內容大鋼
《數學＋＋》是一本簡潔易懂的關於20世紀6個選定數學領域的介紹，提供了許多現代數學工具，這些工具在電腦科學、工程學等領域的當前研究中被廣泛應用。這些領域包括測度論、高維幾何、傅里葉分析、群表示、多元多項式和拓撲學。對每個領域，作者介紹了基本概念、例子和結果。本書闡述清晰易懂，強調直觀理解，並包括精心挑選的練習作為內容的一部分。理論電腦科學和離散數學的一些應用對理論做了補充——有些應用非常令人驚訝。各章相互獨立，讀者可以按任意順序學習。
作者假定讀者已經學習了基礎數學課程。雖然作者是在教授理論電腦科學和離散數學的博士生時構思這本書的，但它適合更廣泛的讀者閱讀，例如其他研究方向的數學家、決定從事專業研究的數學學生或工程學等領域的專家。
目錄
Preface
Chapter 1.Measure and Integral
§1.Measure
§2.The Lebesgue Integral
§3.Foundations of Probability Theory
§4.Literature
Bibliography
Chapter 2.High-Dimensional Geometry and Measure Concentration
§1.Peculiarities of Large Dimensions
§2.The Brunn-Minkowski Inequality and Euclidean Isoperimetry
§3.The Standard Normal Distribution and the Gaussian Measure
§4.Measure Concentration
§5.Literature
Bibliography
Chapter 3.Fourier Analysis
§1.Characters
§2.The Fourier Transform
§3.Two Unexpected Applications
§4.Convolution
§5.Poisson Summation Formula
§6.Influence of Variables
§7.Infinite Groups
§8.Literature
Bibliography
Chapter 4.Representations of Finite Groups
§1.Basic Definitions and Examples
§2.Decompositions into Irreducible Representations
§3.Irreducible Decompositions, Characters, Orthogonality
§4.Irreducible Representations of the Symmetric Group
§5.An Application in Communication Complexity
§6.More Applications and Literature
Bibliography
Chapter 5.Polynomials
§1.Rings, Fields, and Polynomials
§2.The Schwartz-Zippel Theorem
§3.Polynomial Identity Testing
§4.Interpolation, Joints, and Contagious Vanishing
§5.Varieties, Ideals, and the Hilbert Basis Theorem
§6.The Nullstellensatz
§7.Bezout's Inequality in the Plane
§8.More Properties of Varieties
§9.Bezout's Inequality in Higher Dimensions
§10.Bounding the Number of Connected Components
§11.Literature
Bibliography
Chapter 6.Topology
§1.Topological Spaces and Continuous Maps
§2.Bits of General Topology
§3.Compactness
§4.Homotopy and Homotopy Equivalence
§5.The Borsuk-Ulam Theorem
§6.Operations on Topological Spaces
§7.Simplicial Complexes and Relatives
§8.Non-embeddability
§9.Homotopy Groups
§10.Homology of Simplicial Complexes
§11.Simplicial Approximation
§12.Homology Does Not Depend on Triangulation
§13.A Quick Harvest and Two More Theorems
§14.Manifolds
§15.Literature
Bibliography
Index
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