Spectra and Geometric structure of manifolds and graph

我们研究了覆盖空间上热核的渐近性。我们的结果在一定程度上推广了关于交换覆盖的结果。屋顶的某些点也是扩张的，在这个意义上，相关的算子可以用表示理论来分解。然而，我们需要克服以下几个难点。由于离散幂零群不是第I类群，因此没有可用的表示理论。我们把它嵌入到李群中，并利用它的表示理论。然而，在不可约分解中处理无限维表示仍然存在困难。接下来，我们通过迭代积分将上面的内容与算子的分解联系起来。然后利用半经典分析，推导出有限维“分解”在某种意义上。此外，我们还研究了几何不等式(Sakai)、磁场下的散射理论(Tamura)、等谱流形(Ikeda)、非线性半群(Tanaka)、位形空间的拓扑学(Shimakawa)、椭球体上的割轨迹(Kiyohara)、拓扑空间的完备性(Yoshioka)、p-Laplace ian(Takeuchi)。

We investigated the asymptotics of the heat kernels on covering spaces. Our results gives an extent on of the results for abelian coverings. Some point of roof is also extension in the sense that the related operators can be decomposed using representation, theory. However we need to overcome several difficult points as follows. Since the discrete nilpotent group is not type I, there is no available re resentation theory. We embed it into the Lie group and use its representation theory. However there still remain difficulties to treat infinite dimensional representations in the irreducible decomposition. Next we connect the above with decomposition of operators through the iterated integrals. Then we use semi-classical analysis and deduce the finite dimensional "decomposition" in some sense. We have also done the preparation for more detailed asymptotic expansion and an extension to solvable group case.Others investigate the followings projects except for collaborating the above ; Geometric inequalities(Sakai), Scattering theory under magnetic fields(Tamura), Isospectral manifolds(Ikeda), Nonlinear semigroup(Tanaka), Topology of configuration spaces(Shimakawa), Cut locus on ellipsoid(Kiyohara), Completeness of TopologicalSpaces(Yoshioka), p-Laplacians(Takeuchi).