Geometric structure and topology of manifolds and graphs

本课题主要研究流形和图的谱几何。(1)图的谱几何：我们证明了欧几里得空间域上著名的Faber-Krahn不等式在图中的类似性，并找到了关于组合拉普拉斯图的等谱对的两种构造方法。这些是参考资料的内容。此外，我开始研究由Heisenberg群定义在有限图的无限覆盖上的随机漫步的渐近行为。(2) Gel’fand逆光谱问题的稳定性：与Y. V. Kurylev和M. Lassas合作完成。Gel'fand逆谱问题是由谱和诺伊曼拉普拉斯算子的边值确定一个有边界的黎曼流形。Belishev和Kurylev的结果结合Tataru的近似可控性结果解决了这个问题。接下来的挑战之一就是稳定性。我们首先得到了包含曲率导数条件的流形类的稳定性结果，然后成功地去掉了它。此外，我们还得到了有边界流形中调和坐标的存在性结果。这些结果严重依赖于紧性参数，因此无法给出有效的估计。然而，我们也有部分有效估计的结果。

The present project has been devoted to the study on the following subjects related to spectral geometry of manifolds and graphs.(1) Spectral geometry of graphs : We have proved the analog in graphs of the celebrated Faber-Krahn inequlity for domains in Euclidean spaces and find two methods of construction of isospectral pairs of graphs with respect to combinatorial Laplacian. These are contents of References. Moreover, I start to investigate aymptotic behavior of random walks defined on infinite cover of finite graphs by the Heisenberg group.(2) Stability of the Gel'fand inverse spectral problem : This project is joint work with Y. V. Kurylev and M. Lassas. The Gel'fand inverse spectral problem is to determine a Riemannian manifold with boundary from the spectrum and the boundary value of the Neuman Laplacian. This is solved by results Belishev and Kurylev combining the approximate controllabity results by Tataru. Then, one of next challenge is the stability. We first obtained stability results in the class of manifolds including the condition on the derivative of curvature and later, succeed to remove it. Moreover, we have obtained the existence results of harmonic coordinates in manifolds with boundary. These results are heavily depend on compactness arguments and thus, no effective estimate can not be given. However, we also have partial results foreffective estimates.