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

本项目涵盖了低维流形、Donaldson和Casson不变量、代数曲面上的变换群理论、有限群和离散群的上同调等多个主题的研究。D. Wilczynski和R. Lee计划共同研究用2球表示单连通4流形的同调类的问题，包括拓扑范畴和光滑范畴。与S. Weintraub一起，R. Lee计划继续他们对2属西格尔模块化变种的研究。R. Lee， S. Cappell和E. Miller计划利用Casson不变量和Floer同调来研究三维流形的手术理论。李博士计划与I. Hambleton和F. Raymond一起研究代数曲面上的代数群作用，并特别强调其在四维流形理论中的应用。最后，他计划与J. Milgram、J. Maginnis和J. Martino一起研究散发性类群的上同源性。这些都是当前拓扑学研究的主流课题。李的方法的特点是选择非常具体的问题，并从令人印象深刻的庞大和强大的武器库中运用技术。有些问题适用于代数几何，有些适用于复分析，有些适用于群论，等等。这个结果再次清楚地证明了数学的统一性。

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.