Discrete, Finite and Lie Groups: Comprehensive Group Theory in Geometry and Analysis

In a self contained and exhaustive work the author covers Group Theory in its multifaceted aspects, treating its conceptual foundations in a proper logical order. First discrete and finite group theory, that includes the entire chemical-physical field of crystallography is developed self consistently, followed by the structural theory of Lie Algebras with a complete exposition of the roots and Dynkin diagrams lore. A primary on Fibre-Bundles, Connections and Gauge fields, Riemannian Geometry and the theory of Homogeneous Spaces G/H is also included and systematically developed.

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Pietro G. Fré, born in Alessandria in 1952, is Professor Emeritus of Theoretical Physics at the University of Turin. He was Dean of the Department of Physics until his retirement on November 1^st 2022. He was Director of the Arnold-Regge Center for Algebra, Geometry and Theoretical Physics from 2017 to 2022. From 1996 to 2000 he was President of SIGRAV (Italian Society of General Relativity and Physics of Gravitation). He was Professor of Theoretical Physics and Director of the Particle Physics Sector at SISSA from 1990 to 1996. Fellow of the Theoretical Division of CERN, post-doctoral fellow at the California Institute of Technology and the University of Bielefeld in the early years of his career. Author of about 380 scientific publications including 9 monographic books, he has given relevant contributions to the field of Supergravity and String Theories with applications to Cosmology, Black Holes, Holographic Gauge/Gravity Correspondence and to the development of the underlying Mathematical Structures. We mention in particular the introduction of the rheonomic formulation (with R. D'Auria), the invention, independently from mathematicians, under the name of "Cartan Integrable Systems" of Free Differential Algebras and their extension with fermionic p-forms, the D'Auria-Fré algebra which later evolved into the so named A-infinity algebras, the geometric formulation of the theory in 11-dimensions (M-theory), the geometric systematization of the Lagrangian of N = 2 Supergravity (with S. Ferrara et al), the discovery of some of the seven dimensional homogeneous Sasakian varieties and the early development of their holographic equivalents via quivers. Since 2001, Prof. Frè has started an intense collaboration with Ukraine and Russia. In 2002 he spent a sabbatical year in Kiev and taught a long course in Supergravity at the Bogoliubov Institute of the Ukrainian Academy of Sciences. A frequent visitor to the Joint Institute of Nuclear Research in Dubna, in the years from 2003 to 2008, he became Scientific Attaché of the Embassy of Italy in Moscow in 2009 and held this role for eight years until September 2017. During the period in Russia, in addition to his diplomatic activity, he continued his scientific activity, collaborating in particular with Prof. Alexander Sorin, his frequent co-author, and the teaching activity, holding for three subsequent years an advanced course in Modern Mathematical Physics at MEPhI (Moscow Engineering Physical Institute). He has also been awarded two Honorary Degrees: one from the State Technical Chemical University of Ivanovo, the other from the Joint Institute of Nuclear Research in Dubna. Prof. Fré, who has had a considerable number of Ph.D. and Doctoral students, among whom many are already Full and Associate Professors of various Universities, also dedicates himself to Literature and he is the author of some novels and essays, in particular historical.