D-BRANE MODULI SPACES IN MATHEMATICS AND PHYSICS

采用D-膜模空间作为一个统一的主题，这个研究计划包括三个项目躺在代数几何和弦理论之间的接口。第一个是围绕一个新的数学结构（ADHM层不变量）产生新的数学代数关于当地的BPS不变量，量子cohomologyof mesenchants和黑洞对应。第二个项目研究局部表面上Bridgeland稳定物体模空间的cameral结构和跨壁行为，强调Donaldson-Thomas型不变量与黑洞物理之间的联系。第三部分提出了一个系统的方法，通过虚圈和抛物希格斯包的本地化拓扑规范理论中的表面运营商的交换子。更广泛的影响：数学与弦理论之间的相互作用一直是现代数学物理学的主要驱动力。这里计划的研究旨在揭示这种相互作用的新方面，为Donaldson-Thomas理论，量子上同调以及黑洞物理学和弦对偶性等理论物理学的基本领域开辟新的研究方向。这旨在通过解开数学和物理学之间的新联系来刺激数学和物理学的几个领域的发展。在教育方面，该研究计划为各级学生以及博士后研究员提供多种培训机会，促进各种技术和沟通技能的发展，具有广泛的学术环境之外的适用性。

Employing D-brane moduli spaces as a unifying theme, this research program consists of three projects lying at the interface between algebraic geometry and string theory. The first is centered around a new mathematical construction (ADHM sheaf invariants) yielding new mathematical conjectures concerning local BPS invariants, quantum cohomologyof quiver varieties and black hole correspondence. The second project investigates the cameral structure and wallcrossing behavior of moduli spaces of Bridgeland stable objects on local surfaces emphasizing a connection between Donaldson-Thomas type invariants and black hole physics. The third proposes a systematic approach to correlators of surface operators in topological gauge theories via virtual cycles and localization for parabolic Higgs bundles. Broader Impact: The interaction between mathematics and string theory has been a major driving force in modern mathematical physics. The research planned here aims at revealing new aspects of this interaction opening new directions of research in Donaldson-Thomas theory, quantum cohomology of quiver varieties as well as fundamental areas of theoretical physics such as black hole physics and string duality. This aims to stimulate the development of several areas of mathematics and physics by unraveling new connections between them. With respect to education, this research program offers multiple training opportunities for students of all levels as well as postdoctoral fellows, stimulating the development of a variety of technical and communication skills with a wide range of applicability beyond the academic environment.