CAREER: Branes in the Moduli Space of Higgs Bundles

这个国家科学基金会职业奖支持数学领域的研究，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资助的研究基础上，PI将研究被称为希格斯束的拓扑对象及其相应的平坦连接空间。该项目教育部分的驱动主题是纳入代表性不足的群体，在美国建立研究人员社区，并推广到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，并注意欢迎少数群体。 PI和她的合作者将进行研究，以了解支持全纯层（A膜）及其对偶空间（B膜）的希格斯包模空间的拉格朗日子流形的外观。该项目的首要目标是获得一个几何分类，并在相干层的导出范畴和希格斯包的模空间的福谷范畴中对膜进行彻底的研究，将新方法扩展到其他超卡勒空间，并理解它们对几何朗兰兹计划的影响。为此，PI将开发新的工具来构建自然产生的膜族，不仅在平坦连接和希格斯束中，而且在其他超卡勒设置中。此外，PI还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阿贝尔化。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.