Collaborative Research: Homotopy Theory: Applications and New Dimensions

这项提议的工作涉及三名高级调查人员(霍普金斯、鲁里和米勒)和两名初级调查人员(巴里克和贝伦斯)的合作努力。在过去的几年里，为代数拓扑学开辟了革命性的新方向。中心是更高范畴的理论，它以不同的方式出现。Hopkins和Lurie一直使用无穷大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，并提议执行Lurie概述的计划，将其扩展到所有维度。分类术语代表了贝兹-多兰共生假说的精炼。Barwick和Lurie建议开发新的方法来研究无限n范畴，更好地适应许多新方向对主题的要求。Lurie提出了一个利用导出代数几何的程序来研究将仿射代数群方案提升到球谱上定义的派生群方案的问题。Behrens，Lurie和Miller建议在这一背景下研究Goodwillie塔，因为它给出了一个从无穷大1-范畴的无穷2-范畴到稳定多范畴的无穷2-范畴的函子。拓扑学的新方向也因显著的计算进步而被创造出来。Hopkins，Mike Hill和Doug Ravenel在计算与形式群律的奥比福尔族有关的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一个长期存在的“克维埃不变量”问题的解决。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之间的中介，Behrens和Hopkins正计划确定拓扑自同构形的新环。Behrens和Hopkins也在研究确定向量丛所需结构的问题，以便使其在拓扑自同构形理论中定向。这些取向是这些理论的任何几何解释的基础，并代表了与无穷n范畴理论的另一种接口。概括地说，这项建议的工作代表了代数拓扑学中最古老的问题：如何计算方程组的解的数目的深入进展和新的方向。当方程的个数等于未知数的个数时，问题的答案就被称为度，1920年代的S和1930年初的S对度的许多研究成果都源于对度的清晰认识。20世纪30年代中期，当方程的个数小于未知数时，庞特里亚金提出了新的拓扑学方法。这导致了代数拓扑学和几何之间显着的相互关系，并在接下来的50年里在几何的基本问题上取得了戏剧性的进展。重要的“Kervaire不变”问题始于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·拉弗内尔利用这一提议的一些想法解决。这里提出的工作的一部分是使用这些新想法进一步推动这一发展。在20世纪80年代末，Atiyah和Witten根据量子场论的要求，发展了一种不同的计算方程组解的机制。他们引入了“拓扑场论”的概念。将这一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“无穷大$n$-范畴”，它的一部分最好用传统的代数拓扑学方法来探索，而它的一部分在范畴理论的基本组合概念框架中得到最好的理解。雅各布·鲁里是世界上研究无限n范畴理论的顶尖专家之一，他和克拉克·巴威克提议研究这一理论的新方法。在与霍普金斯大学的部分合作中，Lurie在方程式组解的数量的“量子计数”方面取得了巨大的进展。用更多的数学术语，他为拓扑场理论的分类阐明了一个明确的框架，并在其实现上取得了实质性进展。一旦一个人决定了如何计算一组方程的解的个数，就会出现这样一个数学值的数学性质的基本问题。大约十年前，霍普金斯和米勒定义了“拓扑模形式”理论，旨在为这些值提供一个特别有用的容器。最近，Mark Behrens和Tyler Lawson引入了拓扑自同构形式的推广，Behrens提出了与Lurie和Hopkins合作的几个项目，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.