Complex geometry of noncommutative tori and t-structures on derived categories

本文的研究主要集中在两个方面：1)非交换环面上的全纯束；2)派生范畴上的t结构和稳定性条件。作为非交换流形的最简单的例子，非交换环面在非交换几何中占有重要的地位。然而，将复杂结构置于其上并考虑相应的全纯对象是最近才出现的新思想。第一个项目致力于研究非交换环上全纯束的范畴，推广二维情况下的已知结果。第二个项目是关于布里奇兰发现的代数变异上相干束的派生范畴上有趣的新结构（受到一些物理学思想的启发）。也就是说，目标是解决与此类派生类别的稳定性条件有关的一些问题。提案中的另外两个项目涉及雅可比矩阵变体上的重言循环和代数几何中出现的a无穷代数的研究。提出的研究领域是非交换几何和代数几何。非交换几何是一个相对较新的领域，最初是由cones开发的，它结合了非交换代数、微分几何和泛函分析的思想。代数几何是研究由多项式方程和相关数学概念定义的几何对象的数学的一个经典分支。这两个领域的许多最新进展都是由它们在物理学中的应用所推动的。

The proposed research will focus mainly on two topics:1) holomorphic bundles on noncommutative tori, and 2) t-structures and stability conditions on derived categories.Noncommutative tori play an important role in noncommutative geometry being the simplest examples of noncommutative manifolds. However, putting the complex structure and considering corresponding holomorphic objects on them is a recently new idea. The first project is devoted to the study of the categories of holomorphic bundles on noncommutative tori generalizingknown results in the two-dimensional case. The second project is concerned withinteresting new structures on derived categories of coherent sheaves on algebraic varietiesdiscovered by Bridgeland (motivated by some ideas from physics). Namely, the goalis to attack a number of problems related to stability conditions on such derived categories.Two more projects in the proposal are concerned with the study of tautological cycles on Jacobian varieties and with A-infinity algebras arising in algebraic geometry.The proposed research is in the fields of noncommutative geometry and algebraic geometry.Noncommutative geometry is a relatively new field originally developed by Connes that combines ideas from noncommutative algebra, differential geometry and functional analysis. Algebraic geometry is a classical branch of mathematics studying geometric objects defined by polynomial equations and related mathematical concepts. Many recent advances in both fieldswere motivated by their use in physics.