Studies of Algebraic and Analytic Varieties

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Kyoji Saito has described the Teichmuller space for compact Riemann surfaces as the real point variety of affine algebraic scheme defined over Z, whose complex structure is described by an infinite sequence of local systems by generalizing the Eichler integrals. Then certain global automorphic form on the space is introduced by a limit element in the configuration Hopf algebra for the discrete group.Masaki Kashiwara has developed a general frame of bases of quantum groups for the value q = 0 and called it crystal basis. Then together with Tetsuji Miwa and others, he showed that the crystal basis is equivalent to the pathes appeared in one point function for a solvable lattice model. In particular, for the vertex model they established that the one point function becomes character.Morihiko Saito has developed a theory of Hodge module by a use of D-module theory as analogous of the solution of Weil conjecture due to Delinge. Then he gave its several georhetric applications such as a study of b-functions for non-isolated singulaxities and Dimca's conjecture on polynomial maps.Isao Naruki has shown that abelian surfaces without a principal polarization but with a polarization by a Pfaffian of degree 3 admits that associated Kummer surface can be embedded into P^3 as a quartic surface.Takahiro Kawai, together with Takashi Aoki and Yoshitsugu Takei, has given a micro local analytic foundation for the WKB method and Borel transformation. Then gave normal forms for the case of a few simple turning point and make it clear that some strange phenomenon in higher order equation based on certain intersection points.Noboru Nakanishi, together with Izumi Ojima, gave a consistent formulation of local gauge invariance and BRS cohomology in the frame of relativistic quantum theory with indefinite metric and clarified their natural meanings.Huzihiro Araki has generalized the Lieb Thirring inequality on the true of powers of positive self adjoint operators.