The study of Low-dimensicnal manifolds with various geometric structures

首席研究员UE继续研究3维和4维流形。特别是，他考虑了有理同调3-球面的Fukumoto-Furuta不变量和由Heegaard Floer同调定义的Ozsvath-Szabo d-不变量。他证明了这两个不变量对于球面3-流形和某些垂直型3-流形都是一致的。这意味着在这样的条件下，这两个不变量都是经典Rochlin不变量的积分〓，也是同调余边不变量的积分不变量。但它们并不是普遍意义上的，是否存在满足上述两个条件的不变量还是个未知数。他还利用上述不变量给出了边界Seifert有理同调3-球面的4-流形的签名的某些约束。研究人员加藤刚用Yang-Milis规范理论估计了嵌入在K3曲面上的Casson柄的增长，并用映射族分析了迭代的熵。Fujii发现了双曲2锥流形变形引起的常微分方程奇点的汇合现象，并用超几何函数刻画了双曲3锥流形上的调和矢量场。Shin‘ichi Kato建立了Jaquet定理的相对=对称空间版本，该定理声称每个p-adi约化群的不可约可容许表示都嵌入到抛物子群的不可约尖点表示的诱导表示中。Ushiki考虑了Riemann球面上复动力系统诱导的分布空间上的海侵算子，证明了Fredholm型行列式由Artin-Mazurzeta函数表示。

The head investigator Ue continued the research of 3 and 4-manifolds. In particular he considered Fukumoto-Furuta invariant for rational homology 3-spheres coming from Seiberg-Witten theory (which coincides with the Neumann-Siebenmann invariant in case of plumbed 3-manifolds) and Ozsvath-Szabo's d-invariant defined by Heegaard Floer homology. He showed that these two invariants coincide for spherical 3-manifolds, and also for certain plumbed 3-manifolds. This implies that under such conditions both invariants are the integral 〓 of the classicalRochlin invariant and also homology cobordism invariants. But they are not the sane in general, and it is sill open whether there exists an invariant satisfying the above two conditions. He also gave certain constraits for the signature of 4-manifolds bounding Seifert rational homology 3-spheres in terms of the above invariants.The investigator Tsuyoshi Kato estimated the growth of the Casson handles embedded in the K3 surface by Yang-Milis Gauge theory, and also analized the entropy of iterations by families of maps. Fujii found the confluence phenomena of singular points of ordinary differential equations induced by deformations of hyperbolic 2-cone-manifolds, and descried harmonic vector fields on hyperbola 3-cone-manifolds in terms of hypergeometric functions. Shin'ichi Kato established the relative = symmetric space version of Jaquet's theorem, which claims that every irreducible admissible representation of p-adic reductive groups is embedded to an induced representation for irreducible cusp representation of a parabolic subgroup. Ushiki considered transgression operators over the space of distributions induced by complex dynamical systems over the Riemann sphere and showed that the Fredholm determinant is represented by Artin-Mazurzeta function.