Chromatic homotopy - stable and unstable

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Abstract Award: DMS 1611786, Principal Investigator: Mark J. Behrens This project aims to address major problems in the field of algebraic topology. Topology is the study of geometry (in any number of dimensions) where you identify one geometric object with another if one can be deformed into the other. The goal of algebraic topology is to ascribe discrete algebraic invariants to these geometric objects to distinguish their topological types. In this way, distinguishing geometric objects is reduced to algebraic computations. Such algebraic computations are desirable, because they can be handled by a computer, for example. An early success of algebraic topology was the classification of all possible surfaces (2-dimensional objects) by means of Euler characteristic (a number, defined by Euler in the 18th century) and orientability (e.g., a Mobius strip is nonorientable). By contrast, the situation in higher dimensions is much more intractable, and is the subject of active research. Understanding the topological type of geometric objects is a fundamental act of scientific/mathematical inquiry, comparable to the study of prime numbers, or the classification of the fundamental particles that constitute matter and carry forces. However, there are also important applications of algebraic topology. We live in a 3-dimensional universe (4-dimensions if you include time). What is the shape of this universe? Addressing this question requires a working knowledge of the possible shapes in 3 or 4 dimensions. The fundamental interactions of matter and forces in particle physics is governed by quantum field theory. The global behavior of the partition functions of such theories involves topological considerations. Such considerations are impossible to avoid in the context of string theory, as a moving string traces out a surface. Topological computations have recently been applied to solve problems in solid state physics. Also, data involving the interrelation of a large number of variables naturally traces out a high dimensional geometric object in a higher dimensional space. The study of such data-sets using algebraic topology is the subject of the new and active field of topological data analysis. The specific projects proposed are aimed at shedding light on many problems in algebraic topology surrounding the homotopy groups of spheres, chromatic homotopy theory, and topological modular forms (tmf). The principal investigator (PI) and his collaborators will engage in a project to develop the tmf-based Adams Spectral Sequence to the point where it can be actually used for calculations. The classical Adams spectral sequence has succeeded in computing the first 60 stable stems. We expect that since tmf is a much more sensitive cohomology theory, when properly developed, its associated Adams spectral sequence could push these computations into the 90s, which would shed light on the only remaining case of the Kervaire Invariant Problem, in dimension 126. The PI also plans on using the tmf-based Adams spectral sequence to investigate the Telescope Conjecture at chromatic level 2. Unstable homotopy will also be studied through the chromatic lens, using a generalization of Quillen-Sullivan rational homotopy theory based on topological Andre-Quillen cohomology. Computations in stable homotopy theory at generic primes using ultra-filters will be investigated using Drinfeld Modules. The Chromatic Splitting Conjecture will be investigated using Goodwillie calculus. The PI will also study the conjectural relationship between the Ochanine genus, topological modular forms, and smooth structures on loop spaces of spheres.