Poisson Geometry, Quantum Moduli, and Geometric Dualities

这是一项关于代数几何的研究。代数几何领域通过提取几何模型的基本复杂性并将其编码为多项式方程来研究几何模型。该项目融合了量子物理学的想法和对自然基本定律对称性的物理研究，以提取关于空间几何的新的和意想不到的信息。该项目将专注于揭开量子场多维几何模型的隐藏结构，并在一系列多项式不变量中捕捉这种结构。这些不变量将为理解和证明各种经验观测到的物理对偶提供一种新的数学工具，这些对偶有望识别先验的无关量子理论。该项目旨在统一任意维表示的参数空间的解析和几何性质，并为以一种适合于在广泛的应用中实际使用的方式理解模问题的基本结构奠定基础。这项工作将直接涉及辛几何、几何表示理论、弦理论和量子场论中的深层问题。将研究三个方向。第一个是研究有势簇的Hodge理论以及与之相关的倒位滤子和权滤子是如何被T-对偶交换的。在第二个项目中，我们将发展一种新的方法来构造非紧代数流形上的非规则联络的模、构造显式辛叶以及计算它们的叶。这将涉及到一种新的De Rham变种形式边界理论，以及一种从形式几何出发的新的层理论不变量的构造。最后的项目将建立一个频谱覆盖论，用于在与四边形相交相关的丛的模上构建Hecke本征轮。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This is a research in algebraic geometry. The field of algebraic geometry studies geometric models by distilling and encoding their essential complexity in polynomial equations. The project integrates ideas from quantum physics and the physical study of symmetries of the fundamental laws of nature to extract new and unexpected information about the geometry of spaces. The project will focus on unravelling the hidden structure of multi-dimensional geometric models of quantum fields and to capture this structure in a cascade of polynomial invariants. These invariants will give a new mathematical tool for understanding and proving various empirically observed physics dualities which are expected to identify a priori unrelated quantum theories. The project aims to unify the analytic and geometric properties of parameter spaces of representations in arbitrary dimension and sets the stage for understanding the basic structure of moduli problems in a way suitable for pragmatic use in a broad spectrum of applications. The proposed work will be immediately relevant to deep questions in symplectic geometry, geometric representation theory, string theory and quantum field theory.Three directions will be studied. The first is to investigate how the Hodge theory of varieties with potentials and the associated perverse and weight filtrations are exchanged by T-duality. In the second project a new method will be developed for constructing moduli of irregular connections on non-compact algebraic manifolds, for building explicit symplectic foliations, and for computing their leaves. This will involve a new de Rham theory of formal boundaries of varieties, and a new construction of sheaf theoretic invariants from the formal geometry. The final project will build a spectral coverformalism for constructing Hecke eigensheaves on moduli of bundles related to intersections of quadrics.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.