The God has created a man in order that he creates that the God fails to do

Thursday, 27 September 2012

My new book on differential geometry of modules and rings

G. Sardanashvily, “Lectures on Differential Geometry of Modules and Rings. Application to Quantum Theory" (Lambert Academic Publishing, Saarbrucken, 2012) #

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.


1 Commutative geometry: 1.1 Commutative algebra, 1.2 Dierential operators on modules and rings, 1.3 Connections on modules and rings, 1.4 Dierential calculus over a commutative ring, 1.5 Local-ringed spaces, 1.6 Dierential 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 Berry connection.

3 Supergeometry: 3.1 Graded tensor calculus, 3.2 Graded dierential 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 dierential calculus, 4.3 Dierential operators in non-commutative geometry, 4.4 Connections in non-commutative geometry, 4.5 Matrix geometry, 4.6 Connes’ non-commutative geometry, 4.7 Dierential calculus over Hopf algebras.

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

Wednesday, 19 September 2012

New Managing Editor of IJGMMP

A letter to the Editors of International Journal of Geometric Methods in Modern Physics (IJGMMP):

Dear Editors of IJGMMP,
I am going to leave the Journal as a Managing Editor. I prepare v10 (2013) of IJGMMP, but would prefer that new Managing Editor produces v11 (2014). Then he should start in March-April 2013.

Please, think if you could suggest candidates for Managing Editor. Of course, the Publisher decides, but you can recommend one or a few persons. 

The candidates can contact me for information they need.

Gennadi Sardanashvily

Saturday, 8 September 2012

The list of my posts in 2011 - 2012

Why to gauge gravity?

 Discrete space-time (from my Scientific Biography)

 What is a mathematical structure?

 The Higgs boson or the Higgs vacuum?

 Impact Factor 2011 of Journals in Mathematical Physics

 Dmitri Ivanenko’s archive: Nobel Laureates Letters

 My book: Generalized Hamiltonian Formalism for Field Theory

 Nobel laureates inscriptions on the walls of Ivanenko's office in Moscow State University

 My lectures on mathematical physics

 Lagrangian BRST theory (from my Scientific biography)

 Classical mechanics and field theory admit comprehensive geometric formulation

 My Library: Completely integrable and superintegrable Hamiltonian systems with noncompact invariant submanifolds

 Lagrangian dynamics of higher-dimensional submanifolds

 A problem of an inertial reference frame in classical mechanics

 My Library: General Noether theorems

 On a gauge model of the fifth force

 My Library: Jet Manifold Formalism

Freedom is an immanent property of living nature

Review on our book "Geometric and Algebraic Topological Methods in Quantum Mechanics" in Mathematical Reviews

 An energy-momentum is not uniquely defined

 My Library: Time-dependent mechanics

 Review on our book "Advanced Classical Field Theory" in Mathematical Reviews

 My Library: Advanced Classical Field Theory

 My Library: Gauge gravitation theory

Is a momentum space of quantum fields Euclidean?

 Hierarchy of Noether identities (from my Scientific Biography)

 Can contemporary mathematics describe quantum physics?

 Why only electromagnetic and gravitational interactions are in classical physics?

 “Antropomorphic” mathematics and the crisis of science

 What is a reference frame in field theory and mechanics

 Covariant (polysymplectic) Hamiltonian field theory (from my Scientific Biography)

 Why a classical system admits different non-equivalent quantization

Five fundamental problems of contemporary physics

Integrable Hamiltonian systems: generalization to a case of non-compact invariant submanifolds (from my Scientific Biography)

 On a mathematical hypothesis of quantum space-time

 Review on our book “Geometric Formulation of Classical and Quantum mechanics” in Mathematical Reviews

 II. How we developed gauge gravitation theory (from my Scientific Biography)

 I. How we developed gauge gravitation theory (from my Scientific Biography)

On a mathematical hypothesis of the quark confinement

 The prespinor model (from my Scientific Biography)

 Who is who among Universities in 2011

 My Scientific Biography: Student period

 Illusion of matter

 On the strangeness of relativistic mechanics

 ”Quantum” causality of ancient Greeks

 Why a citation list for a theoretician?

 What are classical Higgs fields?

 The generalized Serre – Swan theorem is a cornerstone of classical field theory

Metric gravity as a non-quantized Higgs field

 What are gauge symmetries?

 Why connections in classical field theory?

 Geometry in quantum theory IV: Modern geometries

 Geometry in quantum theory III: Differential geometry of modules and rings

 Does Impact Factor show anything?

 Geometry in quantum theory II: Infinite-dimensional fiber bundles

 Geometry in quantum theory I: Why familiar differential geometry contributes to quantum theory

 Problems of gravitation theory: What is a criterion of gravitational singularities?

 On geometric formulation of mechanics

 Why connections in field theory

 What are general covariant transformations?

 What is classical field theory really about?

 Non-commutative geometry meets a serious problem

 What is meant by supergeometry

 Quantum field theory: Functional integrals are not true integrals?

 What is a discrete space-time?

 What is true Equivalence principle?

 On relativistic mechanics in a very general setting

 Mechanics as particular classical field theory

 What is a fundamental science?

 Well-known mathematics that theoreticians do not know

 What is true Hamiltonian field theory?

 Classical field theory is complete: the strict geometric formulation

 My teacher Dmitri Ivanenko, a great theoretician of XX century