This is a list of important publications in
mathematics, organized by field.
Some reasons why a particular publication
might be regarded as important:
Topic creator – A publication that created
a new topic
Breakthrough – A publication that changed
scientific knowledge significantly
Influence – A publication which has significantly
influenced the world or has had a massive
impact on the teaching of mathematics.Among
published compilations of important publications
in mathematics are Landmark writings in Western
mathematics 1640–1940 by Ivor Grattan-Guinness
and A Source Book in Mathematics by David
Eugene Smith.
== Algebra ==
=== Theory of equations ===
==== Baudhayana Sulba Sutra ====
Baudhayana (8th century BCE)Believed to have
been written around the 8th century BCE, this
is one of the oldest mathematical texts. It
laid the foundations of Indian mathematics
and was influential in South Asia and its
surrounding regions, and perhaps even Greece.
Though this was primarily a geometrical text,
it also contained some important algebraic
developments, including the earliest list
of Pythagorean triples discovered algebraically,
geometric solutions of linear equations, the
earliest use of quadratic equations of the
forms ax2 = c and ax2 + bx = c, and integral
solutions of simultaneous Diophantine equations
with up to four unknowns.
==== The Nine Chapters on the Mathematical
Art ====
The Nine Chapters on the Mathematical Art
from the 10th–2nd century BCE.Contains the
earliest description of Gaussian elimination
for solving
system of linear equations, it also contains
method for finding square root and cubic root.
==== Haidao Suanjing ====
Liu Hui (220-280 CE)Contains the application
of right angle triangles for survey of depth
or height of distant objects.
==== Sunzi Suanjing ====
Sunzi (5th century CE)Contains the earliest
description of Chinese remainder theorem.
==== Aryabhatiya ====
Aryabhata (499 CE)Aryabhata introduced the
method known as "Modus Indorum" or the method
of the Indians that has become our algebra
today. This algebra came along with the Hindu
Number system to Arabia and then migrated
to Europe. The text contains 33 verses covering
mensuration (kṣetra vyāvahāra), arithmetic
and geometric progressions, gnomon / shadows
(shanku-chhAyA), simple, quadratic, simultaneous,
and indeterminate equations. It also gave
the modern standard algorithm for solving
first-order diophantine equations.
==== Jigu Suanjing ====
Jigu Suanjing (626 CE)
This book by Tang dynasty mathematician Wang
Xiaotong Contains the world's earliest third
order equation.
==== Brāhmasphuṭasiddhānta ====
Brahmagupta (628 CE)Contained rules for manipulating
both negative and positive numbers, rules
for dealing the number zero, a method for
computing square roots, and general methods
of solving linear and some quadratic equations,
solution to Pell's equation.
==== Al-Kitāb al-mukhtaṣar fī hīsāb
al-ğabr wa'l-muqābala ====
Muhammad ibn Mūsā al-Khwārizmī (820 CE)The
first book on the systematic algebraic solutions
of linear and quadratic equations by the Persian
scholar Muhammad ibn Mūsā al-Khwārizmī.
The book is considered to be the foundation
of modern algebra and Islamic mathematics.
The word "algebra" itself is derived from
the al-Jabr in the title of the book.
=== Līlāvatī, Siddhānta Shiromani and
Bijaganita ===
One of the major treatises on mathematics
by Bhāskara II provides the solution for
indeterminate equations of 1st and 2nd order.
==== Yigu yanduan ====
Liu Yi (12th century)Contains the earliest
invention of 4th order polynomial equation.
==== Mathematical Treatise in Nine Sections
====
Qin Jiushao (1247)This 13th century book contains
the earliest complete solution of 19th century
Horner's method of solving
high order polynomial equations (up to 10th
order). It also contains a complete solution
of Chinese remainder theorem, which predates
Euler and Gauss by several centuries.
==== Ceyuan haijing ====
Li Zhi (1248)Contains the application of high
order polynomial equation in solving complex
geometry problems.
==== Jade Mirror of the Four Unknowns ====
Zhu Shijie (1303)Contains the method of establishing
system of high order polynomial equations
of up to four unknowns.
==== Ars Magna ====
Gerolamo Cardano (1545)Otherwise known as
The Great Art, provided the first published
methods for solving cubic and quartic equations
(due to Scipione del Ferro, Niccolò Fontana
Tartaglia, and Lodovico Ferrari), and exhibited
the first published calculations involving
non-real complex numbers.
==== Vollständige Anleitung zur Algebra ====
Leonhard Euler (1770)Also known as Elements
of Algebra, Euler's textbook on elementary
algebra is one of the first to set out algebra
in the modern form we would recognize today.
The first volume deals with determinate equations,
while the second part deals with Diophantine
equations. The last section contains a proof
of Fermat's Last Theorem for the case n = 3,
making some valid assumptions regarding Q(√−3)
that Euler did not prove.
==== Demonstratio nova theorematis omnem functionem
algebraicam rationalem integram unius variabilis
in factores reales primi vel secundi gradus
resolvi posse ====
Carl Friedrich Gauss (1799)Gauss' doctoral
dissertation, which contained a widely accepted
(at the time) but incomplete proof of the
fundamental theorem of algebra.
=== Abstract algebra ===
==== Group theory ====
===== Réflexions sur la résolution algébrique
des équations =====
Joseph Louis Lagrange (1770)The title means
"Reflections on the algebraic solutions of
equations". Made the prescient observation
that the roots of the Lagrange resolvent of
a polynomial equation are tied to permutations
of the roots of the original equation, laying
a more general foundation for what had previously
been an ad hoc analysis and helping motivate
the later development of the theory of permutation
groups, group theory, and Galois theory. The
Lagrange resolvent also introduced the discrete
Fourier transform of order 3.
==== Articles Publiés par Galois dans les
Annales de Mathématiques ====
Journal de Mathematiques pures et Appliquées,
II (1846)Posthumous publication of the mathematical
manuscripts of Évariste Galois by Joseph
Liouville. Included are Galois' papers Mémoire
sur les conditions de résolubilité des équations
par radicaux and Des équations primitives
qui sont solubles par radicaux.
==== Traité des substitutions et des équations
algébriques ====
Camille Jordan (1870)Online version: Online
version
Traité des substitutions et des équations
algébriques (Treatise on Substitutions and
Algebraic Equations). The first book on group
theory, giving a then-comprehensive study
of permutation groups and Galois theory. In
this book, Jordan introduced the notion of
a simple group and epimorphism (which he called
l'isomorphisme mériédrique), proved part
of the Jordan–Hölder theorem, and discussed
matrix groups over finite fields as well as
the Jordan normal form.
==== Theorie der Transformationsgruppen ====
Sophus Lie, Friedrich Engel (1888–1893).Publication
data: 3 volumes, B.G. Teubner, Verlagsgesellschaft,
mbH, Leipzig, 1888–1893. Volume 1, Volume
2, Volume 3.
