Geometry of Groups and Moduli Spaces

我们主要从拓扑学的角度研究了黎曼曲面的映射类群和模空间以及相关的群和模空间。在与Penner的一项联合工作中，该代表在TeichMuller空间上构造了一个典型的1-上循环。Kawazumi与Penner和Bene在Teichmuller空间上组合实现了高阶Johnson同态。Furuta发展了与Seiberg-Witten理论相联系的Floer同伦型理论，Tsuboi在实解析微分同胚群的结构方面得到了显着的结果。Fujiwara、Kohno和MatSumoto分别在双曲的组合结构、映射类群、位形空间的研究和算术映射类群方面得到了深入的结果。MitSumatsu、Kitano、Akita with-Kawazumi、HiRose和Murakami分别在三维接触几何、扭曲Alexander多项式、积分Morita-Mumford类、4维流形中的曲面映射类群和双曲几何中的体积等方面得到了有趣的结果。此外，该代表还研究了由曲面的同调生成的无单位张量代数的辛导子组成的李代数的结构，并作为应用，构造了亏格1模空间的不稳定同调类。在这些结果的基础上，对我们的主题进行了进一步的研究。

We investigated mapping class group of surfaces and moduli space of Riemann surfaces as well as related groups and moduli spaces mainly from the viewpoint of topology. In a joint work with Penner, the representative constructed a canonical 1-cocycle on the Teichmuller space. Kawazumi, together with Penner and Bene, realized higher Johnson homomorphisms combinatorially on the Teichmuller space. Furuta developed theory of Floer homotopy type associated to the Seiberg-Witten theory and Tsuboi obtained a remarkable result concerning the structure of real analytic diffeomorphism group of manifolds. Fujiwara, Kohno, and Matsumoto obtained deep results in combinatorial structure of hyperbolic as well as the mapping class group, study of configuration spaces and arithmetic mapping class group, respectively. Mitsumatsu, Kitano, Akita with-Kawazumi, Hirose, and Murakami obtained interesting results, respectively, in the studies of 3-dimensional contact geometry, twisted Alexander polynomials, integral Morita-Mumford classes, mapping class group of surfaces in 4-manifolds, and volumes in hyperbolic geometry. Also the representative studied the structure of the Lie algebra consisting of symplectic derivations of the tensor algebra, without unit, generated by the homology of surfaces and, as an application, constructed unstable homology classes of genus 1 moduli spaces. Based on these results, further study of our theme has come into view.