Algebro-geometric methods in integrable systems, random matrices and spectral geometry

可积系统、随机矩阵和谱几何中的代数几何方法

• 批准号: 227154-2010
• 负责人: Korotkin, Dmitry
• 金额: $ 1.75万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2012
• 资助国家: 加拿大
• 起止时间: 2012-01-01 至 2013-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=508764
• 关键词: Algebro; geometric; methods; integrable; systems

This research proposal is mainly focused on the application of the theory of matrix Riemann-Hilbert problem, algebro-geometric methods and the theory of quantum groups to models of classical and quantum gravity, Frobenius manifolds, the theory of random matrices and spectral theory of Riemann surfaces. We plan to use the algebro-geometric methods developed earlier to solve boundary value problem corresponding to superposition of Kerr black hole with rotating semi-infinite matter disc, and to find a periodic analog of Kerr black hole determining thereby new physically important exact solutions of Einstein's equations.We plan to develop further the theory of quantum Einstein-Rosen waves with two polarization with the aim of making new advances towards a formulation of quantum gravity. We also plan to study physically relevant asymptotic expansions in hermitian matrix models, which could be used in numerous areas of applications of the random matrix theory from statistics to quantum gravity. Finally, we hope to contribute to the spectral theory of Laplace operator on flat Riemann surfaces with trivial and finite holonomy groups.

本研究计划主要集中在矩阵Riemann-Hilbert问题理论、代数几何方法和量子群理论在经典和量子引力模型、Frobenius流形、随机矩阵理论和Riemann曲面谱理论中的应用。我们计划利用已有的代数几何方法求解Kerr黑洞与旋转半无限物质盘叠加的边值问题，寻找Kerr黑洞的周期性类似物，从而确定爱因斯坦方程的新的物理意义的精确解，进一步发展量子爱因斯坦理论，罗森波与两个偏振的目的是使新的进展，制定量子引力。 我们还计划研究厄米矩阵模型中的物理相关渐近展开，这些渐近展开可以用于随机矩阵理论从统计到量子引力的许多应用领域。最后，我们希望对具有平凡和有限完整群的平坦Riemann曲面上的拉普拉斯算子的谱理论有所贡献。