LOCALIZATION, STRING DUALITY AND MODULI SPACES

摘要奖：DMS-0705284主要研究人员：科风刘模空间在从几何、拓扑、代数几何到数论的许多数学学科中都扮演着重要的角色。弦理论中的弦对偶性给出了许多深刻的数学结果和理论。主要研究人员建议将局部化方法与其他新发展的几何技术相结合来深入研究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。在所提出的研究经费的支持下，我们希望解决几个具体的问题，包括作为特例的一般重言类的交数的一般递推公式，一般射影流形的Virasoro猜想，对拓扑学的理解，模空间的代数几何，以及利用我们关于模空间的几何结果来计算全纯反常。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作为自然界所有基本力量大一统的最有希望的候选者，它应该包括所有以前的理论，如杨-米尔斯和陈-西蒙斯理论。弦对偶在弦理论中识别不同的理论，已经产生了许多关于黎曼曲面的模空间的几何和拓扑的令人惊讶的美丽的数学公式。许多猜想公式的数学证明在很大程度上依赖于本地化技术。这个程序不仅将有助于验证重要的物理理论，而且还将产生美丽的基础数学。在实施该项目时，我们还将培训几名年轻的学生和博士后，通过合作和讲座在这些主题上进行研究。

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.