The study of Low-dimensicnal manifolds with various geometric structures

首席研究员我们继续研究3流形和4流形。特别地，他考虑了来自Seiberg-Witten理论的理性同调3球的Fukumoto-Furuta不变量（它与管径3流形的Neumann-Siebenmann不变量相一致）和heeggaard Floer同调定义的Ozsvath-Szabo的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3流形是重合的。这意味着在这种条件下，这两个不变量都是经典rochlin不变量和同调协不变量的积分〓。但它们在一般情况下并不相同，是否存在满足上述两个条件的不变量仍然是一个开放的问题。他还利用上述不变量给出了4流形边界塞费特有理同调3球的签名约束。研究者Tsuyoshi Kato用Yang-Milis规范理论估计了嵌入K3表面的卡森手柄的生长，并分析了图族迭代的熵。Fujii发现了双曲2锥流形变形引起的常微分方程奇点汇合现象，并用超几何函数描述了双曲3锥流形上的调和向量场。加东信一建立了Jaquet定理的相对对称空间版本，该定理认为p进约群的每一个不可约容许表示都嵌入到抛物子群的不可约尖表示的诱导表示中。Ushiki考虑了复杂动力系统在Riemann球上的分布空间上的过侵算符，并证明了Fredholm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.