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

这项研究主要集中在矩阵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曲面的谱理论中的应用。我们计划利用早先发展的代数几何方法来解决克尔黑洞与旋转半无限物质圆盘叠加对应的边值问题，并找到克尔黑洞的周期模拟，从而确定爱因斯坦方程新的物理意义上的精确解。我们计划进一步发展具有两个极化的量子爱因斯坦-罗森波理论，目的是在量子引力的表述方面取得新的进展。我们还计划研究厄米矩阵模型中与物理相关的渐近展开式，这些展开式可用于随机矩阵理论从统计学到量子引力的许多应用领域。最后，我们希望对具有平凡完整群和有限完整群的平坦Riemann曲面上的Laplace算子的谱理论作出贡献。

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.