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Mathematics and Computation

A Theory Revolutionizing Technology and Science

Avi Wigderson

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Princeton University Press img Link Publisher

Naturwissenschaften, Medizin, Informatik, Technik / Mathematik


An introduction to computational complexity theory, its connections and interactions with mathematics, and its central role in the natural and social sciences, technology, and philosophy

Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field, with strong links to most mathematical areas and to a growing number of scientific endeavors.

Avi Wigderson takes a sweeping survey of complexity theory, emphasizing the field’s insights and challenges. He explains the ideas and motivations leading to key models, notions, and results. In particular, he looks at algorithms and complexity, computations and proofs, randomness and interaction, quantum and arithmetic computation, and cryptography and learning, all as parts of a cohesive whole with numerous cross-influences. Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. He ends with a comprehensive look at the theory of computation, its methodology and aspirations, and the unique and fundamental ways in which it has shaped and will further shape science, technology, and society. For further reading, an extensive bibliography is provided for all topics covered.

Mathematics and Computation is useful for undergraduate and graduate students in mathematics, computer science, and related fields, as well as researchers and teachers in these fields. Many parts require little background, and serve as an invitation to newcomers seeking an introduction to the theory of computation.

  • Comprehensive coverage of computational complexity theory, and beyond
  • High-level, intuitive exposition, which brings conceptual clarity to this central and dynamic scientific discipline
  • Historical accounts of the evolution and motivations of central concepts and models
  • A broad view of the theory of computation's influence on science, technology, and society
  • Extensive bibliography

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Linear programming, Binary search algorithm, Sample complexity, Special case, Computational resource, Quantum algorithm, Universal property, Gödel's incompleteness theorems, Auxiliary function, Parameter (computer programming), Metaheuristic, Quantum computing, Subset, Computational complexity theory, Public-key cryptography, Hardness, Computation, Entscheidungsproblem, Randomized algorithm, VC dimension, Hilbert's program, Zermelo–Fraenkel set theory, Communication complexity, Boosting (machine learning), PP (complexity), Semidefinite programming, Deterministic algorithm, Circuit complexity, Ellipsoid method, Logical framework, Sanjeev Arora, Simplex algorithm, Zero-knowledge proof, Computational model, Mathematics, Mathematical optimization, Computational indistinguishability, PSPACE, Pseudorandomness, Central limit theorem, Pseudorandom number generator, Scientific notation, Boolean function, Pigeonhole principle, Probabilistic method, Turing test, BQP, Geometric complexity theory, NP (complexity), Brute-force search, Hardness of approximation, Online algorithm, PSPACE-complete, Upper and lower bounds, Search problem, Cryptographic primitive, Disjunctive normal form, Time complexity, NP-hardness, With high probability, Decision problem, Turing machine, Best, worst and average case, Approximation algorithm, Algorithm, Computational problem, Probability, Commutative property, Roth's theorem, Theorem, Quadratic residue, Randomness, Complexity class, Instance (computer science), Information theory, Random element, Natural proof, Weighted Majority Algorithm, Result, Variable (mathematics), Law of small numbers, Learning with errors, NP-completeness, P versus NP problem, Polynomial, Open problem, Theory of computation, Cryptography, Pseudorandom generator, Boolean circuit, Density matrix renormalization group, Byzantine fault tolerance, Conjecture, Karush–Kuhn–Tucker conditions, Optimization problem, PCP theorem, Proof complexity, Matching (graph theory), One-way function, C0