Geometry of Moduli Spaces

AbstractAward：DMS-0604471项目负责人：孙晓峰在本项目中，项目负责人提出研究黎曼曲面的模空间几何和黎曼曲面上的稳定向量丛。这些模空间是数学和物理学许多分支的中心。PI在前人工作的基础上，首次提出了在黎曼曲面的模空间上证明Ricci度量和Kahler-Einstein度量等几种典型度量的Mumford优性。这些度量的优良性将允许人们将Chern-Weil理论应用于这些非紧的模空间。PI希望通过微分几何的方法，利用优度来研究模空间上的交理论。为了研究Kahler-Einstein度规的优良性，除了连续性方法外，PI还提出了研究模空间上的Kahler-Ricci流。更一般地说，PI提议研究非紧和准投射流形上的Kahler-Ricciflow，其中初始度量的势函数是无界且温和增长的。PI证明了Kahler-Ricci流下的Kahler类和初始度量的优良性。同时，还研究了Kahler-Ricci流的长时间存在性和收敛性.此外，PI还提出了研究模空间上各种度量的负性和调和映射理论，从而得到各种消失定理和刚性定理。PI还提出将其方法推广到稳定丛的模空间，并利用正则度量研究消失定理，黎曼曲面的模空间理论在数学、物理和计算机图形学中有着广泛的应用。模空间上的交理论是现代弦理论的核心内容之一。此外，它在图像识别、压缩和解压缩等方面也有很大的改进。传统的方法是通过拾取数百万个顶点并记录组合数据来对曲面进行三角剖分。因此，需要进行大量的计算。在微分几何中，Torelli定理告诉我们曲面的复杂结构是由它的Hodge结构决定的。因此，人们只需要记录表面的Hodge结构的数据，以便重建表面。Hodge结构的数据量远小于三角剖分的数据量。因此，一种有效的曲面与其Hodge结构的连接方法将大大简化图像处理的过程，提高其效率和精度。

AbstractAward: DMS-0604471Principal Investigator: Xiaofeng SunIn this project, the principal investigator proposed to study thegeometry of moduli spaces of Riemann surfaces and stable vectorbundles over a Riemann surface. These moduli spaces are thecenter of many branches of mathematics and physics. Continuinghis previous works, the PI first proposed to prove the Mumfordgoodness of several canonical metrics including the Ricci metricand the Kahler-Einstein metric on the moduli space of Riemannsurfaces. The goodness of these metrics will allow one to applyChern-Weil theory to these moduli spaces which arenoncompact. The PI hope to develop a method to study theintersection theory on the moduli spaces via differentialgeometry by using the goodness. To study the goodness of theKahler-Einstein metric, except for the continuity method, the PIproposed to investigate the Kahler-Ricci flow on the modulispaces. More generally, the PI proposed to study the Kahler-Ricciflow on non-compact and quasi-projective manifolds where thepotential function of the initial metric is unbounded with mildgrowth. The PI proposed to prove that the Kahler class and thegoodness of the initial metric are preserved under theKahler-Ricci flow. Also, the long time existence and convergenceof the Kahler- Ricci flow will be studied. Furthermore, the PIproposed to investigate the negativity of various metrics on thismoduli space and harmonic map theory which will lead to variousvanishing and rigidity theorems. The PI also proposed to extendhis methods to the moduli space of stable bundles and studyvanishing theorems by using canonical metrics.The theory of moduli space of Riemann surfaces has manyapplication in mathematics, physics and computer graphics. Theintersection theory on the moduli spaces is one of the key partsin the modern string theory. Furthermore, it leads to significantimprovements in image recognition, compression anddecompression. The traditional method is to triangulate a surfaceby picking millions of vertices and record the combinatoricdata. Thus a huge amount of calculations are needed. Indifferential geometry, the Torelli theorem tells us that thecomplex structure of a surface is determined by its Hodgestructure. Thus one only need to record the data of the Hodgestructure of a surface in order to reconstruct the surface. Thesize of the data of the Hodge structure is much smaller than thesize of the data from triangulation. Thus a numerically efficientmethod to link a surface and its Hodge structure willdramatically simplify the procedure of image processing andimprove its efficiency and accuracy.