Riemann surfaces in geometry and analysis

黎曼曲面是一个非常有趣的对象，具有丰富的结构，涉及复杂的分析，拓扑和代数几何。为处理黎曼曲面和相关物体而开发的技术被证明是非常有用的，例如，在寻找和研究物理学中产生的重要模型微分方程的解时。 一方面，我们将通过建立定义在任意黎曼曲面上的所谓σ函数的性质，进一步发展黎曼曲面上的函数理论。 另一方面，我们计划应用现有的黎曼曲面理论的技术来寻找两个经典问题之间的新关系：Painlevé方程和Poncelet定理。Painlevé方程在具有某些自然性质的微分方程的分类中起着重要的作用。庞斯莱定理，在其最简单的情况下，给出了一个标准的台球轨迹内的椭圆是封闭的。我们希望找到这两个问题之间的新关系，然后将其推广，并在更一般的结构之间得出类似的关系。我们希望发现这些关系将使我们更好地了解所涉及的结构。 另一个项目涉及所谓的簇代数，某些组合结构有关的三角曲面。这些代数在数学中具有重要的理论意义，但它们也与现代物理学，即弦理论有联系。我们将进一步探讨它们与黎曼曲面的关系。 我们还计划使用黎曼曲面的理论，在三维空间中找到离散类似的椭球坐标线，这可以在未来导致在计算机图形学中的应用。

Riemann surfaces are fascinating objects with rich structure that involves complex analysis, topology and algebraic geometry. The techniques developed to deal with Riemann surfaces and related objects proved to be very useful, for example, in finding and studying solutions to important model differential equations arising in physics. We are going, on one hand, to develop further the theory of functions on Riemann surfaces by establishing properties of the so-called sigma-functions defined for an arbitrary Riemann surface. On the other hand, we plan to apply existing techniques of Riemann surface theory to find new relationship between two classical problems: Painlevé equations and Poncelet theorem. Painlevé equations play an important role in classification of differential equations having certain natural properties. The Poncelet theorem, in its simplest case, gives a criterion for a billiard trajectory within an ellipse to be closed. We want to find new relationship between these two problems to then generalize it and to conclude a similar relationship between more general structures. We hope that discovering these relationships will lead us to a better understanding of the structures involved. Another project deals with the so-called cluster algebras, certain combinatorial structures related to triangulations of surfaces. These algebras have theoretical importance in mathematics but they also have links to modern physics, namely to string theory. We are going to explore further their relationship with Riemann surfaces. We also plan to use the theory of Riemann surfaces to find discrete analogues of ellipsoidal coordinate lines in three dimensional space, which can in the future lead to applications in computer graphics.