Sophus Lie, a giant in mathematics / Lizhen Ji -- Felix Klein: his life and mathematics / Lizhen Ji -- Klein and the Erlangen Programme / Jeremy J. Gray -- Klein’s “Erlanger Programm”: do traces of it exist in physical theories? / Hubert Goenner -- On Klein’s So-called Non-Euclidean geometry / Norbert A’Campo, Athanase Papadopoulos -- What are symmetries of PDEs and what are PDEs themselves? / Alexandre Vinogradov -- Transformation groups in non-Riemannian geometry / Charles Frances -- Transitional geometry / Norbert A’Campo, Athanase Papadopoulos -- On the projective geometry of constant curvature spaces / Athanase Papadopoulos, Sumio Yamada -- The Erlangen program and discrete differential geometry / Yuri B. Suris -- Three-dimensional gravity – an application of Felix Klein’s ideas in physics / Catherine Meusburger -- Invariances in physics and group theory / Jean-Bernard Zuber.

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The Erlangen program expresses a fundamental point of view on the use of groups and transformation groups in mathematics and physics. The present volume is the first modern comprehensive book on that program and its impact in contemporary mathematics and physics. Klein spelled out the program, and Lie, who contributed to its formulation, is the first mathematician who made it effective in his work. The theories that these two authors developed are also linked to their personal history and to their relations with each other and with other mathematicians, incuding Hermann Weyl, Élie Cartan, Henri Poincaré, and many others. All these facets of the Erlangen program appear in the present volume. The book is written by well-known experts in geometry, physics and history of mathematics and physics. It is addressed to mathematicians, to graduate students, and to all those interested in the development of mathematical ideas.