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This visionary and engaging book provides a mathematical perspective on the fundamental ideas of numbers, space, life, evolution, the brain and the mind. The author suggests how a development of mathematical concepts in the spirit of category theory may lead to unravelling the mystery of the human mind and the design of universal learning algorithms. The book is divided into two parts, the first of which describes the ideas of great mathematicians and scientists, those who saw sparks of light in the dark sea of unknown. The second part, Memorandum Ergo, reflects on how mathematics can contribute to the understanding of the mystery of thought. It argues that the core of the human mind is a structurally elaborated object that needs a creation of a broad mathematical context for its understanding. Readers will discover the main properties of the expected mathematical objects within this context, called ERGO-SYSTEMS, and readers will see how these “systems” may serve as prototypes for design of universal learning computer programs. This is a work of great, poetical insight and is richly illustrated. It is a highly attractive read for all those who welcome a mathematical and scientific way of thinking about the world.

Mathematical logic --- wiskunde --- logica

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Metric theory has undergone a dramatic phase transition in the last decades when its focus moved from the foundations of real analysis to Riemannian geometry and algebraic topology, to the theory of infinite groups and probability theory. The new wave began with seminal papers by Svarc and Milnor on the growth of groups and the spectacular proof of the rigidity of lattices by Mostow. This progress was followed by the creation of the asymptotic metric theory of infinite groups by Gromov. The structural metric approach to the Riemannian category, tracing back to Cheeger's thesis, pivots around the notion of the Gromov–Hausdorff distance between Riemannian manifolds. This distance organizes Riemannian manifolds of all possible topological types into a single connected moduli space, where convergence allows the collapse of dimension with unexpectedly rich geometry, as revealed in the work of Cheeger, Fukaya, Gromov and Perelman. Also, Gromov found metric structure within homotopy theory and thus introduced new invariants controlling combinatorial complexity of maps and spaces, such as the simplicial volume, which is responsible for degrees of maps between manifolds. During the same period, Banach spaces and probability theory underwent a geometric metamorphosis, stimulated by the Levy–Milman concentration phenomenon, encompassing the law of large numbers for metric spaces with measures and dimensions going to infinity. The first stages of the new developments were presented in Gromov's course in Paris, which turned into the famous "Green Book" by Lafontaine and Pansu (1979). The present English translation of that work has been enriched and expanded with new material to reflect recent progress. Additionally, four appendices—by Gromov on Levy's inequality, by Pansu on "quasiconvex" domains, by Katz on systoles of Riemannian manifolds, and by Semmes overviewing analysis on metric spaces with measures—as well as an extensive bibliography and index round out this unique and beautiful book.

Riemannian manifolds --- 514.76 --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Geometry, Differential --- Manifolds (Mathematics) --- 514.76 Geometry of differentiable manifolds and of their submanifolds --- Geometry of differentiable manifolds and of their submanifolds --- Riemannian manifolds. --- Riemann, Variétés de

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Pattern Formation in Morphogenesis is a rich source of interesting and challenging mathematical problems. The volume offers an interdisciplinary interaction space between biologists working in this field and mathematicians, who may propose solutions to the problems put forward by biologists. The main goal is to facilitate the process of cultivating a mutual recognition of the complementary skills between biologists and mathematicians, to the point where the resulting synergy generates new and novel discoveries in the field of Developmental Biology. Lastly, the volume shows how a combination of new discoveries in developmental biology and associated mathematical modeling and computational techniques has stimulated or may stimulate relevant advances in the field.                             .

Pattern formation (Biology) --- Morphogenesis. --- Morphogeny --- Organogenesis --- Biological pattern formation --- Mathematics. --- Bioinformatics. --- Microbial genetics. --- Microbial genomics. --- Applied mathematics. --- Engineering mathematics. --- Applications of Mathematics. --- Biomedicine general. --- Computational Biology/Bioinformatics. --- Microbial Genetics and Genomics. --- Mathematics, general. --- Embryology --- Morphology --- Developmental biology --- Medicine. --- Genomics --- Microbial genetics --- Microorganisms --- Genetics --- Microbiology --- Bio-informatics --- Biological informatics --- Biology --- Information science --- Computational biology --- Systems biology --- Clinical sciences --- Medical profession --- Human biology --- Life sciences --- Medical sciences --- Pathology --- Physicians --- Math --- Science --- Data processing --- Biomedicine, general. --- Health Workforce --- Engineering --- Engineering analysis --- Mathematical analysis --- Mathematics

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"Visions in Mathematics - Towards 2000" was one of the most remarkable mathematical meetings in recent years. It was held in Tel Aviv from August 25th to September 3rd, 1999, and united some of the leading mathematicians worldwide. The goals of the conference were to discuss the importance, the methods, the past and the future of mathematics as we enter the 21st century and to consider the connection between mathematics and related areas. The aims of the conference are reflected in the present set of survey articles, documenting the state of art and future prospects in many branches of mathematics of current interest. This is the first part of a two-volume set that will serve any research mathematician or advanced student as an overview and guideline through the multifaceted body of mathematical research in the present and near future.

Mathematics --- wiskunde

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