Topology of Algebraic Varieties

DMS-010Anatoly S. Libgober本项目提出的具体问题是非奇异和奇异簇的二元椭圆亏格和手征deRham复形的研究，二次量子化椭圆亏格和轨道折叠椭圆亏格的数学基础，镜像对称中出现的常微分方程和偏微分方程，镜像的Chern类和补的拓扑以及补的基本群的研究。本项目的目标是分析从数学的角度来看，理论物理学中使用的几种方法和技术，更确切地说，是弦理论。这样的理论研究应该会在所涉及的物理理论中产生新的结果，并对物理学家所使用的方法的有效性的原因产生更广泛的理解。我们计划解决的特定问题是由弦论中的对偶性激发的，更具体地说，是由镜像对应激发的。问题之一是找到镜像对应的新性质及其表征。这涉及拓扑学的研究，代数几何和微分方程。

DMS-010Anatoly S. LibgoberConcrete issues proposed in this project are the studyof 2-variable elliptic genus and chiral deRham complexfor non-singular and singular varieties,mathematical foundations of the second quantized ellipticgenus and orbifold elliptic genus,ODE and PDE appearing in mirror symmetry, the Chernclasses of mirrors and the topology of thecomplements as well as the study of the fundamental groupsof the complements.The goal of this project is to analyze frommathematical point of view several methods andtechniques which were used in theoretical physics,more precisely in string theory. Such mathematicalinvestigation should lead to new results inthe physical theories involved and produce abroader understanding of the reasons of effectivenessof the methods used by physicists.Particular problems which we are planning to solveare motivated by dualities in string theoryand more specifically by mirror correspondence.One of the problems is to find new propertiesof mirror correspondence and its characterization.This involved the study in topology, algebraic geometryand differential equations.