**Publication data:** c. 300 BC

**Online version:** Interactive Java version

**Description:** This is probably not only the most important work in geometry but the most important work in mathematics. It contains many important results in geometry, number theory and the first algorithm as well. The *Elements* is still a valuable resource and a good introduction to algorithm. More than any specific result in the publication, it seems that the major achievement of this publication is the popularization of logic and mathematical proof as a method of solving problems.

**Importance:** Topic creator, Breakthrough, Influence, Introduction, Latest and greatest (though it is the first, some of the results are still the latest)

**Description:** La 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.

**Importance:** Topic creator, Breakthrough, Influence

**Description:** 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 disappointing way, by Gödel's incompleteness theorem in 1931.

**Importance:** Influence

(Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, Monatshefte für Mathematik und Physik, vol. 38 (1931).)

**Online version:** Online version

**Description:** In mathematical logic, **Gödel's incompleteness theorems** are two celebrated theorems proved by Kurt Gödel in 1930. The first incompleteness theorem states:

For any formal system such that (1) it is ω-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.

**Importance:** Breakthrough, Influence

See the list of publications in information theory.

**Description:** The *Disquisitiones Arithmeticae* is a textbook of 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 important new results of his own.

**Importance:** Breakthrough, Influence

**Description:** **On the Number of Primes Less Than a Given Magnitude** (or **Über die Anzahl der Primzahlen unter einer gegebenen Grösse**) 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.

**Importance:** Breakthrough, Influence

**Description:** *Vorlesungen über Zahlentheorie* (*Lectures on Number Theory*) is a textbook of number theory written by German mathematicians P.G.L. Dirichlet and Richard 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

**Importance:** Breakthrough, Influence

**Description:** The **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 July 5, 1687. Probably 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. He derives Kepler's laws for the motion of the planets (which were first obtained empirically). In formulating his physical theories, Newton had developed a field of mathematics known as calculus.

Up to the publication of this book, mathematics was only used to *describe* nature. This is the first instance when mathematics is used to *explain* nature. 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 other words, the greatness of the Principia is not only in developing a number of fundamental theories in physics and mathematics but first and foremost (amply demonstrated in the title!) in the very linking of science and mathematics. The influence of this book is so deep that nowadays we find this link obvious and cannot imagine doing science in any other way.

**Importance:** Topic creator, Breakthrough, Influence

**Description:** **Method 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.

**Importance:** Topic creator, Breakthrough, Influence

John Maynard Smith

(Theory of Games and Economic Behavior, 3rd ed., Princeton University Press 1953)

**Description:** This book led to the investigation of modern game theory as a prominent branch of mathematics. This profound work contained the method -- alluded to above -- for finding optimal solutions for two-person zero-sum games.

**Importance:** Influence, Topic creator, Breakthrough

**Description:** The 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.

**Description:** A compendium of information on mathematical games. It was first published in 1982 in two volumes, one focusing on Conway's game theory and surreal numbers, and the other concentrating on a number of specific games.

**Description:** A 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.

**Importance:** 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.

## Early manuscripts

These are publications that are not necessarily relevant to a mathematician nowadays, but are nonetheless important publications in the History of mathematics.

**Description:** It is 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. Besides describing how to obtain an approximation of π only missing the mark by under 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.

**Description:** a Chinese mathematics book, probably composed in the 1st century AD, but perhaps as early as 200 BC. Among its content: 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.

**Description:** Although 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. Contrary to historically ignorant statements found in some 20th-century calculus textbooks, he did not use anything like Riemann sums, either in the work embodied in this palimpsest or in any of his other works. For explicit details of the method used, see how Archimedes used infinitesimals.

**Online version:** Online version

**Description:** The first known (European) system of number-naming that can be expanded beyond the needs of everyday life.

## Textbooks

**Description:** A 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.

- Richard Rusczyk and Sandor Lehoczky

**Description:** *The Art of Problem Solving* began as a set of two books coauthored by Richard Rusczyk and Sandor Lehoczky. The books, which are about 750 pages together, are for students who are interested in math and/or compete in math competitions .

**Description:** An excellent introduction to the mathematical theory of logical formal systems, covering completeness-proofs, consistency-proofs, and so on and even set-theory .

## Popular writing

**Description:** Gödel, 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."

**Description**: Written in 1542, it was the first really popular arithmetic book written in the English Language.

**Description**: An early and popular English arithmetic textbook published in America in the eighteenth century. The book reached from the introductory topics to the advanced in five sections.

