Mirror symmetry for flag varieties

在研究多变量多项式方程的解集时，代数和几何结合在一起--所谓的代数族。我的研究重点是国旗品种，这是一种特别漂亮的代数品种，具有非常严格的结构。例如，它们是“同质的”：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。各种国旗从最简单的例子--黎曼球体开始，到任意高度的维度。镜像对称在20世纪90年代初引起了数学家的注意，当时物理学家们对其上的某些三维代数簇和有理曲线的数量做出了惊人而精确的预测。从那时起，解开这些预测背后的东西的过程一直在进步，这个领域已经成为现代数学的一个主要部分。在我的研究中，我建议在旗帜变体的背景下研究镜像对称性。关于FLAG簇的量子上同调理论已经有了大量的工作，它起源于镜像对称，本身就是一个非常丰富的学科。但这项研究几乎完全是从经典的量子上同调的角度来推广普通上同调，根本不涉及镜像对称性。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了这种模型，它以完全不同的形式保存了关于枚举代数几何的信息。在我最近的论文[14]中，我为一般的FLAG簇明确地提出了这样一个模型，并将它与量子上同调联系起来。拟议研究的主要目标是利用这一镜像模型来理解更深层次、更困难的国旗品种格罗莫夫-维滕理论。

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.