Mirror symmetry for flag varieties

当研究多元多项式方程的解集时，代数和几何就结合在一起了，即所谓的代数簇。我的研究集中在旗簇，这是一个非常严格的结构，特别美丽的代数簇。例如，它们是“对称的”：它们有一个大的（矩阵）对称群，可以将任何点转换为任何其他点。旗形的种类从最简单的例子黎曼球面开始，一直到任意高维。镜像对称在20世纪90年代初引起了数学家的注意，当时物理学家对某些三维代数簇和它们上的有理曲线的数量做出了惊人而精确的预测。从那时起，揭示这些预测背后的过程一直在进步，该领域已成为现代数学的重要组成部分。在我的研究中，我建议研究镜像对称的背景下，旗品种。关于旗簇的量子上同调理论，已经有大量的研究工作。旗簇产生于镜像对称，本身就是一个非常丰富的课题。但这项研究几乎完全是从量子上同调的经典角度推广普通上同调，根本没有涉及镜像对称。被遗漏的是神秘的镜像模型，物理学家在计算有理曲线的数量时使用的模型，它以完全不同的形式保存了枚举代数几何的信息。在我最近的论文[14]中，我明确地为一般的旗簇提出了这样一个模型，并将其与量子上同调联系起来。建议的研究的主要目标是使用这个镜像模型，以了解更深层次和更困难的Gromov-Witten理论旗品种。

Algebra and geometry come together when studying the solution sets of polynomial equations in many variables -- so-called algebraic varieties. My research centers on flag varieties, which are particularly beautiful algebraic varieties with a very rigid structure. For example they are 'homogeneous': They have a large (matrix) symmetry group which can translate any point into any other. Flag varieties come in series starting with the simplest example, the Riemann sphere, and reaching arbitrarily high dimension. Mirror symmetry came to the attention of mathematicians in the early 1990's when physicists made astounding and precise predictions about certain 3-dimensional algebraic varieties and numbers of rational curves on them. Since then the process of unraveling what underlies these predictions has been progressing, and the field has become a major part of modern mathematics. In my research I propose to study mirror symmetry in the context of flag varieties. There has already been a great deal of work on the theory of quantum cohomology for flag varieties which arose out of mirror symmetry and is a very rich subject in its own right. But this research has been almost entirely from a classical perspective of quantum cohomology generalizing ordinary cohomology, which has not involved mirror symmetry at all. What has been left out is the mysterious mirror model, what the physicists used in their computations of numbers of rational curves, which holds the information about enumerative algebraic geometry in a completely different form. In my recent paper [14] I propose such a model explicitly for general flag varieties and relate it to quantum cohomology. The main goal of the proposed research is to use this mirror model to understand the deeper and more difficult Gromov-Witten theory for flag varieties.