Integrable systems, moduli spaces and spectral geometry

提出的研究目标是将可积系统理论（黎曼-希尔伯特问题，tau函数）中的方法应用于模空间，经典重力和光谱几何。在模空间中，我们将讨论以下问题：黎曼曲面模空间上各种上同调和同调类（Witten环）之间关系的解析推导；黎曼曲面上二次和n阶微分模空间上tau函数的“过壁”行为及其在五角形运动下的变换；霍奇积分的应用。这也将有助于进一步阐明簇代数在模空间理论中的作用。在经典引力领域，主要目标是找到克尔解的周期模拟；解析上，这种解对应于恩斯特方程的无限孤子链。在光谱几何领域，我们希望发展具有圆柱出口的黎曼曲面上任意平坦度规拉普拉斯算子行列式的数学严谨理论，并研究圆锥奇点存在下拉普拉斯算子行列式的极值问题。

The goal of the proposed research is to apply methods originating in the theory of integrable systems (Riemann-Hilbert problems, tau-functions) to  moduli spaces, classical gravity and spectral geometry. In the area of moduli spaces we are going to address the following problems: analytical derivation of relations between various cohomology and homology classes (Witten cycles) on moduli spaces of Riemann surfaces; understanding of the "wall-crossing" behaviour of tau-functions on moduli spaces of quadratic and n-differentials on Riemann surfaces, and their transformation under the pentagram moves; application to Hodge integrals. This should help also to further elucidate the role of cluster algebras in the theory of moduli spaces. In the area of classical gravity the main goal is to find a periodic analog of Kerr solution; analytically such solution should corresponds to an infinite chain of solitons of Ernst equation.  In the area of spectral geometry we hope to develop a mathematically rigorous theory of determinants of Laplace operators in arbitrary flat metrics over Riemann surfaces with cylindrical outlets and study extremal problems for determinant of Laplacian in presence of conical singularities.