Research Statement – He Wang 1 Overview

My main research interests are in algebraic topology and group theory, and their interactions with algebra, geometry and combinatorics. In early work with X. Liu [18, 19], we studied the stable homotopy theory of spheres using the May spectral sequence and the Adams spectral sequence. In a paper with X. Liu and W. Li [17], we described differentials in a certain spectral sequence converging to the twisted de Rham cohomology of a smooth manifold using higher-order Massey products. In recent work with A. Suciu [23, 24, 26, 25], we investigate several general properties of commutative differential graded algebra (CDGA) models and various Lie algebras of spaces and groups, especially their formality properties, the way such properties behave under change of coefficient field, and the way they relate to cohomology jump loci. Applying the general machinery we have developed to some important families of groups, we have answered several questions regarding the structure of pure virtual and pure welded braid groups. I have also worked with K. Igusa [15] on the classification of fibrations realizing long exact sequences of abelian groups. My research plans involve both a development of present work, and branching out in new directions. I will investigate the many and varied connections among the characteristic varieties and resonance varieties of spaces and CDGA models, and Bieri-Neumann-Strebel invariants of the fundamental groups. and I will explore the relationship between the ranks of the Chen Lie algebras and the dimensions of the resonance varieties of the CDGA models of groups. I will explore these notions in the context of pure braid groups on Riemann surfaces, graphic configuration spaces, picture groups from the quiver representations, etc. In each section of this statement, I will provide some background and motivation, state our results, and follow with some of my plans for future research and expected applications.