Beyond the framework of classical arithmetic geometry-Zeta, arithmetic topology, and categorical arithmetic geometry

作为几何拓扑学在算术几何领域的一个应用，М。Matsumoto和R. Hain得到了以下结果：当且仅当曲线C的雅可比矩阵中对应的代数环C-C^-的伽罗瓦上同调环时，数域的绝对伽罗瓦群对曲线C的单幂基群的作用达到最大值。最近，证明了g>1曲线属映射类群的加权补全的伽罗瓦作用是结晶的和混合Tate型的；而对于g=1，它不是混合的。Tamagawa在具有混合特性的DVR上构造了光滑曲线族的非奇异性分辨率。这篇文章发表在《RIMS出版物》上。这一结果将格罗腾狄克猜想简化为仿射猜想。望月发展了分类算术几何理论和类人猿理论，为ABC猜想提供了一个途径。Tsuzuki引入了刚性上同调的普遍上同调可降性和普遍de Rham可降性的概念，并得到了Kedlaya有限维性的一个推广。Kimura发展了纯动机的态射的有限维的概念，并构造了许多有限维动机的非平凡的例子。Sigeyuki Morita研究了幂零基群的映射类群和导数代数，得到了关于单元象的上同调结果和猜想。该论文发表在Proc. Sympos上。纯粹数学。Matsumoto利用形式幂级数几何和伽罗瓦理论，充分利用当前cpu的并行性，设计了一种新的快速伪随机数生成器。论文发表于《中国科学院院刊》2006。

As an application of geometric topology in the area of arithmetic geometry, М. Matsumoto and R. Hain Obtained the, following result : The action of the absolute Galois group of a number field on the unipotent fundamental group of a curve C attains the maximum size, if and only if the Galois cohomology cocycle corresponding to the algebraic cocycle C-C^- in the jacobian of C. This was published in J. of Institute of Mathematics Jussieu. More recently, the Galois action on the weighted completion of the mapping class group of the genus g>1 curves is proved to be crystalline and of mixed Tate type ; Whereas for g=1 it is not mixed Tate.Tamagawa constructed a resolution of nonsingularities of smooth curve family over DVR with mixed characteristic. This was published in Publications of RIMS. This results reduces the proper Grothendieck conjecture to the affine one. Mochizuki developped the theory of categorical arithmetic geometry and frobenioids, which give an approach to the ABC conjecture.Tsuzuki introduced the notion of the universal cohomology descendability and unimversal de Rham descendability for rigid cohomology, and obtaind an extention of Kedlaya's finite dimerisionality.Kimura developed the notion of finite dimensionality of a morphism of pure motives, and constructed many non-trivial examples of finite dimensional motives.Sigeyuki Morita studied the mapping class group and the derivation algebra of nilpotent fundamental groups, and obtained cohomological results and conjectures on the image of Soule elements. The paper was published in Proc. Sympos. Pure Mathematics.Matsumoto utilized the geometry of formal power series and Galois theory, to designe a new fast pseudorandom number generator taking full advantage of paralellism of recent CPUs. A paper is accepted in Proceedings of MCQMC2006.