A-infinity structures and derived categories in algebraic geometry

这项研究是在代数几何领域，与弦理论和非对易几何有一些联系。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。在经典代数几何中，人们与这样的几何对象(称为代数簇)联系在一起的是一个交换环的函数空间。在这个研究项目中，我们将研究与代数簇相关的更复杂的代数结构，如A-无穷代数和导出的层范畴。研究项目将集中在以下方面：1)与曲线相关的A-无穷结构及其与曲线的模空间的关系；2)有限群作用下等变层的导出范畴的半正交分解；3)与拟齐次多项式相关的上同调场理论；4)NC-加厚上的Saves和Jacobian的一个刻划。第一个项目是关于一些与带标志点的曲线相关的A-无穷代数。本研究将研究这些A-无穷代数直到同伦的标准形，并将它们与曲线的模空间联系起来。在第二个项目中，给出了有限反射群的某些作用的等变相干层的派生范畴的正则半正交分解的构造，并将对其进行研究。第三个项目涉及矩阵分解范畴与计算在具有孤立奇点的拟齐次多项式的上同调场理论中的应用。第四个项目集中于阿贝尔簇上的哪些凝聚层可以扩展为非对易增厚，这是正则函数层的泊松包络的量子化。这可能导致曲线的雅可比曲线的一个新的刻画。

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.