Integrable systems, moduli spaces and spectral geometry

拟议研究的目标是将起源于可积系统理论（Riemann-Hilbert问题，τ函数）的方法应用于模空间，经典引力和谱几何。在该地区的模空间，我们将解决以下问题：分析推导之间的关系各种上同调和同源类（维滕周期）模空间的黎曼曲面;了解“墙交叉”行为的tau函数模空间的二次和n-微分黎曼曲面，和他们的变换下的五角星移动;应用霍奇积分。这也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Kerr解的周期性类似物，解析地说，这样的解应该对应于Ernst方程的无限孤子链。 在谱几何领域，我们希望发展一个数学上严格的理论，任意平坦度量的拉普拉斯算子行列式的黎曼曲面上的圆柱出口和研究极值问题的拉普拉斯算子行列式的存在下的圆锥奇点。

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.