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

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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.