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Description
Geometry is an ancient science, rooted in the thought of Greek philosophers. The book recounts how the ideas and results of geometry were born and how they developed. Beginning with Euclid's Elements and continuing to present days, when geometry becomes abstract and helps us either to understand our multidimensional universe or to organize the multitude of data used in virtual space. The book describes "the new light in the darkness" found by Descartes to describe curves through an equation, which gives rise to modern analytic or algebraic geometry. Then it considers the revolution of Leibniz, Newton and others who, with the invention of Calculus (differential and integral), solved problems arising in physics constructing new curves. The central section is dedicated to Frederic Gauss and Bernard Riemann, who, with incredible imagination and rigor, paved the way for contemporary geometry, a path that would lead to the study of non-Euclidean Geometries, General Relativity, the shape of biological structures (Protein Folding), and much more. The conclusion is dedicated to present various Contemporary Research Programs, which organize many aspects of geometry, according to a celebrated Manifesto by Felix Klein. These include The Program of Minimal Models, by S. Mori, and The Program for the Classification of Topological Manifolds, by W. Thurston and G. Perelmann. This book will be of interest not just for mathematicians but for anyone who has curiosity in the field.
Introduction.- Space, a philosophical problem.- Genesis, Greek geometry.- Galileo, Decartes and Fermat, the overcoming of classicism.- The breakthrough of calculus.- Geometry becomes modern.- II. Curves, In the beginning there was the point.- Back to Greece, the origin of things.- Pythagoras Theorem, we enter in the Plato realm of Forms.- One curve leads to the other.- How to construct a curve.- Decartes and the Ge ome trie.- A problem about tangents.- Galileo, a new way to face problems.- Calculus, more problems, more curves.- Curvature, for the straightforward pathway had been lost.- The search for rational points- How digital natives build curves in their way.- III. Surfaces, Archimedes and the geometry of the sphere. Surfaces can as well be defined by equations. Singularities of surfaces in the guinness of records.- The shortest path. Theorema Egregium, according to Gauss. The curvature of a surface. Cartography and Geographic Maps. Non-Euclidian Geometries.- IV. Nowadays geometry, An Academic Lecture.- From Riemann to General Relativity.- A Program (or a Manifesto) to play Geometry.- Tessellations of the space.- The revolutionary art of Italian Renaissance painters.- Projective Algebraic Geometry. From Erlangen Program to the God Particle.- Topology, extreme Geometry.- The Millennium Problems.- Minimal Biography.- Index of names.
Full Professor of Geometry at the University of Trento, he is a scholar in the field of Projective Algebraic Geometry. He is currently Director of the International Center for Mathematical Research (CIRM) in Trento. He has been a visiting professor at many international research centers, including the Max Planck Institute (Bonn, Germany), the Newton Institute for Math. (Cambridge, UK), and the Math.Sc. Research Institute (Berkley, US). He is President of the Italian Mathematicla Society (UMI) For several years, he is also working in the field of scientific popularization, writing historical and popular books. Among these is Archimedes and the Art of Measurement, which won the National Scientific Popularization Award, Class of Science, 2022. He was President of MUSE, the science museum designed by Renzo Piano, during its creation and for eight years.
