A-infinity structures and derived categories in algebraic geometry

这个研究项目是在代数几何领域与弦理论和非交换几何的一些连接。代数几何是数学的一个分支，研究由多项式方程和相关数学结构定义的几何对象。在古典代数几何中，人们把这样的几何对象（称为代数簇）与一个交换环的函数空间联系起来。本研究计划将研究与代数簇相关的更复杂的代数结构，例如A-无限代数和层的导出范畴。研究计划将集中于以下主题：1）曲线的A-无穷结构及其与曲线模空间的关系，2）有限群作用的等变层的导出范畴的半正交分解，3）与拟齐次多项式有关的上同调场论，4）NC-加厚上的层与Jacobi算子的一个刻画第一个项目是关于带标记点的曲线的A-无穷代数。本研究将研究这些A-无穷代数直到同伦的规范形，并将它们与曲线的模空间联系起来。在第二个项目的建设一个典型的半正交分解的衍生类别的等变相干层的有限反射群的某些行动概述，并将进行研究。第三个项目是关于矩阵分解的范畴在上同调场论中的应用。第四个项目的重点是一个阿贝尔品种的相干层可以扩展到一个非交换增厚，这是一个量子化的泊松包络层的正规函数。这可能会导致一个新的特征的雅可比曲线。

This research project is in the field of algebraic geometry with some connections to string theory and noncommutative geometry. Algebraic geometry is a branch of mathematics studying geometric objects defined by polynomial equations and related mathematical structures. In classical algebraic geometry one associates with such geometric objects (called algebraic varieties) a space of functions that is a commutative ring. In this research project, more sophisticated algebraic structures associated with algebraic varieties, such as A-infinity algebras and derived categories of sheaves, will be studied.The research project will focus on the following topics:1) A-infinity structures associated with curves and their relation to the moduli spaces of curves,2) Semiorthogonal decomposition of the derived categories of equivariant sheaves for finite group actions,3) Cohomological field theories associated with quasihomogeneous polynomials,4) Sheaves on NC-thickenings and a characterization of Jacobians.The first project is about some A-infinity algebras associated with curves with marked points. The research will study normal forms of these A-infinity algebras up to homotopy and to relate them to the moduli spaces of curves. In the second project a construction of a canonical semiorthogonal decomposition of the derived category of equivariant coherent sheaves for some actions of finite reflection groups is outlined and will be studied. The third project is concerned with applications of categories of matrix factorizations with computation in the cohomological field theories attached to quasihomogeneous polynomials with isolated singularities. The fourth project focuses on which coherent sheaves on an abelian variety can be extended to a noncommutative thickening, which is a quantization of the Poisson envelope of the sheaf of regular functions. This may lead to a new characterization of Jacobians of curves.