Mathematics and Computation: A Theory Revolutionizing Technology and Science (Hardcover)

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

《數學與計算》是一本關於計算複雜性理論的廣泛概念性概述，介紹了它與數學的聯繫和互動，以及它在自然科學、社會科學、技術和哲學中的核心作用。

計算複雜性理論是對有效計算的數學研究，對計算機科學和工業具有重要的實際應用。它已經發展成為一個高度跨學科的領域，與大多數數學領域以及越來越多的科學努力有著密切的聯繫。

作者 Avi Wigderson 對複雜性理論進行了廣泛的調查，強調了該領域的見解和挑戰。他解釋了導致關鍵模型、概念和結果的思想和動機。特別是，他關注算法和複雜性、計算和證明、隨機性和交互作用、量子和算術計算以及密碼學和學習，所有這些都是一個有著眾多相互影響的統一整體的一部分。Wigderson展示了這個領域的廣度、美麗和豐富性，以及它與數學其他領域的多樣且不斷增長的互動。他最後全面探討了計算理論、其方法論和抱負，以及它在科學、技術和社會方面所塑造和將繼續塑造的獨特而基礎的方式。書中還提供了廣泛的參考文獻，供讀者進一步閱讀。

《數學與計算》對數學、計算機科學和相關領域的本科生和研究生以及這些領域的研究人員和教師都很有用。許多部分不需要太多背景知識，可以作為對計算理論入門的邀請。

主要特點包括：
- 全面涵蓋計算複雜性理論及其延伸領域
- 高層次、直觀的解釋，為這一中心且動態的科學學科帶來概念上的清晰度
- 對中心概念和模型的演變和動機的歷史描述
- 對計算理論對科學、技術和社會的影響的廣泛觀點
- 廣泛的參考文獻

Avi Wigderson is the Herbert H. Maass Professor in the School of Mathematics at the Institute for Advanced Study in Princeton, New Jersey.

Avi Wigderson是普林斯頓高等研究院數學學院的Herbert H. Maass教授。