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

PI定义和研究几何空间及其量子类似的不变量。这一领域源于对几何或物理系统的对称性及其线性行为(表示论)及其量子化的研究，这是从经典到量子力学(非对易几何)的数学版本。PI定义了推广de Rham上同调的新的同调理论，并利用D-模技巧给出了Hochschild和循环同调的新的解释。这些思想可应用于李群的表示理论、各种代数(Cherednik代数、辛反射代数和W-代数)的研究，以及辛和Calabi-Yau分解。PI将证明他的Poisson-de Rham同调在新的情况下恢复每个辛分解的De Rham同调，例如对于行列式簇和超曲簇。他将从中恢复重要的多项式，如Kostka和Tutte多项式。PI计划进行猜想，将其与全纯纤维闭合形式在分辨率变形上的消失阶数联系起来。主要技术使用D-模来封装哈密顿流，该流是由辛叶上的正则局部系统建立的。他还计划利用循环同调，通过余切丛到旗簇的镜像的非交换变形的Gauss-Manin连接，得到仿射Hecke代数的表示。

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.