The first comprehensive work on transformation
groups, serving as the foundation for the
modern theory of Lie groups.
==== Solvability of groups of odd order ====
Walter Feit and John Thompson (1960)Description:
Gave a complete proof of the solvability of
finite groups of odd order, establishing the
long-standing Burnside conjecture that all
finite non-abelian simple groups are of even
order. Many of the original techniques used
in this paper were used in the eventual classification
of finite simple groups.
==== Homological algebra ====
==== Homological Algebra ====
Henri Cartan and Samuel Eilenberg (1956)Provided
the first fully worked out treatment of abstract
homological algebra, unifying previously disparate
presentations of homology and cohomology for
associative algebras, Lie algebras, and groups
into a single theory.
==== "Sur Quelques Points d'Algèbre Homologique"
====
Alexander Grothendieck (1957)Revolutionized
homological algebra by introducing abelian
categories and providing a general framework
for Cartan and Eilenberg's notion of derived
functors.
== Algebraic geometry ==
=== "Theorie der Abelschen Functionen" ===
Bernhard Riemann (1857)Publication data: Journal
für die Reine und Angewandte Mathematik
Developed the concept of Riemann surfaces
and their topological properties beyond Riemann's
1851 thesis work, proved an index theorem
for the genus (the original formulation of
the Riemann–Hurwitz formula), proved the
Riemann inequality for the dimension of the
space of meromorphic functions with prescribed
poles (the original formulation of the Riemann–Roch
theorem), discussed birational transformations
of a given curve and the dimension of the
corresponding moduli space of inequivalent
curves of a given genus, and solved more general
inversion problems than those investigated
by Abel and Jacobi. André Weil once wrote
that this paper "is one of the greatest pieces
of mathematics that has ever been written;
there is not a single word in it that is not
of consequence."
=== Faisceaux Algébriques Cohérents ===
Jean-Pierre SerrePublication data: Annals
of Mathematics, 1955
FAC, as it is usually called, was foundational
for the use of sheaves in algebraic geometry,
extending beyond the case of complex manifolds.
Serre introduced Čech cohomology of sheaves
in this paper, and, despite some technical
deficiencies, revolutionized formulations
of algebraic geometry. For example, the long
exact sequence in sheaf cohomology allows
one to show that some surjective maps of sheaves
induce surjective maps on sections; specifically,
these are the maps whose kernel (as a sheaf)
has a vanishing first cohomology group. The
dimension of a vector space of sections of
a coherent sheaf is finite, in projective
geometry, and such dimensions include many
discrete invariants of varieties, for example
Hodge numbers. While Grothendieck's derived
functor cohomology has replaced Čech cohomology
for technical reasons, actual calculations,
such as of the cohomology of projective space,
are usually carried out by Čech techniques,
and for this reason Serre's paper remains
important.
=== Géométrie Algébrique et Géométrie
Analytique ===
Jean-Pierre Serre (1956)In mathematics, algebraic
geometry and analytic geometry are closely
related subjects, where analytic geometry
is the theory of complex manifolds and the
more general analytic spaces defined locally
by the vanishing of analytic functions of
several complex variables. A (mathematical)
theory of the relationship between the two
was put in place during the early part of
the 1950s, as part of the business of laying
the foundations of algebraic geometry to include,
for example, techniques from Hodge theory.
(NB While analytic geometry as use of Cartesian
coordinates is also in a sense included in
the scope of algebraic geometry, that is not
the topic being discussed in this article.)
The major paper consolidating the theory was
Géometrie Algébrique et Géométrie Analytique
by Serre, now usually referred to as GAGA.
A GAGA-style result would now mean any theorem
of comparison, allowing passage between a
category of objects from algebraic geometry,
and their morphisms, and a well-defined subcategory
of analytic geometry objects and holomorphic
mappings.
=== "Le théorème de Riemann–Roch, d'après
A. Grothendieck" ===
Armand Borel, Jean-Pierre Serre (1958)Borel
and Serre's exposition of Grothendieck's version
of the Riemann–Roch theorem, published after
Grothendieck made it clear that he was not
interested in writing up his own result. Grothendieck
reinterpreted both sides of the formula that
Hirzebruch proved in 1953 in the framework
of morphisms between varieties, resulting
in a sweeping generalization. In his proof,
Grothendieck broke new ground with his concept
of Grothendieck groups, which led to the development
of K-theory.
=== Éléments de géométrie algébrique
===
Alexander Grothendieck (1960–1967)Written
with the assistance of Jean Dieudonné, this
is Grothendieck's exposition of his reworking
of the foundations of algebraic geometry.
It has become the most important foundational
work in modern algebraic geometry. The approach
expounded in EGA, as these books are known,
transformed the field and led to monumental
advances.
=== Séminaire de géométrie algébrique
===
Alexander Grothendieck et al.These seminar
notes on Grothendieck's reworking of the foundations
of algebraic geometry report on work done
at IHÉS starting in the 1960s. SGA 1 dates
from the seminars of 1960–1961, and the
last in the series, SGA 7, dates from 1967
to 1969. In contrast to EGA, which is intended
to set foundations, SGA describes ongoing
research as it unfolded in Grothendieck's
seminar; as a result, it is quite difficult
to read, since many of the more elementary
and foundational results were relegated to
EGA. One of the major results building on
the results in SGA is Pierre Deligne's proof
of the last of the open Weil conjectures in
the early 1970s. Other authors who worked
on one or several volumes of SGA include Michel
Raynaud, Michael Artin, Jean-Pierre Serre,
Jean-Louis Verdier, Pierre Deligne, and Nicholas
Katz.
== Number theory ==
=== Brāhmasphuṭasiddhānta ===
Brahmagupta (628)Brahmagupta's Brāhmasphuṭasiddhānta
is the first book that mentions zero as a
number, hence Brahmagupta is considered the
first to formulate the concept of zero. The
current system of the four fundamental operations
(addition, subtraction, multiplication and
division) based on the Hindu-Arabic number
system also first appeared in Brahmasphutasiddhanta.
It was also one of the first texts to provide
concrete ideas on positive and negative numbers.
=== De fractionibus continuis dissertatio
===
Leonhard Euler (1744)First presented in 1737,
this paper provided the first then-comprehensive
account of the properties of continued fractions.
It also contains the first proof that the
number e is irrational.
=== Recherches d'Arithmétique ===
Joseph Louis Lagrange (1775)Developed a general
theory of binary quadratic forms to handle
the general problem of when an integer is
representable by the form
a
x
2
+
b
y
2
+
c
x
y
{\displaystyle ax^{2}+by^{2}+cxy}
. This included a reduction theory for binary
quadratic forms, where he proved that every
form is equivalent to a certain canonically
chosen reduced form.
