Infinite-Dimensional Integrable Systems and Moduli Spaces of Riemann Surfaces

主要研究方向为黎曼曲面的模理论、代数数域上定义的代数曲线、厄米矩阵积分几何以及可积非线性偏微分方程。目标是(1)用曲面拓扑模型上的图的变形给出紧致黎曼曲面的极小变形的显式组合描述。这将建立由Strebel理论得到的点黎曼曲面模空间的可微轨道结构与复数域上的代数几何构造之间的正则微分同态。(2)找到条带图与黎曼曲面关系的代数组合描述。提出了研究具有复边长度的带图的代数-几何对应物。这一研究有望导致带状图和黎曼曲面之间关系的代数描述的发现。(3)建立了KP方程的超越解理论。一类矩阵积分给出了KP方程的全新解，这些解是不能用广义高阶krichhever构造得到的。这些矩阵积分包括点黎曼曲面模空间的轨道欧拉特征的生成函数。任何类弦物体在时空中的运动，当它扫过某个区域时，都会导致阿里曼表面。例如，我们体内的DNA链来自我们的祖先。我们可以想象很久以前创造的aDNA循环，一直在时间中旅行，现在储存在我们的细胞中。这个特殊DNA的整个历史是用黎曼曲面来描述的。表面的时间片是这个特定时间的DNA。我们不知道我们基因的确切过去。但我们可以考虑所有可能的情况的集合，最终形成我们DNA的当前结构。这个集合是黎曼曲面的模空间。通过研究模空间，对我们所有可能的基因过去进行更深入的分析，有望使我们更好地理解DNA的分子进化。由于它与高能物理中的弦理论的联系以及在DNA分子进化中的潜在应用，研究生和本科生对这一交织代数、几何和分析的研究表现出浓厚的兴趣。

AbstractAward: DMS-9971371Principal Investigator: Motohico MulaseThe proposed research is directed to problems in moduli theory ofRiemann surfaces, algebraic curves defined over the field ofalgebraic numbers, geometry of Hermitian matrix integrals, andintegrable nonlinear partial differential equations. The goalsare (1) To give an explicit combinatorial description ofinfinitesimal deformations of a compact Riemann surface in termsof deformations of graphs on the topological model of thesurface. This will establish a canonical diffeomorphism betweenthe differentiable orbifold structure of the moduli space ofpointed Riemann surfaces that is obtained by the Strebel theoryand the algebro-geometric construction over the field of complexnumbers. (2) To find an algebraic and combinatorial descriptionof the relation between ribbon graphs and Riemann surfaces. It isproposed to study an algebro-geometric counterpart of a ribbongraph with complex edge length. This study is expected to lead toa discovery of algebraic description of the relation betweenribbon graphs and Riemann surfaces. (3) To establish a theory oftranscendental solutions of the KP equations. A class of matrixintegrals give totally new solutions of the KP equations whichare not obtained by the generalized higher-rank Kricheverconstruction. These matrix integrals include generating functionsof the orbifold Euler characteristics of the moduli spaces ofpointed Riemann surfaces.The motion of any string-like object in space-time leads to aRiemann surface as it sweeps out some region. The DNA strands inour body, for example, come from our ancestors. We can imagine aDNA loop, that was created a long time ago, has been traveling intime and now stored in our cell. The whole history of thisparticular DNA is described by a Riemann surface. The time sliceof the surface is the DNA at this particular time. We do not knowthe exact past of our genes. But we can consider the collectionof all possible scenarios that would end up with the currentstructure of our DNA. This collection is the moduli space ofRiemann surfaces. Deeper analysis of the collection of allpossible past of our genes through studying the moduli spaces isexpected to lead us to a better understanding of molecularevolution of DNA. Because of its connections to string theory inhigh energy physics and potential applications to molecularevolution of DNA, graduate and undergraduate students show stronginterests in this research, which interweaves algebra, geometryand analysis.