典和拓扑自守形式，和贝伦斯和霍普金斯正计划确定新的环的拓扑自守形式。贝伦斯和霍普金斯也致力于问题的确定结构所需的向量束，以便它是面向理论的拓扑自守形式。 这些方向是这些理论的任何几何解释的基础，代表了与无穷n-范畴理论的另一个接口。概括地说，这个提议中的工作代表了代数拓扑学中最古老的问题的深入进展和新方向：如何计算方程组的解的数量。 当方程的数目等于未知数的数目时，问题的答案被称为“度”，20世纪20年代和30年代初这个问题的许多胜利都是由于对度的清楚理解。 在20世纪30年代中期，庞特里亚金介绍了新的拓扑方法的情况下，方程的数量小于数量的未知数。 这导致了一个显着的相互关系代数拓扑和几何和在未来50年内取得巨大进展的基本问题的几何。 重要的“Kervaire不变量”问题的日期从这项工作庞特里亚金和仍然开放，直到最近，当它被解决了迈克希尔，霍普金斯，和道格Ravenel，使用的一些想法，这一建议。 这里提出的工作的一部分是利用这些新的想法进一步推动这一发展。 在20世纪80年代后期，Atiyah和维滕根据量子场论的要求，提出了一种不同的计算方程组解的方法。 他们提出了“拓扑场论"的概念。将这个概念与庞特里亚金斯的工作联系起来，迫使人们重新审视关于“空间”的最基本的概念，出现的是一种混合对象，一种"无穷n范畴“，其中一部分最好用代数拓扑学的传统方法来探讨，另一部分最好用范畴论的基本组合概念框架来理解。 雅各布·卢里是世界上研究无穷n-范畴理论的主要专家之一，他和克拉克·巴维克提出了研究该理论的新方法。 部分工作与霍普金斯，卢里取得了巨大的进展，人们可能会称之为“量子计数“的数量的解决方案，以一个系统的方程。 在更多的数学术语中，他为拓扑场论的分类阐明了一个清晰的框架，并在其实现上取得了实质性的进展。 一旦决定了“如何”计算方程组解的数量，就会出现有关这种计数“值”的数学本质的基本问题。大约十年前，霍普金斯和米勒定义了“拓扑模形式”理论，旨在成为这些值的特别有用的容器。 最近，Mark Behrens和泰勒劳森介绍了一个推广，理论的“拓扑自守形式。“贝伦斯建议工作的几个项目与卢里和霍普金斯这将进一步我们的理解这些拓扑自守形式。