=== Disquisitiones Arithmeticae ===
Carl Friedrich Gauss (1801)The Disquisitiones
Arithmeticae is a profound and masterful book
on number theory written by German mathematician
Carl Friedrich Gauss and first published in
1801 when Gauss was 24. In this book Gauss
brings together results in number theory obtained
by mathematicians such as Fermat, Euler, Lagrange
and Legendre and adds many important new results
of his own. Among his contributions was the
first complete proof known of the Fundamental
theorem of arithmetic, the first two published
proofs of the law of quadratic reciprocity,
a deep investigation of binary quadratic forms
going beyond Lagrange's work in Recherches
d'Arithmétique, a first appearance of Gauss
sums, cyclotomy, and the theory of constructible
polygons with a particular application to
the constructibility of the regular 17-gon.
Of note, in section V, article 303 of Disquisitiones,
Gauss summarized his calculations of class
numbers of imaginary quadratic number fields,
and in fact found all imaginary quadratic
number fields of class numbers 1, 2, and 3
(confirmed in 1986) as he had conjectured.
In section VII, article 358, Gauss proved
what can be interpreted as the first non-trivial
case of the Riemann Hypothesis for curves
over finite fields (the Hasse–Weil theorem).
=== "Beweis des Satzes, daß jede unbegrenzte
arithmetische Progression, deren erstes Glied
und Differenz ganze Zahlen ohne gemeinschaftlichen
Factor sind, unendlich viele Primzahlen enthält"
===
Peter Gustav Lejeune Dirichlet (1837)Pioneering
paper in analytic number theory, which introduced
Dirichlet characters and their L-functions
to establish Dirichlet's theorem on arithmetic
progressions. In subsequent publications,
Dirichlet used these tools to determine, among
other things, the class number for quadratic
forms.
=== "Über die Anzahl der Primzahlen unter
einer gegebenen Grösse" ===
Bernhard Riemann (1859)"Über die Anzahl der
Primzahlen unter einer gegebenen Grösse"
(or "On the Number of Primes Less Than a Given
Magnitude") is a seminal 8-page paper by Bernhard
Riemann published in the November 1859 edition
of the Monthly Reports of the Berlin Academy.
Although it is the only paper he ever published
on number theory, it contains ideas which
influenced dozens of researchers during the
late 19th century and up to the present day.
The paper consists primarily of definitions,
heuristic arguments, sketches of proofs, and
the application of powerful analytic methods;
all of these have become essential concepts
and tools of modern analytic number theory.
It also contains the famous Riemann Hypothesis,
one of the most important open problems in
mathematics.
=== Vorlesungen über Zahlentheorie ===
Peter Gustav Lejeune Dirichlet and Richard
DedekindVorlesungen über Zahlentheorie (Lectures
on Number Theory) is a textbook of number
theory written by German mathematicians P.
G. Lejeune Dirichlet and R. Dedekind, and
published in 1863.
The Vorlesungen can be seen as a watershed
between the classical number theory of Fermat,
Jacobi and Gauss, and the modern number theory
of Dedekind, Riemann and Hilbert. Dirichlet
does not explicitly recognise the concept
of the group that is central to modern algebra,
but many of his proofs show an implicit understanding
of group theory
=== Zahlbericht ===
David Hilbert (1897)Unified and made accessible
many of the developments in algebraic number
theory made during the nineteenth century.
Although criticized by André Weil (who stated
"more than half of his famous Zahlbericht
is little more than an account of Kummer's
number-theoretical work, with inessential
improvements") and Emmy Noether, it was highly
influential for many years following its publication.
=== Fourier Analysis in Number Fields and
Hecke's Zeta-Functions ===
John Tate (1950)Generally referred to simply
as Tate's Thesis, Tate's Princeton Ph.D. thesis,
under Emil Artin, is a reworking of Erich
Hecke's theory of zeta- and L-functions in
terms of Fourier analysis on the adeles. The
introduction of these methods into number
theory made it possible to formulate extensions
of Hecke's results to more general L-functions
such as those arising from automorphic forms.
=== "Automorphic Forms on GL(2)" ===
Hervé Jacquet and Robert Langlands (1970)This
publication offers evidence towards Langlands'
conjectures by reworking and expanding the
classical theory of modular forms and their
L-functions through the introduction of representation
theory.
=== "La conjecture de Weil. I." ===
Pierre Deligne (1974)Proved the Riemann hypothesis
for varieties over finite fields, settling
the last of the open Weil conjectures.
=== "Endlichkeitssätze für abelsche Varietäten
über Zahlkörpern" ===
Gerd Faltings (1983)Faltings proves a collection
of important results in this paper, the most
famous of which is the first proof of the
Mordell conjecture (a conjecture dating back
to 1922). Other theorems proved in this paper
include an instance of the Tate conjecture
(relating the homomorphisms between two abelian
varieties over a number field to the homomorphisms
between their Tate modules) and some finiteness
results concerning abelian varieties over
number fields with certain properties.
=== "Modular Elliptic Curves and Fermat's
Last Theorem" ===
Andrew Wiles (1995)This article proceeds to
prove a special case of the Shimura–Taniyama
conjecture through the study of the deformation
theory of Galois representations. This in
turn implies the famed Fermat's Last Theorem.
The proof's method of identification of a
deformation ring with a Hecke algebra (now
referred to as an R=T theorem) to prove modularity
lifting theorems has been an influential development
in algebraic number theory.
=== The geometry and cohomology of some simple
Shimura varieties ===
Michael Harris and Richard Taylor (2001)Harris
and Taylor provide the first proof of the
local Langlands conjecture for GL(n). As part
of the proof, this monograph also makes an
in depth study of the geometry and cohomology
of certain Shimura varieties at primes of
bad reduction.
=== "Le lemme fondamental pour les algèbres
de Lie" ===
Ngô Bảo ChâuNgô Bảo Châu proved a
long-standing unsolved problem in the classical
Langlands program, using methods from the
Geometric Langlands program.
== Analysis ==
=== Introductio in analysin infinitorum ===
Leonhard Euler (1748)The eminent historian
of mathematics Carl Boyer once called Euler's
Introductio in analysin infinitorum the greatest
modern textbook in mathematics. Published
in two volumes, this book more than any other
work succeeded in establishing analysis as
a major branch of mathematics, with a focus
and approach distinct from that used in geometry
and algebra. Notably, Euler identified functions
rather than curves to be the central focus
in his book. Logarithmic, exponential, trigonometric,
and transcendental functions were covered,
as were expansions into partial fractions,
evaluations of ζ(2k) for k a positive integer
between 1 and 13, infinite series-infinite
product formulas, continued fractions, and
partitions of integers. In this work, Euler
proved that every rational number can be written
as a finite continued fraction, that the continued
fraction of an irrational number is infinite,
and derived continued fraction expansions
for e and
e
{\displaystyle \textstyle {\sqrt {e}}}
. This work also contains a statement of Euler's
formula and a statement of the pentagonal
number theorem, which he had discovered earlier
and would publish a proof for in 1751.
=== Calculus ===
==== Yuktibhāṣā ====
Jyeshtadeva (1501)Written in India in 1530,
this was the world's first calculus text.
"This work laid the foundation for a complete
system of fluxions" and served as a summary
of the Kerala School's achievements in calculus,
trigonometry and mathematical analysis, most
of which were earlier discovered by the 14th
century mathematician Madhava. It is possible
that this text influenced the later development
of calculus in Europe. Some of its important
developments in calculus include: the fundamental
ideas of differentiation and integration,
the derivative, differential equations, term
by term integration, numerical integration
by means of infinite series, the relationship
between the area of a curve and its integral,
and the mean value theorem.
==== Nova methodus pro maximis et minimis,
itemque tangentibus, quae nec fractas nec
irrationales quantitates moratur, et singulare
pro illi calculi genus ====
Gottfried Leibniz (1684)Leibniz's first publication
on differential calculus, containing the now
familiar notation for differentials as well
as rules for computing the derivatives of
powers, products and quotients.
==== Philosophiae Naturalis Principia Mathematica
====
Isaac NewtonThe Philosophiae Naturalis Principia
Mathematica (Latin: "mathematical principles
of natural philosophy", often Principia or
Principia Mathematica for short) is a three-volume
work by Isaac Newton published on 5 July 1687.
Perhaps the most influential scientific book
ever published, it contains the statement
of Newton's laws of motion forming the foundation
of classical mechanics as well as his law
of universal gravitation, and derives Kepler's
laws for the motion of the planets (which
were first obtained empirically). Here was
born the practice, now so standard we identify
it with science, of explaining nature by postulating
mathematical axioms and demonstrating that
their conclusion are observable phenomena.
In formulating his physical theories, Newton
freely used his unpublished work on calculus.
When he submitted Principia for publication,
however, Newton chose to recast the majority
of his proofs as geometric arguments.
==== Institutiones calculi differentialis
cum eius usu in analysi finitorum ac doctrina
serierum ====
Leonhard Euler (1755)Published in two books,
Euler's textbook on differential calculus
presented the subject in terms of the function
concept, which he had introduced in his 1748
Introductio in analysin infinitorum. This
work opens with a study of the calculus of
finite differences and makes a thorough investigation
of how differentiation behaves under substitutions.
Also included is a systematic study of Bernoulli
polynomials and the Bernoulli numbers (naming
them as such), a demonstration of how the
Bernoulli numbers are related to the coefficients
in the Euler–Maclaurin formula and the values
of ζ(2n), a further study of Euler's constant
(including its connection to the gamma function),
and an application of partial fractions to
differentiation.
==== Über die Darstellbarkeit einer Function
durch eine trigonometrische Reihe ====
Bernhard Riemann (1867)Written in 1853, Riemann's
work on trigonometric series was published
posthumously. In it, he extended Cauchy's
definition of the integral to that of the
Riemann integral, allowing some functions
with dense subsets of discontinuities on an
interval to be integrated (which he demonstrated
by an example). He also stated the Riemann
series theorem, proved the Riemann–Lebesgue
lemma for the case of bounded Riemann integrable
functions, and developed the Riemann localization
principle.
==== Intégrale, longueur, aire ====
Henri Lebesgue (1901)Lebesgue's doctoral dissertation,
summarizing and extending his research to
date regarding his development of measure
theory and the Lebesgue integral.
=== Complex analysis ===
==== Grundlagen für eine allgemeine Theorie
der Functionen einer veränderlichen complexen
Grösse ====
Bernhard Riemann (1851)Riemann's doctoral
dissertation introduced the notion of a Riemann
surface, conformal mapping, simple connectivity,
the Riemann sphere, the Laurent series expansion
for functions having poles and branch points,
and the Riemann mapping theorem.
=== Functional analysis ===
==== Théorie des opérations linéaires ====
Stefan Banach (1932; originally published
1931 in Polish under the title Teorja operacyj.)The
first mathematical monograph on the subject
of linear metric spaces, bringing the abstract
study of functional analysis to the wider
mathematical community. The book introduced
the ideas of a normed space and the notion
of a so-called B-space, a complete normed
space. The B-spaces are now called Banach
spaces and are one of the basic objects of
study in all areas of modern mathematical
analysis. Banach also gave proofs of versions
of the open mapping theorem, closed graph
theorem, and Hahn–Banach theorem.
=== Fourier analysis ===
==== Mémoire sur la propagation de la chaleur
dans les corps solides ====
Joseph Fourier (1807)Introduced Fourier analysis,
specifically Fourier series. Key contribution
was to not simply use trigonometric series,
but to model all functions by trigonometric
series.
When Fourier submitted his paper in 1807,
the committee (which included Lagrange, Laplace,
Malus and Legendre, among others) concluded:
...the manner in which the author arrives
at these equations is not exempt of difficulties
and [...] his analysis to integrate them still
leaves something to be desired on the score
of generality and even rigour. Making Fourier
series rigorous, which in detail took over
a century, led directly to a number of developments
in analysis, notably the rigorous statement
of the integral via the Dirichlet integral
and later the Lebesgue integral.
==== Sur la convergence des séries trigonométriques
qui servent à représenter une fonction arbitraire
entre des limites données ====
Peter Gustav Lejeune Dirichlet (1829, expanded
German edition in 1837)In his habilitation
thesis on Fourier series, Riemann characterized
this work of Dirichlet as "the first profound
paper about the subject". This paper gave
the first rigorous proof of the convergence
of Fourier series under fairly general conditions
(piecewise continuity and monotonicity) by
considering partial sums, which Dirichlet
transformed into a particular Dirichlet integral
involving what is now called the Dirichlet
kernel. This paper introduced the nowhere
continuous Dirichlet function and an early
version of the Riemann–Lebesgue lemma.
==== On convergence and growth of partial
sums of Fourier series ====
Lennart Carleson (1966)Settled Lusin's conjecture
that the Fourier expansion of any
L
2
{\displaystyle L^{2}}
function converges almost everywhere.
== Geometry ==
=== Baudhayana Sulba Sutra ===
BaudhayanaWritten around the 8th century BC,
this is one of the oldest geometrical texts.
It laid the foundations of Indian mathematics
and was influential in South Asia and its
surrounding regions, and perhaps even Greece.
Among the important geometrical discoveries
included in this text are: the earliest list
of Pythagorean triples discovered algebraically,
the earliest statement of the Pythagorean
theorem, geometric solutions of linear equations,
several approximations of π, the first use
of irrational numbers, and an accurate computation
of the square root of 2, correct to a remarkable
five decimal places. Though this was primarily
a geometrical text, it also contained some
important algebraic developments, including
the earliest use of quadratic equations of
the forms ax2 = c and ax2 + bx = c, and integral
solutions of simultaneous Diophantine equations
with up to four unknowns.
=== Euclid's Elements ===
EuclidPublication data: c. 300 BC
Online version: Interactive Java version
This is often regarded as not only the most
important work in geometry but one of the
most important works in mathematics. It contains
many important results in plane and solid
geometry, algebra (books II and V), and number
theory (book VII, VIII, and IX). More than
any specific result in the publication, it
seems that the major achievement of this publication
is the promotion of an axiomatic approach
as a means for proving results. Euclid's Elements
has been referred to as the most successful
and influential textbook ever written.
=== The Nine Chapters on the Mathematical
Art ===
Unknown authorThis was a Chinese mathematics
book, mostly geometric, composed during the
Han Dynasty, perhaps as early as 200 BC. It
remained the most important textbook in China
and East Asia for over a thousand years, similar
to the position of Euclid's Elements in Europe.
Among its contents: Linear problems solved
using the principle known later in the West
as the rule of false position. Problems with
several unknowns, solved by a principle similar
to Gaussian elimination. Problems involving
the principle known in the West as the Pythagorean
theorem. The earliest solution of a matrix
using a method equivalent to the modern method.
=== The Conics ===
Apollonius of PergaThe Conics was written
by Apollonius of Perga, a Greek mathematician.
His innovative methodology and terminology,
especially in the field of conics, influenced
many later scholars including Ptolemy, Francesco
Maurolico, Isaac Newton, and René Descartes.
It was Apollonius who gave the ellipse, the
parabola, and the hyperbola the names by which
we know them.
=== Surya Siddhanta ===
Unknown (400 CE)Contains the roots of modern
trigonometry. It describes the archeo-astronomy
theories, principles and methods of the ancient
Hindus. This siddhanta is supposed to be the
knowledge that the Sun god gave to an Asura
called Maya. It uses sine (jya), cosine (kojya
or "perpendicular sine") and inverse sine
(otkram jya) for the first time, and also
contains the earliest use of the tangent and
secant. Later Indian mathematicians such as
Aryabhata made references to this text, while
later Arabic and Latin translations were very
influential in Europe and the Middle East.
=== Aryabhatiya ===
Aryabhata (499 CE)This was a highly influential
text during the Golden Age of mathematics
in India. The text was highly concise and
therefore elaborated upon in commentaries
by later mathematicians. It made significant
contributions to geometry and astronomy, including
introduction of sine/ cosine, determination
of the approximate value of pi and accurate
calculation of the earth's circumference.
=== La Géométrie ===
René DescartesLa Géométrie was published
in 1637 and written by René Descartes. The
book was influential in developing the Cartesian
coordinate system and specifically discussed
the representation of points of a plane, via
real numbers; and the representation of curves,
via equations.
=== Grundlagen der Geometrie ===
David HilbertOnline version: English
Publication data: Hilbert, David (1899). Grundlagen
der Geometrie. Teubner-Verlag Leipzig. ISBN
978-1-4020-2777-2.Hilbert's axiomatization
of geometry, whose primary influence was in
its pioneering approach to metamathematical
questions including the use of models to prove
axiom independence and the importance of establishing
the consistency and completeness of an axiomatic
system.
=== Regular Polytopes ===
H.S.M. CoxeterRegular Polytopes is a comprehensive
survey of the geometry of regular polytopes,
the generalisation of regular polygons and
regular polyhedra to higher dimensions. Originating
with an essay entitled Dimensional Analogy
written in 1923, the first edition of the
book took Coxeter 24 years to complete. Originally
written in 1947, the book was updated and
republished in 1963 and 1973.
=== Differential geometry ===
==== Recherches sur la courbure des surfaces
====
Leonhard Euler (1760)Publication data: Mémoires
de l'académie des sciences de Berlin 16 (1760)
pp. 119–143; published 1767. (Full text
and an English translation available from
the Dartmouth Euler archive.)
Established the theory of surfaces, and introduced
the idea of principal curvatures, laying the
foundation for subsequent developments in
the differential geometry of surfaces.
==== Disquisitiones generales circa superficies
curvas ====
Carl Friedrich Gauss (1827)Publication data:
"Disquisitiones generales circa superficies
curvas", Commentationes Societatis Regiae
Scientiarum Gottingesis Recentiores Vol. VI
(1827), pp. 99–146; "General Investigations
of Curved Surfaces" (published 1965) Raven
Press, New York, translated by A.M.Hiltebeitel
and J.C.Morehead.
Groundbreaking work in differential geometry,
introducing the notion of Gaussian curvature
and Gauss' celebrated Theorema Egregium.
==== Über die Hypothesen, welche der Geometrie
zu Grunde Liegen ====
Bernhard Riemann (1854)Publication data: "Über
die Hypothesen, welche der Geometrie zu Grunde
Liegen", Abhandlungen der Königlichen Gesellschaft
der Wissenschaften zu Göttingen, Vol. 13,
1867. English translation
Riemann's famous Habiltationsvortrag, in which
he introduced the notions of a manifold, Riemannian
metric, and curvature tensor.
==== Leçons sur la théorie génerale des
surfaces et les applications géométriques
du calcul infinitésimal ====
Gaston DarbouxPublication data: Darboux, Gaston
(1887,1889,1896). Leçons sur la théorie
génerale des surfaces. Gauthier-Villars.
Volume I, Volume II, Volume III, Volume IV
Leçons sur la théorie génerale des surfaces
et les applications géométriques du calcul
infinitésimal (on the General Theory of Surfaces
and the Geometric Applications of Infinitesimal
Calculus). A treatise covering virtually every
aspect of the 19th century differential geometry
of surfaces.
== Topology ==
=== Analysis situs ===
Henri Poincaré (1895, 1899–1905)Description:
Poincaré's Analysis Situs and his Compléments
à l'Analysis Situs laid the general foundations
for algebraic topology. In these papers, Poincaré
introduced the notions of homology and the
fundamental group, provided an early formulation
of Poincaré duality, gave the Euler–Poincaré
characteristic for chain complexes, and mentioned
several important conjectures including the
Poincaré conjecture.
=== L'anneau d'homologie d'une représentation,
Structure de l'anneau d'homologie d'une représentation
===
Jean Leray (1946)These two Comptes Rendus
notes of Leray from 1946 introduced the novel
concepts of sheafs, sheaf cohomology, and
spectral sequences, which he had developed
during his years of captivity as a prisoner
of war. Leray's announcements and applications
(published in other Comptes Rendus notes from
1946) drew immediate attention from other
mathematicians. Subsequent clarification,
development, and generalization by Henri Cartan,
Jean-Louis Koszul, Armand Borel, Jean-Pierre
Serre, and Leray himself allowed these concepts
to be understood and applied to many other
areas of mathematics. Dieudonné would later
write that these notions created by Leray
"undoubtedly rank at the same level in the
history of mathematics as the methods invented
by Poincaré and Brouwer".
=== Quelques propriétés globales des variétés
differentiables ===
René Thom (1954)In this paper, Thom proved
the Thom transversality theorem, introduced
the notions of oriented and unoriented cobordism,
and demonstrated that cobordism groups could
be computed as the homotopy groups of certain
Thom spaces. Thom completely characterized
the unoriented cobordism ring and achieved
strong results for several problems, including
Steenrod's problem on the realization of cycles.
== Category theory ==
=== "General Theory of Natural Equivalences"
===
Samuel Eilenberg and Saunders Mac Lane (1945)The
first paper on category theory. Mac Lane later
wrote in Categories for the Working Mathematician
that he and Eilenberg introduced categories
so that they could introduce functors, and
they introduced functors so that they could
introduce natural equivalences. Prior to this
paper, "natural" was used in an informal and
imprecise way to designate constructions that
could be made without making any choices.
Afterwards, "natural" had a precise meaning
which occurred in a wide variety of contexts
and had powerful and important consequences.
=== Categories for the Working Mathematician
===
Saunders Mac Lane (1971, second edition 1998)Saunders
Mac Lane, one of the founders of category
theory, wrote this exposition to bring categories
to the masses. Mac Lane brings to the fore
the important concepts that make category
theory useful, such as adjoint functors and
universal properties.
=== Higher Topos Theory ===
Jacob Lurie (2010)This purpose of this book
is twofold: to provide a general introduction
to higher category theory (using the formalism
of "quasicategories" or "weak Kan complexes"),
and to apply this theory to the study of higher
versions of Grothendieck topoi. A few applications
to classical topology are included. (see arXiv.)
== Set theory ==
=== "Über eine Eigenschaft des Inbegriffes
aller reellen algebraischen Zahlen" ===
Georg Cantor (1874)Online version: Online
version
Contains the first proof that the set of all
real numbers is uncountable; also contains
a proof that the set of algebraic numbers
is countable. (See Georg Cantor's first set
theory article.)
=== Grundzüge der Mengenlehre ===
Felix HausdorffFirst published in 1914, this
was the first comprehensive introduction to
set theory. Besides the systematic treatment
of known results in set theory, the book also
contains chapters on measure theory and topology,
which were then still considered parts of
set theory. Here Hausdorff presents and develops
highly original material which was later to
become the basis for those areas.
=== "The consistency of the axiom of choice
and of the generalized continuum-hypothesis
with the axioms of set theory" ===
Kurt Gödel (1938)Gödel proves the results
of the title. Also, in the process, introduces
the class L of constructible sets, a major
influence in the development of axiomatic
set theory.
=== "The Independence of the Continuum Hypothesis"
===
Paul J. Cohen (1963, 1964)Cohen's breakthrough
work proved the independence of the continuum
hypothesis and axiom of choice with respect
to Zermelo–Fraenkel set theory. In proving
this Cohen introduced the concept of forcing
which led to many other major results in axiomatic
set theory.
== Logic ==
=== The Laws of Thought ===
George Boole (1854)Published in 1854, The
Laws of Thought was the first book to provide
a mathematical foundation for logic. Its aim
was a complete re-expression and extension
of Aristotle's logic in the language of mathematics.
Boole's work founded the discipline of algebraic
logic and would later be central for Claude
Shannon in the development of digital logic.
=== Begriffsschrift ===
Gottlob Frege (1879)Published in 1879, the
title Begriffsschrift is usually translated
as concept writing or concept notation; the
full title of the book identifies it as "a
formula language, modelled on that of arithmetic,
of pure thought". Frege's motivation for developing
his formal logical system was similar to Leibniz's
desire for a calculus ratiocinator. Frege
defines a logical calculus to support his
research in the foundations of mathematics.
Begriffsschrift is both the name of the book
and the calculus defined therein. It was arguably
the most significant publication in logic
since Aristotle.
=== Formulario mathematico ===
Giuseppe Peano (1895)First published in 1895,
the Formulario mathematico was the first mathematical
book written entirely in a formalized language.
It contained a description of mathematical
logic and many important theorems in other
branches of mathematics. Many of the notations
introduced in the book are now in common use.
=== Principia Mathematica ===
Bertrand Russell and Alfred North Whitehead
(1910–1913)The Principia Mathematica is
a three-volume work on the foundations of
mathematics, written by Bertrand Russell and
Alfred North Whitehead and published in 1910–1913.
It is an attempt to derive all mathematical
truths from a well-defined set of axioms and
inference rules in symbolic logic. The questions
remained whether a contradiction could be
derived from the Principia's axioms, and whether
there exists a mathematical statement which
could neither be proven nor disproven in the
system. These questions were settled, in a
rather surprising way, by Gödel's incompleteness
theorem in 1931.
=== Systems of Logic Based on Ordinals ===
Alan Turing's Ph.D. thesis
=== "Über formal unentscheidbare Sätze der
Principia Mathematica und verwandter Systeme,
I" ===
(On Formally Undecidable Propositions of Principia
Mathematica and Related Systems)
Kurt Gödel (1931)Online version: Online version
In mathematical logic, Gödel's incompleteness
theorems are two celebrated theorems proved
by Kurt Gödel in 1931.
The first incompleteness theorem states:
For any formal system such that (1) it is
ω
{\displaystyle \omega }
-consistent (omega-consistent), (2) it has
a recursively definable set of axioms and
rules of derivation, and (3) every recursive
relation of natural numbers is definable in
it, there exists a formula of the system such
that, according to the intended interpretation
of the system, it expresses a truth about
natural numbers and yet it is not a theorem
of the system.
== Combinatorics ==
=== "On sets of integers containing no k elements
in arithmetic progression" ===
Endre Szemerédi (1975)Settled a conjecture
of Paul Erdős and Pál Turán (now known
as Szemerédi's theorem) that if a sequence
of natural numbers has positive upper density
then it contains arbitrarily long arithmetic
progressions. Szemerédi's solution has been
described as a "masterpiece of combinatorics"
and it introduced new ideas and tools to the
field including a weak form of the Szemerédi
regularity lemma.
=== Graph theory ===
==== Solutio problematis ad geometriam situs
pertinentis ====
Leonhard Euler (1741)
Euler's original publication (in Latin)Euler's
solution of the Königsberg bridge problem
in Solutio problematis ad geometriam situs
pertinentis (The solution of a problem relating
to the geometry of position) is considered
to be the first theorem of graph theory.
==== "On the evolution of random graphs" ====
Paul Erdős and Alfréd Rényi (1960)Provides
a detailed discussion of sparse random graphs,
including distribution of components, occurrence
of small subgraphs, and phase transitions.
==== "Network Flows and General Matchings"
====
L. R. Ford, Jr. & D. R. Fulkerson
Flows in Networks. Prentice-Hall, 1962.Presents
the Ford-Fulkerson algorithm for solving the
maximum flow problem, along with many ideas
on flow-based models.
== Computational complexity theory ==
See List of important publications in theoretical
computer science.
== Probability theory and statistics ==
See list of important publications in statistics.
== Game theory ==
=== "Zur Theorie der Gesellschaftsspiele"
===
John von Neumann (1928)Went well beyond Émile
Borel's initial investigations into strategic
two-person game theory by proving the minimax
theorem for two-person, zero-sum games.
=== Theory of Games and Economic Behavior
===
Oskar Morgenstern, John von Neumann (1944)This
book led to the investigation of modern game
theory as a prominent branch of mathematics.
This work contained the method for finding
optimal solutions for two-person zero-sum
games.
=== "Equilibrium Points in N-person Games"
===
Nash, John F. (January 1950). "Equilibrium
Points in N-person Games". Proceedings of
the National Academy of Sciences of the United
States of America. 36 (1): 48–9. Bibcode:1950PNAS...36...48N.
doi:10.1073/pnas.36.1.48. MR 0031701. PMC
1063129. PMID 16588946.Nash equilibrium
=== On Numbers and Games ===
John Horton ConwayThe book is in two, {0,1|},
parts. The zeroth part is about numbers, the
first part about games – both the values
of games and also some real games that can
be played such as Nim, Hackenbush, Col and
Snort amongst the many described.
=== Winning Ways for your Mathematical Plays
===
Elwyn Berlekamp, John Conway and Richard K.
GuyA compendium of information on mathematical
games. It was first published in 1982 in two
volumes, one focusing on Combinatorial game
theory and surreal numbers, and the other
concentrating on a number of specific games.
== Fractals ==
=== How Long Is the Coast of Britain? Statistical
Self-Similarity and Fractional Dimension ===
Benoît MandelbrotA discussion of self-similar
curves that have fractional dimensions between
1 and 2. These curves are examples of fractals,
although Mandelbrot does not use this term
in the paper, as he did not coin it until
1975.
Shows Mandelbrot's early thinking on fractals,
and is an example of the linking of mathematical
objects with natural forms that was a theme
of much of his later work.
== Numerical analysis ==
=== Optimization ===
==== Method of Fluxions ====
Isaac NewtonMethod of Fluxions was a book
written by Isaac Newton. The book was completed
in 1671, and published in 1736. Within this
book, Newton describes a method (the Newton–Raphson
method) for finding the real zeroes of a function.
==== Essai d'une nouvelle méthode pour déterminer
les maxima et les minima des formules intégrales
indéfinies ====
Joseph Louis Lagrange (1761)Major early work
on the calculus of variations, building upon
some of Lagrange's prior investigations as
well as those of Euler. Contains investigations
of minimal surface determination as well as
the initial appearance of Lagrange multipliers.
==== "Математические методы
организации и планирования
производства" ====
Leonid Kantorovich (1939) "[The Mathematical
Method of Production Planning and Organization]"
(in Russian).Kantorovich wrote the first paper
on production planning, which used Linear
Programs as the model. He received the Nobel
prize for this work in 1975.
==== "Decomposition Principle for Linear Programs"
====
George Dantzig and P. Wolfe
Operations Research 8:101–111, 1960.Dantzig's
is considered the father of linear programming
in the western world. He independently invented
the simplex algorithm. Dantzig and Wolfe worked
on decomposition algorithms for large-scale
linear programs in factory and production
planning.
==== "How Good is the Simplex Algorithm?"
====
Victor Klee and George J. Minty
Klee, Victor; Minty, George J. (1972). "How
good is the simplex algorithm?". In Shisha,
Oved (ed.). Inequalities III (Proceedings
of the Third Symposium on Inequalities held
at the University of California, Los Angeles,
Calif., September 1–9, 1969, dedicated to
the memory of Theodore S. Motzkin). New York-London:
Academic Press. pp. 159–175. MR 0332165.Klee
and Minty gave an example showing that the
simplex algorithm can take exponentially many
steps to solve a linear program.
==== "Полиномиальный алгоритм
в линейном программировании"
====
Khachiyan, Leonid Genrikhovich (1979). Полиномиальный
алгоритм в линейном программировании
[A polynomial algorithm for linear programming].
Doklady Akademii Nauk SSSR (in Russian). 244:
1093–1096..Khachiyan's work on the ellipsoid
method. This was the first polynomial time
algorithm for linear programming.
== Early manuscripts ==
These are publications that are not necessarily
relevant to a mathematician nowadays, but
are nonetheless important publications in
the history of mathematics.
=== Rhind Mathematical Papyrus ===
Ahmes (scribe)One of the oldest mathematical
texts, dating to the Second Intermediate Period
of ancient Egypt. It was copied by the scribe
Ahmes (properly Ahmose) from an older Middle
Kingdom papyrus. It laid the foundations of
Egyptian mathematics and in turn, later influenced
Greek and Hellenistic mathematics. Besides
describing how to obtain an approximation
of π only missing the mark by less than one
per cent, it is describes one of the earliest
attempts at squaring the circle and in the
process provides persuasive evidence against
the theory that the Egyptians deliberately
built their pyramids to enshrine the value
of π in the proportions. Even though it would
be a strong overstatement to suggest that
the papyrus represents even rudimentary attempts
at analytical geometry, Ahmes did make use
of a kind of an analogue of the cotangent.
=== Archimedes Palimpsest ===
Archimedes of SyracuseAlthough the only mathematical
tools at its author's disposal were what we
might now consider secondary-school geometry,
he used those methods with rare brilliance,
explicitly using infinitesimals to solve problems
that would now be treated by integral calculus.
Among those problems were that of the center
of gravity of a solid hemisphere, that of
the center of gravity of a frustum of a circular
paraboloid, and that of the area of a region
bounded by a parabola and one of its secant
lines. For explicit details of the method
used, see Archimedes' use of infinitesimals.
=== The Sand Reckoner ===
Archimedes of SyracuseOnline version: Online
version
The first known (European) system of number-naming
that can be expanded beyond the needs of everyday
life.
== Textbooks ==
=== Abstract Algebra ===
David Dummit and Richard Foote"Dummit and
Foote has become the modern dominant abstract
algebra textbook following Jacobson's Basic
Algebra.
=== Synopsis of Pure Mathematics ===
G. S. CarrContains over 6000 theorems of mathematics,
assembled by George Shoobridge Carr for the
purpose of training his students for the Cambridge
Mathematical Tripos exams. Studied extensively
by Ramanujan. (first half here)
=== Éléments de mathématique ===
Nicolas BourbakiOne of the most influential
books in French mathematical literature. It
introduces some of the notations and definitions
that are now usual (the symbol ∅ or the
term bijective for example). Characterized
by an extreme level of rigour, formalism and
generality (up to the point of being highly
criticized for that), its publication started
in 1939 and is still unfinished today.
=== Arithmetick: or, The Grounde of Arts ===
Robert RecordeWritten in 1542, it was the
first really popular arithmetic book written
in the English Language.
=== Cocker's Arithmetick ===
Edward Cocker (authorship disputed)Textbook
of arithmetic published in 1678 by John Hawkins,
who claimed to have edited manuscripts left
by Edward Cocker, who had died in 1676. This
influential mathematics textbook used to teach
arithmetic in schools in the United Kingdom
for over 150 years.
=== The Schoolmaster's Assistant, Being a
Compendium of Arithmetic both Practical and
Theoretical ===
Thomas DilworthAn early and popular English
arithmetic textbook published in America in
the 18th century. The book reached from the
introductory topics to the advanced in five
sections.
=== Geometry ===
Andrei KiselyovPublication data: 1892
The most widely used and influential textbook
in Russian mathematics. (See Kiselyov page
and MAA review.)
=== A Course of Pure Mathematics ===
G. H. HardyA classic textbook in introductory
mathematical analysis, written by G. H. Hardy.
It was first published in 1908, and went through
many editions. It was intended to help reform
mathematics teaching in the UK, and more specifically
in the University of Cambridge, and in schools
preparing pupils to study mathematics at Cambridge.
As such, it was aimed directly at "scholarship
level" students — the top 10% to 20% by
ability. The book contains a large number
of difficult problems. The content covers
introductory calculus and the theory of infinite
series.
=== Moderne Algebra ===
B. L. van der WaerdenThe first introductory
textbook (graduate level) expounding the abstract
approach to algebra developed by Emil Artin
and Emmy Noether. First published in German
in 1931 by Springer Verlag. A later English
translation was published in 1949 by Frederick
Ungar Publishing Company.
=== Algebra ===
Saunders Mac Lane and Garrett BirkhoffA definitive
introductory text for abstract algebra using
a category theoretic approach. Both a rigorous
introduction from first principles, and a
reasonably comprehensive survey of the field.
=== Calculus, Vol. 1 ===
Tom M. Apostol
=== Algebraic Geometry ===
Robin HartshorneThe first comprehensive introductory
(graduate level) text in algebraic geometry
that used the language of schemes and cohomology.
Published in 1977, it lacks aspects of the
scheme language which are nowadays considered
central, like the functor of points.
=== Naive Set Theory ===
Paul HalmosAn undergraduate introduction to
not-very-naive set theory which has lasted
for decades. It is still considered by many
to be the best introduction to set theory
for beginners. While the title states that
it is naive, which is usually taken to mean
without axioms, the book does introduce all
the axioms of Zermelo–Fraenkel set theory
and gives correct and rigorous definitions
for basic objects. Where it differs from a
"true" axiomatic set theory book is its character:
There are no long-winded discussions of axiomatic
minutiae, and there is next to nothing about
topics like large cardinals. Instead it aims,
and succeeds, in being intelligible to someone
who has never thought about set theory before.
=== Cardinal and Ordinal Numbers ===
Wacław SierpińskiThe nec plus ultra reference
for basic facts about cardinal and ordinal
numbers. If you have a question about the
cardinality of sets occurring in everyday
mathematics, the first place to look is this
book, first published in the early 1950s but
based on the author's lectures on the subject
over the preceding 40 years.
=== Set Theory: An Introduction to Independence
Proofs ===
Kenneth KunenThis book is not really for beginners,
but graduate students with some minimal experience
in set theory and formal logic will find it
a valuable self-teaching tool, particularly
in regard to forcing. It is far easier to
read than a true reference work such as Jech,
Set Theory. It may be the best textbook from
which to learn forcing, though it has the
disadvantage that the exposition of forcing
relies somewhat on the earlier presentation
of Martin's axiom.
=== Topologie ===
Pavel Sergeevich Alexandrov
Heinz HopfFirst published round 1935, this
text was a pioneering "reference" text book
in topology, already incorporating many modern
concepts from set-theoretic topology, homological
algebra and homotopy theory.
=== General Topology ===
John L. KelleyFirst published in 1955, for
many years the only introductory graduate
level textbook in the US, teaching the basics
of point set, as opposed to algebraic, topology.
Prior to this the material, essential for
advanced study in many fields, was only available
in bits and pieces from texts on other topics
or journal articles.
=== Topology from the Differentiable Viewpoint
===
John MilnorThis short book introduces the
main concepts of differential topology in
Milnor's lucid and concise style. While the
book does not cover very much, its topics
are explained beautifully in a way that illuminates
all their details.
=== Number Theory, An approach through history
from Hammurapi to Legendre ===
André WeilAn historical study of number theory,
written by one of the 20th century's greatest
researchers in the field. The book covers
some thirty six centuries of arithmetical
work but the bulk of it is devoted to a detailed
study and exposition of the work of Fermat,
Euler, Lagrange, and Legendre. The author
wishes to take the reader into the workshop
of his subjects to share their successes and
failures. A rare opportunity to see the historical
development of a subject through the mind
of one of its greatest practitioners.
=== An Introduction to the Theory of Numbers
===
G. H. Hardy and E. M. WrightAn Introduction
to the Theory of Numbers was first published
in 1938, and is still in print, with the latest
edition being the 6th (2008). It is likely
that almost every serious student and researcher
into number theory has consulted this book,
and probably has it on their bookshelf. It
was not intended to be a textbook, and is
rather an introduction to a wide range of
differing areas of number theory which would
now almost certainly be covered in separate
volumes. The writing style has long been regarded
as exemplary, and 
the approach gives insight into a variety
of areas without requiring much more than
a good grounding in algebra, calculus and
complex numbers.
=== Foundations of Differential Geometry ===
Shoshichi Kobayashi and Katsumi Nomizu
=== Hodge Theory and Complex Algebraic Geometry
I ===
=== Hodge Theory and Complex Algebraic Geometry
II ===
Claire Voisin
== Popular writings ==
=== Gödel, Escher, Bach ===
Douglas HofstadterGödel, Escher, Bach: an
Eternal Golden Braid is a Pulitzer Prize-winning
book, first published in 1979 by Basic Books.
It is 
a book about how the creative achievements
of logician Kurt Gödel, artist M. C. Escher
and composer Johann Sebastian Bach interweave.
As the author states: "I realized that to
me, Gödel and Escher and Bach were only shadows
cast in different directions by some central
solid essence. I tried to reconstruct the
central object, and came up with this book."
=== The World of Mathematics ===
James R. NewmanThe World of Mathematics was
specially designed to make mathematics more
accessible to the inexperienced. It comprises
nontechnical essays on every aspect of the
vast subject, including articles by and about
scores of eminent mathematicians, as well
as literary figures, economists, biologists,
and 
many other eminent thinkers. Includes the
work of Archimedes, Galileo, Descartes, Newton,
Gregor Mendel, Edmund Halley, Jonathan Swift,
John Maynard Keynes, Henri Poincaré, Lewis
Carroll, George Boole, Bertrand Russell, Alfred
North Whitehead, John von Neumann, and 
many others. In addition, an informative commentary
by distinguished scholar James R. Newman precedes
each essay or group of essays, explaining
their relevance and context in 
the history and development of mathematics.
Originally published in 1956, it does not
include many of the exciting discoveries of
the later years of the 20th century but it
has no equal as a general historical survey
of important topics and applications
