CAREER: Branes in the Moduli Space of Higgs Bundles

这项美国国家科学基金会职业奖支持在数学领域的研究，这些研究为解决几何和数学物理问题提供了强大的工具。在她成功的nsf资助研究的基础上，PI将研究被称为希格斯束的拓扑对象，以及它们相应的平面连接空间。该项目的教育部分的驱动主题是纳入代表性不足的群体，在美国建立一个研究人员社区，并扩展到K-12。PI将组织几次针对研究生和年轻研究人员的年度研讨会，重点是欢迎少数民族。PI将与她的合作者一起研究支持全纯束（a膜）及其对偶空间（b膜）的希格斯束模空间的拉格朗日子流形的外观。该项目的总体目标是获得一种几何分类，并对希格斯束模空间的相干束的派生类别和深谷类别中的膜进行深入研究，将新方法扩展到其他超kahler空间，并了解它们对几何朗兰兹程序的含义。为此，PI将开发新的工具来构建自然产生的膜族，不仅在平面连接和希格斯束中，而且在其他超kahler环境中。此外，PI将探索通过膜出现的不同几何结构，包括超多边形和自同构群的研究，以及空间的进一步非贝尔化。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This National Science Foundation CAREER award supports research in an area of mathematics that provides powerful tools to tackle problems in geometry and mathematical physics. Building on her successful NSF-funded research, the PI will work with topological objects known as Higgs bundles, and their corresponding spaces of flat connections. The driving themes for the educational component of the project are inclusion of underrepresented groups, building a community of researchers in the USA, and outreach to K-12. The PI will organize several yearly workshops aimed at graduate students and young researchers, with attention paid to welcoming minorities. The PI, together with her collaborators, will undertake research towards understanding the appearance of Lagrangian submanifolds of the moduli space of Higgs bundles supporting holomorphic sheaves (A-branes) and their dual spaces (B-branes). The overarching goal of the project is to obtain a geometric classification and to perform a thorough study of branes in the derived category of coherent sheaves and the Fukaya category of the moduli spaces of Higgs bundles, to extend the novel methods to other hyperkahler spaces, and to understand their implications for the geometric Langlands program. To this aim, the PI shall develop new tools to construct naturally arising families of branes not only within flat connections and Higgs bundles, but also in other hyperkahler settings. Moreover, the PI shall explore different geometric structures appearing through branes, including the study of hyperpolygons and automorphism groups, and further nonabelianization of spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.