Computer Algebra of Vector Bundles, Foliations and Zeta Functions and a Context of Noncommutative Geometry

有限域上K3范畴的zeta函数

Zeta functions of K3 categories over finite fields

本文定义了有限域上非交换K3曲面的zeta函数，作为Fourier-Mukai等价下的不变量，用于非交换点计数，并研究了其对几何性的障碍作用及Honda-Tate类比，给出了三次四重K3范畴Weil多项式的限制。

Noncommutative geometry of algebraic curves

A covariant functor from the category of generic complex algebraic curves to a category of the AF-algebras is constructed. The construction is based on a representation of the Teichmueller space of a curve by the measured foliations due to Douady, Hubbard, Masur and Thurston. The functor maps isomorphic algebraic curves to the stably isomorphic AF-algebras.

Noncommutative Geometry in the Framework of Differential Graded Categories

In this survey article we discuss a framework of noncommutative geometry with differential graded categories as models for spaces. We outline a construction of the category of noncommutative spaces and also include a discussion on noncommutative motives. We propose a motivic measure with values in a motivic ring. This enables us to introduce certain zeta functions of noncommutative spaces.

Noncommutative geometry and motives: the thermodynamics of endomotives

We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of $L$-functions. The analogue in characteristic zero of the action of the Frobenius on l-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative …

叶状结构与稳定映射

Foliations and stable maps

本文利用稳定映射导出栈研究全局向量场的代数轨迹，构造可复合的有理轨迹范畴结构，并提出计数周期轨道的代数 Ruelle Zeta 函数。

Moduli space and structure of noncommutative 3-spheres

We analyse the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and prove a general algebraic result which considerably refines the classical homomorphism from a quadratic algebra to a cross-product algebra associated to the characteristic variety and lands in a richer cross-product. It allows to control the $C^\ast$-norm on involutive quadratic algebras and to construct the differential calculus in the desired generality. The moduli space of noncommutative 3-spheres is identified with equivalence classes of pai…