Anastasios Mallios - Böcker

This two-volume monograph obtains fundamental notions and results of the standard differential geometry of smooth manifolds, without using differential calculus. Here, the sheaf-theoretic character is emphasized. This has theoretical advantages such as greater perspective, clarity and unification, but also practical benefits ranging from elementary particle physics, via gauge theories and theoretical cosmology ("differential spaces"), to non-linear PDEs (generalized functions). Thus, more general applications, which are no longer "smooth" in the classical sense, can be coped with. The treatise might also be construed as a new systematic endeavour to confront the ever-increasing notion that the "world around us is far from being smooth enough". This work is intended for postgraduate students and researchers whose work involves differential geometry, global analysis, analysis on manifolds, algebraic topology, sheaf theory, cohomology, functional analysis or abstract harmonic analysis.

Differential geometry, in the classical sense, is developed through the theory of smooth manifolds. In the early 1990s, the author initiated a new kind of differential geometry in which all the machinary of classical differential geometry can be explained without any notion of smoothness, that enables unexpected potential applicability since anomalies can now be incorporated in the calculations. This was acheived via sheaf theory (geometry) and sheaf cohomology (analysis). "Modern Differential Geometry in Gauge Theories" is a two volume research monograph, which systematically applies his sheaf-theoretic approach to such physical theories as gauge theory. Beginning in Volume 1, the focus is on Maxwell fields. All the basic concepts of his mathematical approach are formulated and used thereafter to describe elementary particles, electromagnetism, and geometric prequantization. Maxwell fields are examined and classified in the language of sheaf theory and sheaf cohomology.This text contains a wealth of detailed and rigorous computations, and will appeal to mathematicians and physicists along with advanced undergraduate and graduate students studying applications of differential geometry to physical theories.

Differential geometry, in the classical sense, is developed through the theory of smooth manifolds. In the early 1990s, the author initiated a new kind of differential geometry in which all the machinary of classical differential geometry can be explained without any notion of smoothness, that enables unexpected potential applicability since anomalies can now be incorporated in the calculations. This was acheived via sheaf theory (geometry) and sheaf cohomology (analysis). "Modern Differential Geometry in Gauge Theories" is a two volume research monograph, which systematically applies his sheaf-theoretic approach to such physical theories as gauge theory. Continuing his point of view, started in the first volume of this work, the author extends the application of his sheaf-theoretic approach to Yang-Mills fields in general. The important topics include cohomological classification of Yang-Mills fields, the geometry of Yang-Mills A-connections and moduli space of a vector sheaf, as well as Einstein's equation in vacuum.This text contains a wealth of detailed and rigorous computations, and will appeal to mathematicians and physicists along with advanced undergraduate and graduate students studying applications of differential geometry to physical theories.

This two-volume monograph obtains fundamental notions and results of the standard differential geometry of smooth (CINFINITY) manifolds, without using differential calculus. Here, the sheaf-theoretic character is emphasised. This has theoretical advantages such as greater perspective, clarity and unification, but also practical benefits ranging from elementary particle physics, via gauge theories and theoretical cosmology (`differential spaces'), to non-linear PDEs (generalised functions). Thus, more general applications, which are no longer `smooth' in the classical sense, can be coped with. The treatise might also be construed as a new systematic endeavour to confront the ever-increasing notion that the `world around us is far from being smooth enough'. Audience: This work is intended for postgraduate students and researchers whose work involves differential geometry, global analysis, analysis on manifolds, algebraic topology, sheaf theory, cohomology, functional analysis or abstract harmonic analysis.