Mathematical Sciences: Manifolds and Algebraic Topology

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This project involves four senior research topologists (William Browder, Wu-chung Hsiang, John N. Mather, and William P. Thurston), one postdoctoral associate (Lang-Fang Wu), and three advanced graduate students. Their research topics are quite various. Browder will study free actions of certain topological groups on finite-dimensional spaces. He will also study fixed point theorems and regularity of minimal surfaces. Hsiang will work on a number of problems related to the cyclotomic trace map, including the notorious Novikov conjecture in L-theory as well as some more tractable ones. Mather's research deals with area-preserving diffeomorphisms of infinite cylinders. There are several hard conjectures about invariant curves under diffeomorphisms and their rotation numbers. Thurston will work toward his famous geometrization conjecture for 3-manifolds. In addition, he will work on automatic groups, self-similar tilings, one-dimensional real and complex dynamical systems, and other topics. Wu's research concerns the Ricci Flow equations on manifolds and orbifolds. All these topics are in the mainstream of research in topology. It is appropriate to involve graduate students in their study. For example, the geometrization conjecture of Thurston is a notoriously difficult problem. It subsumes the Poincare Conjecture, which dates to the turn of the century. Nevertheless, pieces of it have been the stuff of numerous Ph.D theses by Thurston's students for more than a decade.