Noncommutative and Hamiltonian geometry, symplectic resolutions, and D-modules

PI 定义并研究几何空间的不变量及其量子类似物。 该领域源于对几何或物理系统的对称性及其线性作用（表示论）及其量子化的研究，即从经典力学到量子力学（非交换几何）的数学版本。 两者之间存在丰富的相互作用，在许多数学领域都有联系和应用，例如组合学、可积系统、实代数几何、箭袋簇和辛奇点的解析。PI 定义了新的同调理论，它推广了 de Rham 上同调，并使用 D 模技术对 Hochschild 和循环同调给出了新的解释。 这些思想可应用于李群的表示论、各种代数（Cherednik、辛反射和 W 代数）的研究以及辛和 Calabi-Yau 解析。 PI 将证明他的 Poisson-de Rham 同调恢复了新情况下每个辛解析的 de Rham 同调，例如行列簇和超斜簇。 他将从中恢复重要的多项式，例如 Kostka 和 Tutte 多项式。 PI 计划将这一猜想与分辨率变形时全纯纤维闭合形式的消失顺序相关联。 主要技术使用封装哈密顿流的 D 模块，该模块由辛叶上的规范局部系统构建。 他还计划使用循环同调，通过余切丛镜像的非交换变形上的高斯-马宁联系来获得仿射赫克代数的表示，以标记簇。

The PI defines and studies invariants of geometric spaces and quantum analogues thereof. This area arises from the study of symmetries of geometric or physical systems and their linear actions (representation theory) and their quantization, the mathematical version of passing from classical to quantum mechanics (noncommutative geometry). There is a rich interplay between the two, which has connections and applications to many areas of mathematics, such as combinatorics, integrable systems, real algebraic geometry, quiver varieties, and resolutions of symplectic singularities.The PI defines new homology theories which generalize de Rham cohomology and gives new interpretations of Hochschild and cyclic homology using D-module techniques. These ideas have applications to the representation theory of Lie groups, to the study of various algebras (Cherednik, symplectic reflection, and W-algebras), and to symplectic and Calabi-Yau resolutions. The PI will prove that his Poisson-de Rham homology recovers the de Rham homology of every symplectic resolution in new cases, such as for determinantal varieties and hypertoric varieties. He will recover from it important polynomials such as Kostka and Tutte polynomials. The PI plans to pursue conjectures relating this to the orders of vanishing of holomorphic fiberwise-closed forms on the deformation of the resolution. The main technique uses D-modules which encapsulate the Hamiltonian flow, built of canonical local systems on symplectic leaves. He also plans to use cyclic homology to obtain representations of affine Hecke algebras via the Gauss-Manin connection on noncommutative deformations of the mirror of cotangent bundles to flag varieties.