3867 East Hall
Phone: (734) 647-4468
Geometry and Topology
- Differential Equations
My main area of interest is differential topology. More specifically, I am interested in studying manifolds via their symmetry groups--the theory of transformation groups.
I have lately been working on desingularization of group actions on manifolds and residue theorems,c.f., Relations among characteristic classes and fixed points I, the recognition principle in Topology and its Applications; there should be many characteristic class consequences to these results.
I am also interested in knowing when a smooth action of a compact Lie group on a manifold is diffeomorphic to an algebraic action on a smooth (real) variety, c.f., Extending algebraic actions in Revista Matematica de la Universidad Complutense de Madrid is a first step. A forthcoming article with Heiner Dovermann on Algebraic models for G-vector bundles continues that subject.
Topological representation theory is another interest, c.f., Lee & Wasserman, On the Groups JO(G), Mem. AMS 159.
Equivariant cobordism theory is another interest, c.f., Cobordism of regular O(n) manifolds (with C. Lazarov), Ann. Math., 93, 229-251.
The study of isovariant maps is another aspect of transformation groups, c.f., Wasserman, Isovariant Maps and the Borsuk-Ulam Theorem, Top. and its Applications, 1990.
Most of my work is now centered on some equations of mathematical physics, namely the Einstein equations of general relativity coupled with other fields, e.g.,the Yang Mills equations, c.f., Existence of Infinitely Many Smooth, Static, Global Solutions of E/Y-M Equations (with J.A. Smoller), Comm. Math Physics, 151, 303-325 (1993) or Existence of black-hole solutions for the Einstein-Yang/Mills equations (with J. Smoller and S. -T. Yau), Comm. Math. Physics 154 (1993). 377-401.or boson fields, c.f., On existence of mini-boson stars, (with Piotr BizonÂ´), gr-qc /0002034.