Studies of Algebraic and Analytic Varieties

Kyoji Saito将紧致Riemann曲面的Teichmiller空间描述为定义在Z上的仿射代数概型的真实的点簇，其复杂结构通过推广Eichler积分由局部系统的无穷序列描述。Masaki Kashiwara提出了q = 0时量子群基的一般框架，称之为晶体基。然后与三和哲二等人一起，他证明了晶体基等价于可解晶格模型的一个点函数中出现的路径。特别是，对于顶点模型，他们建立了一个点函数成为字符。斋藤盛彦开发了一个理论的霍奇模块使用D-模块理论作为类似的解决方案的韦尔猜想由于Delinge。然后，他给了它的几个几何应用，如非孤立奇点的b-函数的研究和Dimca的猜想多项式映射。Isao Naruki已经表明，交换曲面没有一个主要的极化，但与极化的Pfenchan度3承认，相关的库默表面可以嵌入到P^3作为一个四次曲面。给出了WKB方法和Borel变换的微观局部分析基础。然后给出了几个简单转折点情况下的规范形，并说明了高阶方程中基于某些交点的一些奇异现象。Araki在不定度规的相对论量子理论框架下给出了局域规范不变性和BRS上同调的一个一致的表述，并阐明了它们的自然意义.正自伴算子幂的真值

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.