Lectures on Noncommutative Geometry

TL;DR

This paper provides an overview of noncommutative geometry, covering standard topics and introducing new results like noncommutative Chern-Weil theory and symplectic geometry, with applications to algebraic and geometric structures.

Contribution

It presents a comprehensive lecture-based exposition on noncommutative geometry, including novel results in noncommutative Chern-Weil theory and symplectic geometry.

Findings

Introduction of noncommutative Chern-Weil theory

Development of noncommutative symplectic geometry

Discussion of noncommutative differential forms

Abstract

These Lectures are based on a course on noncommutative geometry given by the author in 2003 at the University of Chicago. The lectures contain some standard material, such as Poisson and Gerstenhaber algebras, deformations, Hochschild cohomology, Serre functors, etc. We also discuss many less known as well as some new results, in particular, noncommutative Chern-Weil theory, noncommutative symplectic geometry, noncommutative differential forms and double-tangent bundles.