LOCALIZATION, STRING DUALITY AND MODULI SPACES

AbstractAward：DMS-0705284首席研究员：刘克峰模空间在从几何、拓扑、代数几何到数论的许多数学学科中起着基础性的作用。弦论的弦对偶性激发了许多深刻的数学结果和理论。主要研究者建议深入研究应用局部化方法结合其他新发展的几何技术来解决由弦对偶引起的关于黎曼曲面和稳定映射的模空间的几何和拓扑的基本问题。我们希望在拟议的研究基金的支持下解决几个具体问题，其中包括一般重言类的交集数的一般递归公式，其中包括作为特例的Faber猜想、一般射影流形的Virasoro猜想、拓扑学的理解、模空间的代数几何，数学和物理的相互作用推动了这两个学科的许多基本进展，弦论作为自然界中所有基本力的大统一的最有希望的候选者，应该包括所有以前的理论，如Yang-Mills理论和Chern-Simons理论。弦论中的弦对偶性是不同理论之间的区别，它产生了许多关于黎曼曲面模空间的几何和拓扑的令人惊讶的美丽的数学公式。这些数学公式的数学证明主要依赖于局部化技术。这一计划不仅有助于验证重要的物理理论，而且也产生美丽的和基本的数学。在实施该项目的过程中，我们还将培养一些年轻的学生和博士后，通过合作和讲座的方式进行这些学科的研究。

AbstractAward: DMS-0705284Principal Investigator: Kefeng LiuModuli spaces have played fundamental roles in many subjects ofmathematics from geometry, topology, algebraic geometry, to numbertheory. String duality from string theory has motivated many deepmathematical results and theories. The principal investigatorproposes to have an intensive study on applying localization methodcombined with other newly developed geometric techniques to solvefundamental problems arisen from string duality about the geometryand topology of moduli spaces of Riemann surfaces and stable maps.There are several specific problems that we hope to solve under thesupport of the proposed research grant, these include the generalrecursion formulas for intersection numbers of general tautologicalclasses which include the Faber conjecture as special case, the Virasoro conjecture for general projective manifolds,the understanding of topology, algebraic geometry of moduli spaces,and the computing of the holomorphic anomaly by using our geometricresults about moduli spaces.The interactions of mathematics and physics have motivated many fundamental advances in both disciplines.String Theory, as the most promising candidate for the grandunification of all fundamental forces in the nature, should includeall previous theories like the Yang-Mills and the Chern-Simonstheory. String duality which identifies different theories in stringtheory has produced many surprisingly beautiful mathematicalformulas about the geometry and topology of moduli spaces of Riemannsurfaces. The mathematical proofs of many of these conjecturalformulas depend crucially on localization techniques. This programwill not only help verify important physical theories, but alsoproduce beautiful and fundamental mathematics. In carrying out theproject we will also train several young students and post-doctors to conduct research inthese subjects through collaboration and lectures.