The goal of this project is to explore some applications and potential applications of homotopy algebras. In algebraic K-theory, the proposal describes ideas applicable to the conjectures of Rognes and the conjecture of Waldhausen on the K-theory chromatic tower, relating arithmetic and geometry. In unstable homotopy theory, the proposal describes how E-infinity differential graded algebras and related homotopy algebras can be used to study the homotopy theory of spaces. In stable homotopy theory, the proposal discusses obstruction theory for E-n algebra structures, and the relationship of E-n algebra structures with structures on categories of modules.
Homotopy theory studies those properties of mathematical objects that do not change under small deformations. These mathematical objects are often of a geometric nature but the methods of homotopy theory have been increasingly applied to objects of an algebraic nature as well. Homotopy theoretic properties tend to be accessible to computation by taking advantage of the invariance under small changes. Since they also generally retain important information about the original mathematical objects, homotopy theory is an effective tool for a wide range of mathematical problems.
Homotopy theory attempts to reduce questions about space and geometry to questions in algebra. Often this involves assigning some kind of algebraic object like a vector space or numerical invariant to a geometric object or space to capture or measure some intrinsic feature. These invariants can then potentially be used to distinguish between different spaces. The invariants studied are typically discrete, and so they do not change under continuous deformation or "homotopy". This freedom often allows topologists to deform complicated problems into simpler ones that have the same invariants, which are then easier to compute. One basic and easily visualized example is the "Euler characteristic" of a surface: Take any surface in space or any abstract surface and draw edges on it to divide it up into facets; let χ be the number of facets minus the number of edges plus the number of vertices (where the edges begin and end). The only requirement is that the edges intersect at the vertices and that each edge be part of the boundary of a facet. The edges do not need to be straight in any sense, and because of this, χ only depends on the abstract drawing on the abstract surface. So χ does not change when the surface is bent or stretched or twisted in any way, as long as it is not punctured or torn, and is independent of how the surface sits in the ambient space. Leonhard Euler observed in the 18th century that the number is also independent of how the edges are drawn on the given surface; no matter how the given surface is divided up, the same number χ always results. This invariant of the surface is called the Euler characteristic and is effective for distinguishing between surfaces of different types. For example, for the surface of a ball, the Euler characteristic is 2, but for the surface of an innertube, the Euler characteristic 0. As a consequence, the Euler characteristic can distinguish between these two surfaces, even if the ball or innertube is tied in knots or contorted into a complicated configuration. The Euler characteristic is a precursor to a theory called "algebraic K-theory" which can be visualized as doing the same kind of thing as the Euler characteristic: It converts the operation of gluing together of basic pieces into addition and subtraction. Algebraic K-theory has applications all through algebra and geometry. For example, K_0 of a number field gives information about factorization and whether unique factorization holds; K_1 of a group ring gives information about the difference between homotopy equivalence and homeomorphism for manifolds with that fundamental group. Several of the results of this project advance our knowledge of algebraic K-theory. For example, the paper "Algebraic K-theory and abstract homotopy theory" tells how to break up Algebraic K-theory into pieces that can be described by a mixture of global and local data. The paper "An inverse K-theory functor" illuminates the relationship between algebraic K-theory and generalized homology theories. The paper "Permutative categories, multicategories, and algebraic K-Theory" makes some progress in understanding more about the global multiplicative structure of algebraic K-theory. Algebraic K-theory is non-linear; its linearization is called "topological Hochschild homology" (THH). Much of what we know calculationally about algebraic K-theory has arisen by studying THH and a related theory called "topological cyclic homology" (TC). The paper "Localization theorems in topological Hochschild homology and topological cyclic homology" and another paper "Localization for THH(ku) and the topological Hochschild and cyclic homology of Waldhausen categories" in progress contribute to the foundations of this subject. Finally, another large component of the project involves homotopy algebras, and in particular, a homotopy algebra called the "Brown-Peterson spectrum" (BP). A homotopy algebra is a certain kind of generalized homology theory, and the Brown-Peterson spectrum in particular has wide applications in homotopy theory (for example, the "chromatic picture" organizes stable homotopy theory around the properties of this homology theory). The paper "The multiplication on BP" proves that the Brown-Peterson spectrum has an "E_4 structure", a type of strongly coherent multiplication. It had been previously conjectured for decades but not previously known that BP had this type of structure. The paper "The smash product for derived categories in stable homotopy theory" describes some new results and constructions that result from having an E_4 structure. The paper "The homology of E_n ring spectra and iterated THH" provides some of the basic tools for constructing the E_4 structure.
