Infinite-Dimensional Integrable Systems and Moduli Spaces of Riemann Surfaces

AbstractAward：DMS-9971371首席研究员：Motohico Mulase拟议的研究是针对黎曼曲面的模理论问题，代数数域上定义的代数曲线，Hermitian矩阵积分的几何，以及可积的非线性偏微分方程。目的是：（1）在紧致Riemann曲面的拓扑模型上，用图的变形给出曲面的无穷小变形的显式组合描述。这将建立由Strebel理论得到的点Riemann曲面的模空间的可微orbifold结构与复数域上的代数几何构造之间的规范的同构。(2)为了找到带状图和黎曼曲面之间关系的代数和组合描述。本文研究复边长带图的代数几何对应。这一研究有望发现带状图与黎曼曲面之间关系的代数描述。(3)建立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，这些解不是由广义高阶Krichever构造得到的.这些矩阵积分包含了尖黎曼曲面模空间的轨道欧拉特征的母函数，时空中任何弦状物体的运动在扫过某个区域时都导致黎曼曲面。例如，我们体内的DNA链来自我们的祖先。我们可以想象一个DNA环，它是很久以前创造的，一直在时间中旅行，现在存储在我们的细胞中。这个特殊DNA的整个历史是由黎曼曲面描述的。表面的时间切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，最终与我们的DNA的当前结构的集合。这个集合就是黎曼曲面的模空间。通过对模空间的研究，更深入地分析我们基因的所有可能的过去，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项交织着代数、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.