数学++: 基础课程之外的精选论题(影印版)

《数学＋＋》是一本简 洁易懂的关于20世纪6个选 定数学领域的介绍，提供了 许多现代数学工具，这些工 具在计算机科学、工程学等 领域的当前研究中被广泛应 用。这些领域包括测度论、 高维几何、傅里叶分析、群 表示、多元多项式和拓扑学 。对每个领域，作者介绍了 基本概念、例子和结果。本 书阐述清晰易懂，强调直观 理解，并包括精心挑选的练 习作为内容的一部分。理论 计算机科学和离散数学的一 些应用对理论做了补充—— 有些应用非常令人惊讶。各 章相互独立，读者可以按任 意顺序学习。 作者假定读者已经学习 了基础数学课程。虽然作者 是在教授理论计算机科学和 离散数学的博士生时构思这 本书的，但它适合更广泛的 读者阅读，例如其他研究方 向的数学家、决定从事专业 研究的数学学生或工程学等 领域的专家。

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