Construction, singularities and porlarlization of moduli space of metric spaces via Quantum statistical mechanics

Research Project

Project/Area Number	26610017
Research Category	Grant-in-Aid for Challenging Exploratory Research
Allocation Type	Multi-year Fund
Research Field	Geometry
Research Institution	Kyushu University
Principal Investigator	OTSU Yukio 九州大学, 数理学研究院, 准教授 (80233170)
Project Period (FY)	2014-04-01 – 2018-03-31
Project Status	Completed (Fiscal Year 2017)
Budget Amount *help	¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000) Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000) Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000) Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords	Differential geometry / モジュライ空間 / Gromov-Hausdorff 距離 / 量子統計力学 / リーマン幾何 / ラプラシアン / 幾何構造の変形 / グロモフハウスドルフ距離 / ランダム離散化 / グロモフ･ハウスドルフ距離 / 経路積分 / 温度グリーン関数
Outline of Final Research Achievements	By the method of random discretization we considered the discrete Laplacian of nets on a compact Riemannian manifold or Alexandorv space. Using the Laplacian we constructed the Hamiltonian ofvibrations on net and its quantum statistical mechanics. By perturbation expansion of its thermodynamical functions we introduced a family of the Riemannian metrics on the bulk moduli space of the pairs of space and net. Then we studied its relation to the Gromov-Hausdorff distance and several classical moduli spaces.