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物的不变量。该领域源于对几何或物理系统的对称性及其线性作用（表示理论）及其量子化的研究，即从经典力学过渡到量子力学的数学版本（非交换几何）。这两者之间有着丰富的相互作用，在数学的许多领域都有联系和应用，如组合学、可积系统、实代数几何、颤振变分和辛奇点的解析。PI定义了新的同调理论，推广了de Rham上同调，并利用d模技术给出了Hochschild和循环同调的新解释。这些思想可以应用于李群的表示理论、各种代数（Cherednik、辛反射和w -代数）的研究，以及辛和Calabi-Yau解析。PI将证明他的泊松-德-朗同调在新的情况下恢复了每一个辛分解的德-朗同调，例如对于行定式和高曲式。他将从中恢复重要的多项式，如Kostka和Tutte多项式。PI计划对全纯纤维封闭形式在分辨率变形中消失的顺序进行推测。主要技术使用了d模，它封装了哈密顿流，建立在辛叶上的正则局部系统上。他还计划利用循环同调，通过对余切束的镜像的非交换变形的gaas - 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.