The PI intends to study problems with origins in classical homotopy theory as well as their overlap with function spaces, cohomology of groups, and properties of classical configuration spaces. The specific problems are as follows: (1) Extend the P.I.'s recent solution of a 25 year old conjecture of M.G. Barratt on the growth of torsion in the homotopy groups of certain finite complexes. (2) Analyze these structures for other finite complexes through simplicial models for iterated loop spaces as developed by the P.I., and R. Levi. (3) In joint work of the P.I., J. Berrick, Y. Wong, and J. Wu, continue to analyze the interplay between braid groups, and the homotopy groups of the 2-sphere. These connections extend in joint work with D. Cohen to analogues of classical modular forms obtained from the cohomology of the braid groups with coefficients in a symplectic representation. (4) In joint work with A. Adem, and D. Cohen, continue to analyze the structure of spaces of representations of fundamental groups of complements of complex hyperplane arrangements, and their associated vector bundles.
One of the main directions here is a connection between Artin's braid groups, and the classical problem of enumerating ways of continuously wrapping spheres of any dimension around other geometric objects. The methods can be thought of as enumerating topological properties of many particles, not allowed to collide, moving through time. The methods overlap with several mathematical structures given combinatorially as well as number theoretically. These methods involve the "geometry of braids", and how this structure encodes counting problems through connections to other mathematical objects.
