Spectral theory of flat surfaces

该项目致力于研究平面的谱理论（即，即具有圆锥形、圆柱形或欧几里得端点的平坦二维黎曼流形）。平坦表面提供了一个非平凡的例子来源，在那里可以测试和改进具有奇点的流形和散射理论的一般分析方法。我们的项目的主要目标之一是调查的各种频谱特性的标量拉普拉斯在奇异平坦度量（特殊值的ζ函数，个人特征值，行列式，散射矩阵等）的基础黎曼曲面的模或相应的平坦度量的选择的关系。我们还计划研究这些谱特征（主要是拉普拉斯行列式）的极值问题，这些谱特征被认为是模空间上的泛函。潜在地，我们的项目的结果可以被用来获得关于各种模空间的几何结构的重要信息，特别是Hurwitz空间（黎曼曲面上的亚纯函数的模空间）和黎曼曲面上的亚纯微分的模空间。.

The project is devoted to the study of the spectral theory of flat surfaces (i. e.flat two-dimensional Riemannian manifolds with conical, cylindrical or Euclidean ends). Flat surfaces provide a source of nontrivial examples, where the general methods of the analysis on manifolds with singularities and the scattering theory can be tested and improved. Among the main objectives of our project is the investigation of the relation of various spectral characteristics of the scalar Laplacians in singular flat metrics (special values of the zeta-function, individual eigenvalues, determinant, scattering matrix, etc) to the moduli of the underlying Riemann surface or to a choice of the corresponding flat metric. We are also planning to study the extremal problems for these spectral characteristics (mostly the determinant of the Laplacian) considered as functionals on the moduli spaces. Potentially, the results of our project can be used to get important information about the geometric structure of various moduli spaces, in particular, the Hurwitz space (the moduli space of meromorphic functions over Riemann surfaces) and the moduli space of meromorphic differentials over Riemann surfaces. .