9401755 Hsiang This project involves research in topological manifolds and differential geometry. The principal investigator, Wu-Chung Hsiang, has some interesting new ideas relating to the Novikov conjecture. Together with Cynthia Curtis, Hsiang is also interested in work on four-manifolds related to the Poincare' conjecture. Curtis and Hsiang recall a theorem of Wall stating that two h-cobordant four-manifolds may be "stabilized" by adding copies of S^2 x S^2 in such a way that the stabilized manifolds are diffeomorphic. They study obstructions to "destabilization"; that is, they attempt to determine whether the original manifolds were themselves diffeomorphic. Curtis also studies generalizations of Casson's invariant for three-manifolds and related invariants. Meanwhile, Zhuang-dan Guan studies geometric structures on manifolds. He is interested in examples of compact holomorphic symplectic manifolds which are not Kaehlerian, quasi-Einstein metrics, singularities on rational curves, and homogeneous spaces. The investigators will study the properties of objects which are 3-, 4-, or higher-dimensional. Specifically, they will develop means of determining when two objects of the same dimension are equivalent in some sense. Curtis and Hsiang are working together on a certain class of four-dimensional objects. Their work relates to a conjecture of Poincare' dating from the turn of the century. Guan works to determine if two same-dimensional objects are equivalent in a stricter sense, taking more geometry into account. The work of all three researchers relates in a general sense to new and exciting developments in physics. ***
