Studies of Algebraic and Analytic Varieties

Kyoji Saito将紧致Riemann曲面的Teichmuller空间描述为定义在Z上的仿射代数格式的实点变数，其复杂结构通过推广Eichler积分用局部系统的无穷序列来描述。然后利用组形Hopf代数中的一个极限元，在空间上引入离散群的全局自同构形式。Masaki Kashiwara提出了q = 0的量子群基的一般框架，并称之为晶体基。随后，他与三和哲司等人一起证明了晶体基等价于可解晶格模型中出现在一个点函数中的路径。特别地，对于他们建立的顶点模型，一点函数变成了特征。Morihiko Saito利用d模理论作为deinge引起的Weil猜想的解的类比，发展了Hodge模理论。然后，他给出了它的几个几何应用，如非孤立奇点的b函数的研究和多项式映射上的Dimca猜想。Isao Naruki已经证明了没有主极化但具有3次Pfaffian极化的阿贝尔曲面承认相关Kummer曲面可以作为四次曲面嵌入到P^3中。Takahiro Kawai与Takashi Aoki和Yoshitsugu Takei共同为WKB方法和Borel变换提供了微观局部分析基础。然后给出了几个简单拐点情况的标准形式，并明确了基于某些交点的高阶方程中的一些奇怪现象。Noboru Nakanishi和Izumi Ojima在不确定度量的相对论量子理论框架下给出了局部规范不变性和BRS上同调的一致表述，并阐明了它们的自然意义。Huzihiro Araki推广了关于正自伴随算子幂为真的Lieb Thirring不等式。

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.