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。 这一领域产生于对几何或物理系统的对称性及其线性作用（表示论）和量子化的研究，从经典力学到量子力学的数学版本（非对易几何学）。 这两者之间有着丰富的相互作用，它与数学的许多领域都有联系和应用，如组合学、可积系统、真实的代数几何、代数簇和辛奇异性的解决方案。PI定义了新的同调理论，推广了德·拉姆上同调，并使用D-模技术对Hochschild和循环同调给出了新的解释。 这些想法有应用的代表性理论李群，研究各种代数（切列德尼克，辛反射，和W-代数），并辛和卡-丘决议。 PI将证明他的Poisson-de Rham同调在新的情况下恢复了每个辛分解的de Rham同调，例如对于行列式簇和超环面簇。 他将从它恢复重要的多项式，如Kostka和Tutte多项式。 PI计划追求与全纯纤维封闭形式在分辨率变形上消失的顺序有关的知识。 主要的技术使用D-模块封装的哈密顿流，建立了典型的局部系统的辛叶。 他还计划使用循环的同源性，以获得代表仿射赫克代数通过高斯-马宁连接非交换变形的镜像余切束旗品种。

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.