CAREER: Branes in the Moduli Space of Higgs Bundles

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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.