*Faisceaux Algébriques Cohérents*

**Publication data:** *Annals of Mathematics*, 1955

**Description:** *FAC*, as it is usually called, first introduced the use of sheaves into algebraic geometry. Serre introduced Cech cohomology of sheaves in this paper, and, despite its technical deficiencies, revolutionized 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. Before *FAC*, this was next to impossible. While Grothendieck's derived functor cohomology has replaced Cech cohomology for technical reasons, actual calculations, such as of the cohomology of projective space, are usually carried out by Cech techniques, and for this reason Serre's paper remains important even today.

**Importance:** Topic creator, Breakthrough, Influence

**Description:** 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.

**Importance:** Topic creator, Breakthrough, Influence

*Categories for the Working Mathematician*

**Description:** Saunders Mac Lane, one of the founders of category theory, wrote this exposition to bring categories to the masses. Mac Lane does not get lost in pointless abstraction, but instead brings to the fore the important concepts that make category theory useful, such as adjoint functors and universal objects . His text is more comprehensive than most mathematicians will ever need, and consequently is also an excellent reference.

**Importance:** Introduction

*Category Theory for Computing Science*

- Michael Barr and Charles Wells

**Description:** Slower-paced introduction that Mac Lane's, assuming much less math background. Suitable for budding computer-scientists, logicians, linguists, etc. 1999 edition contains extensive exercises and solutions.

**Importance:** Introduction

*Topology from the Differentiable Viewpoint*

**Description:** This 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.

**Importance:** Influence

*Algebraic Topology*

**Publication data:** Cambridge University Press, 2002.

**Online version:** http://www.math.cornell.edu/~hatcher/AT/ATpage.html

**Description:** This is the first in a series of three textbooks in algebraic topology having the goal of covering all the basics while remaining readable by newcomers seeing the subject for the first time. The first book contains the basic core material along with a number of optional topics of a relatively elementary nature.

**Importance:** Introduction

**Description:** An 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 advanced topics like large cardinals. Instead it tried, and succeeds, in being intelligible to someone who has never thought about set theory before.

**Importance**: Influence, Introduction

**Kantorovich** wrote the first paper on production planning, which used Linear Programs as the model. He proposed the simplex algorithm as a systematic procedure to solve these Linear Programs. He received Nobel prize for this work in 1975. This work was published in 1930's in the erstwhile Soviet Republic.

**Dantzig** Dantzig's is considered the father of Linear Programming in the western world. He independently invented the simplex method. Dantzig and Wolfe worked on decomposition algorithms for large scale linear programs in factory and production planning.

G. B. Dantzig and P. Wolfe. Decomposition Principle for Linear Programs. Operations Research 8:101–111, 1960.

**Network Flows and General Matchings** Ford and Fulkerson paper on Network Flows. The algorithm along with many ideas on flow-based models can be found in their book. This book is supposedly very well written.

Ford, L., & Fulkerson, D. 1962. Flows in Networks. Prentice-Hall.

Jack Edmonds work on General Matchings.

J. Edmonds. Paths, trees and Flowers. Canadian Journal of Mathematics, 17:449–467, 1965.

**NP Hardness and Computational Complexity**

S. Cook. The complexity of theorem-proving procedures. In Proc. 3rd ACM Symp. Theory of Computing, pages 151–158, 1971.

R. Karp. Reducibility among combinatorial problems. In Complexity of Computer Computations, pages 85–104. Plenum Press, New York, 1972.

Klee and Minty example showing that simplex method can take exponentially many steps to solve a linear program if it chooses the greedy ascent rule.

V. Klee and G. J. Minty: How good is the simplex algorithm? In: O. Shisha (ed.) Inequalities III, Academic Press (1972) 159–175.

**Linear Programming and Polynomial time algorithms**

Khachiyan's work on Ellipsoid method. This was the first polynomial time algorithm for Linear programming.

L. Khachiyan. A polynomial Algorithm in Linear Programming. Doklady Akademii Nauk SSSR 244 (1979) pp. 1093–1096 (Russian).

Karmarkars path-breaking work on Interior-Point algorithms for Linear Programming.

Karmarkar, N. (1984), ‘New polynomial-time algorithm for linear programming’, Combinatorica 4, 373–395.

**Convex optimization and Polynomial time algorithms**

Nesterov and Nemirovski's work on Self-concordant barriers and Interior-Point Methods for general convex programming. All their series of papers (both individual and combined) is compiled more coherently in the following "bible" of convex optimization.

Interior Point Polynomial Algorithms in Convex Programming / Yurii NESTEROV and A. NEMIROVSKY. Philadelphia : Society for Industrial and Applied Mathematics, 1994. (SIAM Studies in Applied Mathematics).