Differential geometry of smooth vector bundles can be formulated in algebraic terms of modules over rings of smooth functions. 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.

Autorentext

Gennadi Sardanashvily, theoretician and mathematical physicist, graduated from Moscow State University in 1973, Ph.D. (1980) and D.Sc. (1998), Principal research scientist at the Department of Theoretical Physics of Moscow State University, published more than 300 scientific works and 23 text books and monographs.