Computation is the process of applying a procedure or algorithm to the solution of a mathematical problem. Mathematicians and physicists have been occupied for many decades pondering which problems can be solved by which procedures, and, for those that can be solved, how this can most efficiently be done. In recent years, quantum mechanics has augmented our understanding of the process of computation and of its limitations.
"Perspectives in Computation" covers three broad topics: the computation process and its limitations, the search for computational efficiency, and the role of quantum...

Robert Geroch's lecture notes on general relativity are unique in three main respects. First, the physics of general relativity and the mathematics, which describes it, are masterfully intertwined in such a way that both reinforce each other to facilitate the understanding of the most abstract and subtle issues. Second, the physical phenomena are first properly explained in terms of spacetime and then it is shown how they can be "decomposed" into familiar quantities, expressed in terms of space and time, which are measured by an observer. Third, Geroch's successful pedagogical approach to...

Geroch's lecture notes on geometrical quantum mechanics are divided into three parts Differential Geometry, Mechanics, and Quantum Mechanics. The necessary geometrical ideas are presented in the first part of the book and are applied to mechanics and quantum mechanics in the second and third part. What also makes this book a valuable contribution to the existing textbooks on quantum physics is Geroch's unique approach to teaching theoretical and mathematical physics the physical concepts and the mathematics, which describes them, are masterfully intertwined in such a way that both reinforce...

Robert Geroch's lecture notes "Infinite-Dimensional Manifolds" provide a concise, clear, and helpful introduction to a wide range of subjects, which are essential in mathematical and theoretical physics - Banach spaces, open mapping theorem, splitting, bounded linear mappings, derivatives, mean value theorem, manifolds, mappings of manifolds, scalar and vector fields, tensor products, tensor spaces, natural tensors, tensor fields, tensor bundles, Lie derivatives, integral curves, geometry of Lie derivatives, exterior derivatives, derivative operators, partial differential equations, and...

The subject of computation deals with solutions to mathematical problems by procedures, i.e., solutions that could be generated by a machine; that require no original thought. For which problems are there such procedures, and when they do exist how efficient can they be?In recent years, the landscape of this subject has changed somewhat by the introduction of "machines" that utilize quantum mechanics in their operation. Perspectives in Computation covers three broad topics: the computation process and its limitations, the search for computational efficiency, and the role of quantum mechanics...