Moduli Spaces, Quivers, and Duality

对偶性在数学和物理学中都起着重要的作用。对偶性提供了一对典型的完全不同的理论的各种对象之间的对称对应。这一点很重要，因为在对偶理论下，一个理论中的难题往往对应于对偶理论中的一个简单问题。两个重要的和具有挑战性的对偶性是朗兰兹对偶性，它起源于数论，镜像对称性，它首先由理论物理学家发现，并在数学中发挥着越来越重要的作用。这两种二重性都没有得到充分理解。本项目将研究这两种双重性。预计这项工作将在看似无关的主题之间建立新的和意想不到的联系。本研究计画将探索一种新的方法来研究与广泛类别相关的模空间上的指数和与积分。在模空间的情况下，该项目的目的是开发一个新的证明卡茨猜想以及一个新的公式，其中涉及一个潜在的，这是密切相关的唐纳森托马斯不变量介绍的Kontsevich和Soibelman。该项目还将研究一个类似的计数问题的模空间的抛物（向量）丛的投影代数曲线与穿孔。

Dualities play a fundamental role in both mathematics and physics. A duality provides a symmetric correspondence between various objects of a pair of typically quite different theories. This is important because a difficult problem in one theory often corresponds, under the duality, to a simpler problem in the dual theory. Two important and challenging dualities are Langlands duality, which has origins in number theory, and mirror symmetry, which was first discovered by theoretical physicists and is playing an increasingly important role in mathematics as well. Neither of these two dualities is fully understood. Both of these dualities will be studied in this project. It is anticipated that the work will establish new and unexpected links between seemingly unrelated topics. This research project will explore a new approach to the study of exponential sums and integrals over moduli spaces associated to a wide class of categories. In the case of the moduli space of quiver representations, the project aims to develop a new proof of the Kac conjecture as well as a new formula, which involves a potential and which is closely related to the Donaldson-Thomas invariants introduced by Kontsevich and Soibelman. The project will also investigate a similar counting problem for the moduli space of parabolic (vector) bundles on a projective algebraic curve with punctures.