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Quasimaps to Nakajima Varieties

Quasimaps to Nakajima Varieties
中岛品种的准地图
批准号:
2401380
负责人:
Andrey Smirnov
金额:
$24.48万
依托单位国家:
美国
项目类别:
Continuing Grant
财政年份:
2024
资助国家:
美国
项目状态:
未结题
起止时间:
2024-06-01 至 2027-05-31

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中文摘要
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英文摘要
Counting curves in a given space is a fundamental problem of enumerative geometry. The origin of this problem can be traced back to quantum physics, and especially string theory, where the curve counting provides transition amplitudes for elementary particles. In this project the PI will study this problem for spaces that arise as Nakajima quiver varieties. These spaces are equipped with internal symmetries encoded in representations of quantum loop groups. Thanks to these symmetries, the enumerative geometry of Nakajima quiver varieties is extremely rich and connected with many areas of mathematics. A better understanding of the enumerative geometry of Nakajima quiver varieties will lead to new results in representation theory, algebraic geometry, number theory, combinatorics and theoretical physics. Many open questions in this field are suitable for graduate research projects and will provide ideal opportunities for students' rapid introduction to many advanced areas of contemporary mathematics. More specifically, this project will investigate and compute the generating functions of quasimaps to Nakajima quiver varieties with various boundary conditions, uncover new dualities between these functions, and prove open conjectures inspired by 3D-mirror symmetry. The project will also reveal new arithmetic properties of the generating functions via the analysis of quantum differential equations over p-adic fields. The main technical tools to be used include the (algebraic) geometry of quasimap moduli spaces, equivariant elliptic cohomology, representation theory of quantum loop groups, and integral representations of solutions of the quantum Knizhnik-Zamolodchikov equations.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.
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会议论文
Dualities in Enumerative Geometry and Representation Theory
国内基金
海外基金
Kronheimer-Nakajima quiver 模空间与有理曲面
  • 批准号:
    11401489
  • 项目类别:
    青年科学基金项目
  • 资助金额:
    22.0万元
  • 批准年份:
    2014
  • 负责人:
    徐芒
  • 依托单位: