III.38 Homology and Cohomology

Published by Princeton University Press 2009

III.38 Homology and Cohomology

From the book The Princeton Companion to Mathematics

https://doi.org/10.1515/9781400830398.221a

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From the book

The Princeton Companion to Mathematics

Chapters in this book (293)

Frontmatter

Contents

Preface

Contributors

Part I. Introduction

I.1 What Is Mathematics About?

I.2 The Language and Grammar of Mathematics

I.3 Some Fundamental Mathematical Definitions

I.4 The General Goals of Mathematical Research

Part II. The Origins of Modern Mathematics

II.1 From Numbers to Number Systems

II.2 Geometry

II.3 The Development of Abstract Algebra

II.4 Algorithms

II.5 The Development of Rigor in Mathematical Analysis

II.6 The Development of the Idea of Proof

II.7 The Crisis in the Foundations of Mathematics

Part III. Mathematical Concepts

III.1 The Axiom of Choice

III.2 The Axiom of Determinacy

III.3 Bayesian Analysis

III.4 Braid Groups

III.5 Buildings

III.6 Calabi–Yau Manifolds

III.7 Cardinals

III.8 Categories

III.9 Compactness and Compactification

III.10 Computational Complexity Classes

III.11 Countable and Uncountable Sets

III.12 C*-Algebras

III.13 Curvature

III.14 Designs

III.15 Determinants

III.16 Differential Forms and Integration

III.17 Dimension

III.18 Distributions

III.19 Duality

III.20 Dynamical Systems and Chaos

III.21 Elliptic Curves

III.22 The Euclidean Algorithm and Continued Fractions

III.23 The Euler and Navier–Stokes Equations

III.24 Expanders

III.25 The Exponential and Logarithmic Functions

III.26 The Fast Fourier Transform

III.27 The Fourier Transform

III.28 Fuchsian Groups

III.29 Function Spaces

III.30 Galois Groups

III.31 The Gamma Function

III.32 Generating Functions

III.33 Genus

III.34 Graphs

III.35 Hamiltonians

III.36 The Heat Equation

III.37 Hilbert Spaces

III.38 Homology and Cohomology

III.39 Homotopy Groups

III.40 The Ideal Class Group

III.41 Irrational and Transcendental Numbers

III.42 The Ising Model

III.43 Jordan Normal Form

III.44 Knot Polynomials

III.45 K-Theory

III.46 The Leech Lattice

III.47 L-Functions

III.48 Lie Theory

III.49 Linear and Nonlinear Waves and Solitons

III.50 Linear Operators and Their Properties

III.51 Local and Global in Number Theory

III.52 The Mandelbrot Set

III.53 Manifolds

III.54 Matroids

III.55 Measures

III.56 Metric Spaces

III.57 Models of Set Theory

III.58 Modular Arithmetic

III.59 Modular Forms

III.60 Moduli Spaces

III.61 The Monster Group

III.62 Normed Spaces and Banach Spaces

III.63 Number Fields

III.64 Optimization and Lagrange Multipliers

III.65 Orbifolds

III.66 Ordinals

III.67 The Peano Axioms

III.68 Permutation Groups

III.69 Phase Transitions

III.70 π

III.71 Probability Distributions

III.72 Projective Space

III.73 Quadratic Forms

III.74 Quantum Computation

III.75 Quantum Groups

III.76 Quaternions, Octonions, and Normed Division Algebras

III.77 Representations

III.78 Ricci Flow

III.79 Riemann Surfaces

III.80 The Riemann Zeta Function

III.81 Rings, Ideals, and Modules

III.82 Schemes

III.83 The Schrödinger Equation

III.84 The Simplex Algorithm

III.85 Special Functions

III.86 The Spectrum

III.87 Spherical Harmonics

III.88 Symplectic Manifolds

III.89 Tensor Products

III.90 Topological Spaces

III.91 Transforms

III.92 Trigonometric Functions

III.93 Universal Covers

III.94 Variational Methods

III.95 Varieties

III.96 Vector Bundles

III.97 Von Neumann Algebras

III.98 Wavelets

III.99 The Zermelo–Fraenkel Axioms

Part IV. Branches of Mathematics

IV.1 Algebraic Numbers

IV.2 Analytic Number Theory

IV.3 Computational Number Theory

IV.4 Algebraic Geometry

IV.5 Arithmetic Geometry

IV.6 Algebraic Topology

IV.7 Differential Topology

IV.8 Moduli Spaces

IV.9 Representation Theory

IV.10 Geometric and Combinatorial Group Theory

IV.11 Harmonic Analysis

IV.12 Partial Differential Equations

IV.13 General Relativity and the Einstein Equations

IV.14 Dynamics

IV.15 Operator Algebras

IV.16 Mirror Symmetry

IV.17 Vertex Operator Algebras

IV.18 Enumerative and Algebraic Combinatorics

IV.19 Extremal and Probabilistic Combinatorics

IV.20 Computational Complexity

IV.21 Numerical Analysis

IV.22 Set Theory

IV.23 Logic and Model Theory

IV.24 Stochastic Processes

IV.25 Probabilistic Models of Critical Phenomena

IV.26 High-Dimensional Geometry and Its Probabilistic Analogues

Part V. Theorems and Problems

V.1 The ABC Conjecture

V.2 The Atiyah–Singer Index Theorem

V.3 The Banach–Tarski Paradox

V.4 The Birch–Swinnerton-Dyer Conjecture

V.5 Carleson’s Theorem

V.6 The Central Limit Theorem

V.7 The Classification of Finite Simple Groups

V.8 Dirichlet’s Theorem

V.9 Ergodic Theorems

V.10 Fermat’s Last Theorem

V.11 Fixed Point Theorems

V.12 The Four-Color Theorem

V.13 The Fundamental Theorem of Algebra

V.14 The Fundamental Theorem of Arithmetic

V.15 Gödel’s Theorem

V.16 Gromov’s Polynomial-Growth Theorem

V.17 Hilbert’s Nullstellensatz

V.18 The Independence of the Continuum Hypothesis

V.19 Inequalities

V.20 The Insolubility of the Halting Problem

V.21 The Insolubility of the Quintic

V.22 Liouville’s Theorem and Roth’s Theorem

V.23 Mostow’s Strong Rigidity Theorem

V.24 The P versus NP Problem

V.25 The Poincaré Conjecture

V.26 The Prime Number Theorem and the Riemann Hypothesis

V.27 Problems and Results in Additive Number Theory

V.28 From Quadratic Reciprocity to Class Field Theory

V.29 Rational Points on Curves and the Mordell Conjecture

V.30 The Resolution of Singularities

V.31 The Riemann–Roch Theorem

V.32 The Robertson–Seymour Theorem

V.33 The Three-Body Problem

V.34 The Uniformization Theorem

V.35 The Weil Conjectures

Part VI. Mathematicians

VI.1 Pythagoras

VI.2 Euclid

VI.3 Archimedes

VI.4 Apollonius

VI.5 Abu Ja’far Muhammad ibn Mūsā al-Khwārizmī

VI.6 Leonardo of Pisa (known as Fibonacci)

VI.7 Girolamo Cardano

VI.8 Rafael Bombelli

VI.9 François Viète

VI.10 Simon Stevin

VI.11 René Descartes

VI.12 Pierre Fermat

VI.13 Blaise Pascal

VI.14 Isaac Newton

VI.15 Gottfried Wilhelm Leibniz

VI.16 Brook Taylor

VI.17 Christian Goldbach

VI.18 The Bernoullis

VI.19 Leonhard Euler

VI.20 Jean Le Rond d’Alembert

VI.21 Edward Waring

VI.22 Joseph Louis Lagrange

VI.23 Pierre-Simon Laplace

VI.24 Adrien-Marie Legendre

VI.25 Jean-Baptiste Joseph Fourier

VI.26 Carl Friedrich Gauss

VI.27 Siméon-Denis Poisson

VI.28 Bernard Bolzano

VI.29 Augustin-Louis Cauchy

VI.30 August Ferdinand Möbius

VI.31 Nicolai Ivanovich Lobachevskii

VI.32 George Green

VI.33 Niels Henrik Abel

VI.34 János Bolyai

VI.35 Carl Gustav Jacob Jacobi

VI.36 Peter Gustav Lejeune Dirichlet

VI.37 William Rowan Hamilton

VI.38 Augustus De Morgan

VI.39 Joseph Liouville

VI.40 Ernst Eduard Kummer

VI.41 Évariste Galois

VI.42 James Joseph Sylvester

VI.43 George Boole

VI.44 Karl Weierstrass

VI.45 Pafnuty Chebyshev

VI.46 Arthur Cayley

VI.47 Charles Hermite

VI.48 Leopold Kronecker

VI.49 Georg Friedrich Bernhard Riemann

VI.50 Julius Wilhelm Richard Dedekind

VI.51 Émile Léonard Mathieu

VI.52 Camille Jordan

VI.53 Sophus Lie

VI.54 Georg Cantor

VI.55 William Kingdon Clifford

VI.56 Gottlob Frege

VI.57 Christian Felix Klein

VI.58 Ferdinand Georg Frobenius

VI.59 Sofya (Sonya) Kovalevskaya

VI.60 William Burnside

VI.61 Jules Henri Poincaré

VI.62 Giuseppe Peano

VI.63 David Hilbert

VI.64 Hermann Minkowski

VI.65 Jacques Hadamard

VI.66 Ivar Fredholm

VI.67 Charles-Jean de la Vallée Poussin

VI.68 Felix Hausdorff

VI.69 Élie Joseph Cartan

VI.70 Emile Borel

VI.71 Bertrand Arthur William Russell

VI.72 Henri Lebesgue

VI.73 Godfrey Harold Hardy

VI.74 Frigyes (Frédéric) Riesz

VI.75 Luitzen Egbertus Jan Brouwer

VI.76 Emmy Noether

VI.77 Wacław Sierpiński

VI.78 George Birkhoff

VI.79 John Edensor Littlewood

VI.80 Hermann Weyl

VI.81 Thoralf Skolem

VI.82 Srinivasa Ramanujan

VI.83 Richard Courant

VI.84 Stefan Banach

VI.85 Norbert Wiener

VI.86 Emil Artin

VI.87 Alfred Tarski

VI.88 Andrei Nikolaevich Kolmogorov

VI.89 Alonzo Church

VI.90 William Vallance Douglas Hodge

VI.91 John von Neumann

VI.92 Kurt Gödel

VI.93 André Weil

VI.94 Alan Turing

VI.95 Abraham Robinson

VI.96 Nicolas Bourbaki

Part VII. The Influence of Mathematics

VII.1 Mathematics and Chemistry

VII.2 Mathematical Biology

VII.3 Wavelets and Applications

VII.4 The Mathematics of Traffic in Networks

VII.5 The Mathematics of Algorithm Design

VII.6 Reliable Transmission of Information

VII.7 Mathematics and Cryptography

VII.8 Mathematics and Economic Reasoning

VII.9 The Mathematics of Money

VII.10 Mathematical Statistics

VII.11 Mathematics and Medical Statistics

VII.12 Analysis, Mathematical and Philosophical

VII.13 Mathematics and Music

VII.14 Mathematics and Art

Part VIII. Final Perspectives

VIII.1 The Art of Problem Solving

VIII.2 “Why Mathematics?” You Might Ask

VIII.3 The Ubiquity of Mathematics

VIII.4 Numeracy

VIII.5 Mathematics: An Experimental Science

VIII.6 Advice to a Young Mathematician

VIII.7 A Chronology of Mathematical Events

Index