Collaborative Research: Homotopy Theory: Applications and New Dimensions

这项提案的工作涉及三位高级研究员（霍普金斯，Lurie和米勒）和两位初级研究员（巴维克和贝伦斯）的合作努力。在过去的几年里，代数拓扑学开辟了革命性的新方向。其核心是高级范畴理论，它以多种方式出现。Hopkins和Lurie一直在使用无穷n范畴的同伦理论对拓扑量子场论进行分类。他们已经在小于或等于2的维度上做到了这一点，并建议按照Lurie概述的计划将其扩展到所有维度。分类的术语代表了对贝兹-多兰协同假设的改进。Barwick和Lurie建议开发新的方法来研究无穷n类，以更好地适应许多新方向对该主题的要求。Lurie提出了一个应用派生代数几何的程序来研究仿射代数群格式提升到在球谱上定义的派生群格式的问题。Behrens， Lurie和Miller提出在这种背景下研究Goodwillie塔，作为给出一个函子，从无穷2-范畴的无穷1-范畴，到稳定多范畴的无穷2-范畴。重大的计算进步也创造了拓扑学的新方向。Hopkins， Mike Hill和Doug Ravenel在计算与形式群律的轨道族相关的Hopkins- miller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。计算本身是经典和拓扑自同构形式之间的中介，贝伦斯和霍普金斯正计划确定拓扑自同构形式的新环。Behrens和Hopkins也在研究如何确定矢量束所需的结构，从而使其在拓扑自同构形式理论中定向。这些方向是这些理论的任何几何解释的基础，并且代表了无限n类理论的另一个接口。总的来说，这个建议的工作代表了代数拓扑学中最古老的问题的深刻进展和新的方向：如何计算一个方程组的解的数量。当方程的数量等于未知数的数量时，这个问题的答案就被称为“度”，20世纪20年代和30年代初，这门学科的许多胜利都源于对度的清晰理解。在20世纪30年代中期，庞特里亚金引入了新的拓扑方法，用于解决方程数量小于未知量的情况。这导致了代数拓扑和几何之间的显著相互关系，并在接下来的50年里在几何的基本问题上取得了巨大的进展。重要的“Kervaire不变量”问题可以追溯到庞特里亚金的这项工作，直到最近才被迈克·希尔、霍普金斯和道格·拉文内尔解决，他们使用了这个提议的一些想法。这里提出的部分工作是利用这些新思想进一步推动这一发展。在20世纪80年代末，为了响应量子场论的要求，Atiyah和Witten开发了一种不同的机制来计算方程组的解。他们引入了“拓扑场论”的概念。将这一概念与庞特里亚金的工作联系起来，迫使人们重新审视关于“空间”的最基本概念，由此产生了一种混合对象，一种“无限范畴”，其中一部分最好用代数拓扑的传统方法来探索，另一部分最好用范畴论的本质组合概念框架来理解。雅各布·卢里（Jacob Lurie）是世界上研究无穷n类理论的顶尖专家之一，他和克拉克·巴维克（Clark Barwick）提议研究该理论的新方法。在与霍普金斯大学的部分合作中，Lurie在一个方程组的解的数量的“量子计数”方面取得了巨大的进展。从更数学的角度来说，他提出了一个清晰的拓扑场论分类框架，并在实现这一框架方面取得了实质性进展。一旦决定了“如何”计算一个方程系统的解的数量，关于这种计数“值”的数学本质的基本问题就出现了。大约十年前，霍普金斯和米勒定义了“拓扑模块形式”理论，旨在成为这些值的特别有用的容器。最近，马克·贝伦斯和泰勒·劳森介绍了一种概括，即“拓扑自同构形式”理论。Behrens建议与Lurie和Hopkins合作几个项目，这将进一步加深我们对这些拓扑自同构形式的理解。

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.