Applications of Modular Invariance in Geometry and Topology

模形式和模不变性最近出现在几何和拓扑的许多领域。例如，它们出现在循环空间的指标理论中；Donaldson和seiberg - witten不变量的计算；镜面对称有理曲线的计数问题。它们也出现在数学物理和3流形拓扑不变量的构造中。研究者认为，系统地研究这些新现象，并找到模形式和模不变性在几何和拓扑中的进一步应用是非常重要的。模不变性应该是理解有限维流形（如循环空间）的几何和拓扑的关键。研究者希望将模不变性与几何学和拓扑学中的许多经典技术结合起来，解决几何和拓扑学中与模形式相关的几个问题。模不变性，被爱德华·威滕（EdwardWitten， I.A.S.）称为自然界的对称性，经常产生于数学物理，特别是弦理论。有趣的是，弦理论中最美丽的数学对象总是具有模不变性的性质。虽然模不变性已经在几何和拓扑中有了一些非常有趣的应用，但深入的几何理解可能会告诉我们，模不变性也是数学和物理之间的对称性

9803234Liu Modular forms and modular invariance have recently appeared in manyareas of geometry and topology. For examples, they showed up in theindex theory of loop space; in the computation of Donaldson andSeiberg-Witten invariants; in the counting problems of rational curvesin mirror symmetry. They have also appeared in mathematical physicsand in the construction of topological invariants of 3-manifolds. Theinvestigator considers it very important to study these new phenomenasystematically and to find further applications of modular forms andmodular invariance in geometry and topology. Modular invarianceshould be the key to the understanding of the geometry and topology ofinfinite dimensional manifolds, such as loop spaces. The investigatorhopes to combine modular invariance and many classical techniques ingeometry and topology to solve several problems in geometry andtopology related to modular forms. Modular invariance, which is called the symmetry of nature by EdwardWitten (I.A.S.), quite often arises from mathematical physics, inparticular, from string theory. It is interesting to notice that themost beautiful mathematical objects in string theory always have theproperty of modular invariance. While modular invariance already hassome very interesting applications in geometry and topology, a deepand geometric understanding may tell us that modular invariance isalso the symmetry between mathematics and physics.***