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

这个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以及有限群和离散群的上同调的各种主题的研究。D.Wilczynski和R.Lee计划共同研究在拓扑范畴和光滑范畴中用2-球面表示单连通4-流形的同调类的问题。R.Lee计划继续他们在亏格2的Siegel模簇上的工作。利用Casson不变量和Floer同调，R.Lee，S.Cappell和E.Miller计划研究三维流形上的外科理论。R.Lee计划与I.Hambleton和F.Raymond一起研究代数曲面上的代数群作用，重点是在4维流形理论中的应用。最后，他计划与J·米尔格拉姆、J·马金尼斯和J·马蒂诺一起研究零星群体的上同调。这些都是当前拓扑学研究的主流课题。R·李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。有些问题归结于代数几何，有些问题归结于复杂分析，有些问题归结于群论，等等。这一结果又一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.