Non-Commutative Geometry Methods for Group C*-Algebras

Riemann-Roch Theorem and Index Theorem in Non-commutative Geometry

In this lecture we review apprearance of the Riemann-Roch Theorem in classical function theory, Algebraic topology, in theory of pseudo-differential operators and finally in noncommutative geometry. We show also it usefulness in many problem of quantization and harmonic analysis on Lie groups.

Maximal commutative subalgebras, Poisson geometry and Hochschild homology

A Poisson geometry arising from maximal commutative subalgebras is studied. A spectral sequence convergent to Hochschild homology with coefficients in a bimodule is presented. It depends on the choice of a maximal commutative subalgebra inducing appropriate filtrations. Its E^{2}_{p,q}-groups are computed in terms of canonical homology with values in a Poisson module defined by a given bimodule and a maximal commutative subalgebra.

Algebra cohomology over a commutative algebra revisited

The aim of this paper is to give a relatively easy bicomplex which computes the Shukla, or Quillen cohomology in the category of associative algebras over a commutative algebra $A$, in the case when $A$ is an algebra over a field.

Commutative Lie algebras and commutative cohomology in characteristic $2$

We discuss a version of the Chevalley--Eilenberg cohomology in characteristic $2$, where the alternating cochains are replaced by symmetric ones.

关于Siegfried Bosch《代数几何与交换代数》的笔记（献给格罗滕迪克）

Notes on Siegfried Bosch's Algebraic Geometry and Commutative Algebra (dedicated to Grothendieck)

本文系统整理交换代数与代数几何的核心内容，涵盖环模层概形及平展上同调等进阶主题，作为对格罗滕迪克的致敬。

Non-commutative formality implies commutative and Lie formality

Over a field of characteristic zero we prove two formality conditions. We prove that a dg Lie algebra is formal if and only if its universal enveloping algebra is formal. We also prove that a commutative dg algebra is formal as a dg associative algebra if and only if it is formal as a commutative dg algebra. We present some consequences of these theorems in rational homotopy theory.