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.