Geometric structure and topology of manifolds and graphs

本文致力于与流形和图的谱几何相关的下列课题的研究：(1)图的谱几何：我们证明了欧氏空间中域的著名的Faber-Krahn不等式在图中的类似，并找到了构造关于组合拉普拉斯的等谱对的两种方法。这些都是参考文献的内容。此外，我开始研究Heisenberg群在有限图的无限覆盖上定义的随机游动的渐近行为。(2)Gel‘fand逆谱问题的稳定性：这个项目是Y.V.Kurylev和M.Lassas共同完成的。Gel‘fand逆谱问题是由纽曼-拉普拉斯流形的谱和边值确定一个具有边界的黎曼流形。这是通过Belishev和Kurylev的结果结合Tataru的近似可控性结果来解决的。那么，下一个挑战之一就是稳定性。我们首先得到了这类流形的稳定性结果，包括曲率导数的条件，然后成功地去掉了它。此外，我们还得到了带边界流形中调和坐标的存在性结果。这些结果在很大程度上依赖于紧性论证，因此不能给出有效的估计。然而，我们也有有效估计的部分结果。

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.