For decades, proponents of artificial intelligence have argued that computers will soon be doing everything that a human mind can do. Admittedly, computers now play chess at the grandmaster level, but do they understand the game as we do? Can a computer eventually do everything a human mind can do? In this absorbing and frequently contentious book, Roger Penrose--eminent physicist and winner, with Stephen Hawking, of the prestigious Wolf prize--puts forward his view that there are some facets of human thinking that can never be emulated by a machine. Penrose examines what physics and mathematics can tell us about how the mind works, what they can't, and what we need to know to understand the physical processes of consciousness. He is among a growing number of physicists who think Einstein wasn't being stubborn when he said his "little finger" told him that quantum mechanics is incomplete, and he concludes that laws even deeper than quantum mechanics are essential for the operation of a mind. To support this contention, Penrose takes the reader on a dazzling tour that covers such topics as complex numbers, Turing machines, complexity theory, quantum mechanics, formal systems, Godel undecidability, phase spaces, Hilbert spaces, black holes, white holes, Hawking radiation, entropy, quasicrystals, the structure of the brain, and scores of other subjects. The Emperor's New Mind will appeal to anyone with a serious interest in modern physics and its relation to philosophical issues, as well as to physicists, mathematicians, philosophers and those on either side of the AI debate.

"As data harvesting continues its exponential growth and computing power increases beyond what we can even imagine, AI and Big Data will be able to predict what we want for breakfast, what shoes we will buy, which political party we will vote for, and who we will fancy in a bar. Indeed, these deep-learning machines will know us better than we know ourselves. The day awaits when academia and big business will be able to quantify just how predictable each of us are. If indeed we are predictable like machines, to what extent are we alive, and under what definition? In The Human Test, quantum physicist Ron Folman unites findings from cognitive science, quantum physics, philosophy, and technology to offer a prescient look into this startling new era of human existence. If we are indeed found to be predictable, it will change everything we thought we knew about human nature. The Human Test strives for a new paradigm, and this new paradigm is found in human predictability. Existing monitors of general brain activity measure only very rudimentary aspects of the human experience, or what may be called human life; could the advent of a disruptive new technology bring about a much more insightful measure of who we are? While The Human Test describes a new and profound near-future impact of AI, it is rooted in a topic of paramount importance to humans: the enigma of consciousness and how it defines human life-or, what differentiates humans from all other life forms. Ultimately, The Human Test addresses some of the most fundamental questions about our species, starting with the most crucial: do we have free will, or are we merely machines? Fortunately, there is an antidote to predictability: creativity, that wondrous state of being original and inherently not predictable. Ultimately, Folman argues that if we can understand predictability, we can learn how to distance ourselves from it, and transcend to a purer idea of what human life should look like"-- Provided by publisher.

0 Background.- 1 Fundamental Concepts.- 2 The Degree of a Map.- 3 The Fundamental Group and Covering Spaces.- 4 Manifolds.- 5 Differential Forms and Homology in Euclidean Space.- 6 Homology and Cohomology.- 7 Homotopy Classification of Maps of the Sphere.- 8 Homotopy Groups.- 9 Fibered Spaces.- 10 Fibrations and Homotopy Groups.- 11 Homotopy Theory of Fibrations.- 12 Lie Groups.- 13 Lie Algebras.- 14 Topology of Lie Groups and Homogeneous Manifolds.- 15 Geometry of Gauge Fields.- Index of Notation.

The present book deals with the construction of solutions of linear partial differential equations by means of integral operators which transform analytic functions of a complex variable into such solutions. The theory of analytic functions has achieved a high degree of deve lopment and simplicity, and the operator method permits us to exploit this theory in the study of differential equations. Although the study of existence and uniqueness of solutions has been highly developed, much less attention has been paid to the investigation of function theo retical properties and to the explicit construction of regular and singular solutions using a unified general procedure. This book attempts to fill in the gap in this direction. Integral operators of various types have been used for a long time in the mathematical literature. In this connection one needs only to mention Euler and Laplace. The author has not attempted to give a complete account of all known operators, but rather has aimed at developing a unified approach. For this purpose he uses special operators which preserve various function theoretical properties of analytic functions, such as domains of regularity, validity of series development, connection between the coefficients of these developments and location and character of singularities, etc. However, all efforts were made to give a complete bibliography to help the reader to find more detailed information.