G. Sardanashvily, “Saarbrucken ,
2012)

**" (Lambert Academic Publishing,***Lectures on Differential Geometry of Modules and Rings. Application to Quantum Theory***#**
Differential geometry of
smooth vector bundles can be formulated in algebraic terms of modules over
rings of smooth function. Generalizing this construction, one can define the
differential calculus, differential operators and connections on modules over
arbitrary commutative, graded commutative and non-commutative rings. For
instance, this is the case of quantum theory, supergeometry and non-commutative
geometry, respectively. The book aims to summarize the relevant material on
this subject. Some basic applications to quantum theory are considered.

The book is based on the
graduate and post graduate courses of lectures given at the Department of
Theoretical Physics of Moscow State University (Russia )
and the Department of Mathematics and Physics of University of Camerino (Italy ). It
addresses to a wide audience of mathematicians, mathematical physicists and
theoreticians.

**Contents**

**1 Commutative geometry**: 1.1 Commutative algebra, 1.2 Diﬀerential operators on modules and rings, 1.3 Connections on modules and rings, 1.4 Diﬀerential calculus over a commutative ring, 1.5 Local-ringed spaces, 1.6 Diﬀerential geometry of C(X)-modules, 1.7 Connections on local-ringed spaces.

**2 Geometry of quantum systems**: 2.1 Geometry of Banach manifolds, 2.2 Geometry of Hilbert manifolds, 2.3 Hilbert and C*-algebra bundles, 2.4 Connections on Hilbert and C*--algebra bundles, 2.5 Instantwise quantization, 2.6

**3 Supergeometry**: 3.1 Graded tensor calculus, 3.2 Graded diﬀerential calculus and connections, 3.3 Geometry of graded manifolds, 3.4 Supermanifolds, 3.5 Supervector bundles, 3.6 Superconnections.

**4 Non-commutative geometry**: 4.1 Modules over C*-algebras, 4.2 Non-commutative diﬀerential calculus, 4.3 Diﬀerential operators in non-commutative geometry, 4.4 Connections in non-commutative geometry, 4.5 Matrix geometry, 4.6 Connes’ non-commutative geometry, 4.7 Diﬀerential calculus over Hopf algebras.

**5 Appendix. Cohomology**: 5.1 Cohomology of complexes, 5.2 Cohomology of Lie algebras, 5.3 Sheaf cohomology.