A Friendly Overview of Noncommutative Geometry

Fuzzy Physics: A Brief Overview of Noncommutative Geometry in Physics

Noncommutative geometry (NCG) is a mathematical tool which has been used in the search for a quantum theory of gravity. However, its application is not limited to this field. In this brief note we present different uses of NCG in Theoretical Physics.

Noncommutative geometry

Noncommutative geometry as a functor

In this note the noncommutative geometry is interpreted as a functor, whose range is a family of the operator algebras. Some examples are given and a program is sketched.

Noncommutative Geometry of Foliations

We review basic notions and methods of noncommutative geometry and their applications to analysis and geometry on foliated manifolds.

A-branes and Noncommutative Geometry

We argue that for a certain class of symplectic manifolds the category of A-branes (which includes the Fukaya category as a full subcategory) is equivalent to a noncommutative deformation of the category of B-branes (which is equivalent to the derived category of coherent sheaves) on the same manifold. This equivalence is different from Mirror Symmetry and arises from the Seiberg-Witten transfo...