Mathematical Sciences: Research in Low Dimensional Manifolds, Transformation Groups, and Cohomology of Finite and Discrete Groups

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This project covers research on a variety of topics in low dimensional manifolds, Donaldson and Casson invariants, theory of transformation groups on algebraic surfaces, and cohomology of finite and discrete groups. Jointly D. Wilczynski and R. Lee plan to investigate the problem of representing homology classes of simply connected 4-manifolds by 2-spheres, in both the topological and the smooth category. With S. Weintraub, R. Lee plans to continue their work on Siegel modular varieties of genus 2. Using Casson's invariant and Floer homology, R. Lee, S. Cappell and E. Miller plan to study the theory of surgery on 3- dimensional manifolds. With I. Hambleton and F. Raymond, R. Lee plans to investigate algebraic group actions on algebraic surfaces with special emphasis on the applications to 4- dimensional manifold theory. Finally with J. Milgram, J. Maginnis and J. Martino, he plans to study the cohomology of sporadic groups. These are all topics in the mainstream of current topological research. The hallmark of R. Lee's approach is to choose very concrete problems and bring to bear techniques from an impressively large and high-powered arsenal. Some problems yield to algebraic geometry, some to complex analysis, some to group theory , and so forth. The result is another clear demonstration of the unity of mathematics.