TOPOLOGICAL MIRROR SYMMETRY

理论物理中的对偶性是从提出的与另一个通常更容易理解的物理理论的对偶性中获得关于物理模型的信息的重要工具。其中一种二元性是镜像对称性，这是一种源于弦理论的二元性理论。这种二元性的数学含义是多方面的。在拟议的项目中，我们感兴趣的是将这些数学含义与来自数论和表象理论的想法联系起来。也就是说，我们建议在一些有限矩阵群的特征标表中寻找模式，从朗兰兹对偶的角度解释这种镜像对称性。后者是现代数论中的一个庞大的程序，在特殊情况下，安德鲁·威尔斯的工作包含了费马大定理。在这个方案中，我们通过研究有限矩阵群的特征标表，将这两个看似遥远的对偶联系在一起：弦理论中的镜像对称性和数论中的朗兰兹对偶。

Dualities in theoretical physics are important tools to gain information on a physical model from its proposed duality with another typically more accessible physical theory. One such duality is mirror symmetry, which is a duality theory stemming from string theory. The mathematical implications of this duality are manifold. In the proposed project we are interested in relating these mathematical implications with ideas coming from number theory and representation theory. Namely, we propose to find patterns in the character tables of some finite matrix groups which explain this mirror symmetry from the perspective of Langlands duality. This latter is a vast program in modern number theory which in a special case implies Fermat's Last Theorem by the work of Andrew Wiles. In this proposal we are connecting via the study of the character tables of finite matrix groups, these two seemingly far dualities: mirror symmetry in string theory, and Langlands duality in number theory.