Geometry of Moduli Spaces

摘要奖：DMS-0604471主要研究人员：孙晓峰在这个项目中，主要研究黎曼曲面的模空间和黎曼曲面上稳定向量丛的几何。这些模空间是许多数学和物理分支的中心。在前人工作的基础上，PI首次提出证明黎曼曲面的模空间上的几种正则度量的Mumford善，其中包括Ricci度量和Kahler-Einstein度量。这些度量的好性将允许人们将Chern-Weil理论应用于这些非紧的模空间。PI希望发展一种利用好的微分几何来研究模空间上的交理论的方法。为了研究Kahler-Einstein度规的优性，除了连续性方法，PI建议研究模空间上的Kahler-Ricci流。更一般地，PI建议研究非紧拟射影流形上的Kahler-Ricciflow，其中初始度量的势函数是无界的，且具有温和的增长。PI建议证明在Kahler-Ricci流下，Kahler类和初始度量的良性是保持的。此外，还将研究Kahler-Ricci流的长期存在性和收敛性。此外，PI还建议研究这个模空间上的各种度量的负性和调和映射理论，这将导致各种零化和刚性定理。PI还建议将他的方法推广到稳定丛的模空间上，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都有广泛的应用。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、压缩和解压缩方面也有显著的改进。传统的方法是通过拾取数百万个顶点来对曲面进行三角剖分，并记录组合数据。因此需要进行大量的计算。在微分几何中，托雷利定理告诉我们，曲面的复数结构是由它的霍奇结构决定的。因此，人们只需要记录曲面的霍奇结构数据，就可以重建曲面。霍奇结构的数据量比三角测量的数据量要小得多。因此，用一种高效的数值方法将曲面与其霍奇结构连接起来，将极大地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.