Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within society at large; the press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton, its first book was a new 1912 edition of John Witherspoon's Lectures on Moral Philosophy. Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two existing local publishers, that of the Princeton Alumni Weekly and the Princeton Press; the new press printed both local newspapers, university documents, The Daily Princetonian, added book publishing to its activities. Beginning as a small, for-profit printer, Princeton University Press was reincorporated as a nonprofit in 1910.
Since 1911, the press has been headquartered in a purpose-built gothic-style building designed by Ernest Flagg. The design of press’s building, named the Scribner Building in 1965, was inspired by the Plantin-Moretus Museum, a printing museum in Antwerp, Belgium. Princeton University Press established a European office, in Woodstock, north of Oxford, in 1999, opened an additional office, in Beijing, in early 2017. Six books from Princeton University Press have won Pulitzer Prizes: Russia Leaves the War by George F. Kennan Banks and Politics in America from the Revolution to the Civil War by Bray Hammond Between War and Peace by Herbert Feis Washington: Village and Capital by Constance McLaughlin Green The Greenback Era by Irwin Unger Machiavelli in Hell by Sebastian de Grazia Books from Princeton University Press have been awarded the Bancroft Prize, the Nautilus Book Award, the National Book Award. Multi-volume historical documents projects undertaken by the Press include: The Collected Papers of Albert Einstein The Writings of Henry D. Thoreau The Papers of Woodrow Wilson The Papers of Thomas Jefferson Kierkegaard's WritingsThe Papers of Woodrow Wilson has been called "one of the great editorial achievements in all history."
Princeton University Press's Bollingen Series had its beginnings in the Bollingen Foundation, a 1943 project of Paul Mellon's Old Dominion Foundation. From 1945, the foundation had independent status and providing fellowships and grants in several areas of study, including archaeology and psychology; the Bollingen Series was given to the university in 1969. Annals of Mathematics Studies Princeton Series in Astrophysics Princeton Series in Complexity Princeton Series in Evolutionary Biology Princeton Series in International Economics Princeton Modern Greek Studies The Whites of Their Eyes: The Tea Party's Revolution and the Battle over American History, by Jill Lepore The Meaning of Relativity by Albert Einstein Atomic Energy for Military Purposes by Henry DeWolf Smyth How to Solve It by George Polya The Open Society and Its Enemies by Karl Popper The Hero With a Thousand Faces by Joseph Campbell The Wilhelm/Baynes translation of the I Ching, Bollingen Series XIX. First copyright 1950, 27th printing 1997.
Anatomy of Criticism by Northrop Frye Philosophy and the Mirror of Nature by Richard Rorty QED: The Strange Theory of Light and Matter by Richard Feynman The Great Contraction 1929–1933 by Milton Friedman and Anna Jacobson Schwartz with a new Introduction by Peter L. Bernstein Military Power: Explaining Victory and Defeat in Modern Battle by Stephen Biddle Banks, Eric. "Book of Lists: Princeton University Press at 100". Artforum International. Staff of Princeton University Press. A Century in Books: Princeton University Press, 1905–2005. ISBN 9780691122922. CS1 maint: Uses authors parameter Official website Princeton University Press: Albert Einstein Web Page Princeton University Press: Bollingen Series Princeton University Press: Annals of Mathematics Studies Princeton University Press Centenary Princeton University Press: New in Print
Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry, algebraic number theory. He was awarded the Fields Medal in 1954 and the inaugural Abel Prize in 2003. Born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and from 1945 to 1948 at the École Normale Supérieure in Paris, he was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France, a position he held until his retirement in 1994, his wife, Professor Josiane Heulot-Serre, was a chemist. Their daughter is the former French diplomat and writer Claudine Monteil; the French mathematician Denis Serre is his nephew. He practices skiing, table tennis, rock climbing. From a young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and commutative algebra and algebraic geometry, where he introduced sheaf theory and homological algebra techniques.
Serre's thesis concerned the Leray–Serre spectral sequence associated to a fibration. Together with Cartan, Serre established the technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres, which at that time was one of the major problems in topology. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl gave high praise to Serre, made the point that the award was for the first time awarded to a non-analyst. Serre subsequently changed his research focus. In the 1950s and 1960s, a fruitful collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were Faisceaux Algébriques Cohérents, on coherent cohomology, Géometrie Algébrique et Géométrie Analytique. At an early stage in his work Serre had perceived a need to construct more general and refined cohomology theories to tackle the Weil conjectures; the problem was that the cohomology of a coherent sheaf over a finite field couldn't capture as much topology as singular cohomology with integer coefficients.
Amongst Serre's early candidate theories of 1954–55 was one based on Witt vector coefficients. Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties – those that become trivial after pullback by a finite étale map – are important; this acted as one important source of inspiration for Grothendieck to develop étale topology and the corresponding theory of étale cohomology. These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique 4 and SGA 5, provided the tools for the eventual proof of the Weil conjectures by Pierre Deligne. From 1959 onward Serre's interests turned towards group theory, number theory, in particular Galois representations and modular forms. Amongst his most original contributions were: his "Conjecture II" on Galois cohomology. In his paper FAC, Serre asked whether a finitely generated projective module over a polynomial ring is free; this question led to a great deal of activity in commutative algebra, was answered in the affirmative by Daniel Quillen and Andrei Suslin independently in 1976.
This result is now known as the Quillen–Suslin theorem. Serre, at twenty-seven in 1954, is the youngest to be awarded the Fields Medal, he went on to win the Balzan Prize in 1985, the Steele Prize in 1995, the Wolf Prize in Mathematics in 2000, was the first recipient of the Abel Prize in 2003. He has been awarded other prizes, such as the Gold Medal of the French National Scientific Research Centre, he has received many honorary degrees. In 2012 he became a fellow of the American Mathematical Society. Serre has been awarded the highest honors in France as Grand Cross of the Legion of Honour and Grand Cross of the Legion of Merit. List of things named after Jean-Pierre Serre Nicolas Bourbaki Groupes Algébriques et Corps de Classes, translated into English as Algebraic Groups and Class Fields, Springer-Verlag Corps Locaux, Hermann, as Local Fields, Springer-Verlag Cohomologie Galoisienne Collège de France course 1962–63, as Galois Cohomology, Springer-Verlag Algèbre Locale, Multiplicités Collège de France course 1957–58, as Local Algebra, Springer-Verlag "Lie algebras and Lie groups" Harvard Lectures, Springer-Verlag.
Algèbres de Lie Semi-simples Complexes, as Complex Semisimple Lie Algebras, Springer-Verlag Abelian ℓ-Adic Representations and Elliptic Curves, CRC Press, reissue. Addison-Wesley. 1989. Cours d'arithmétique, PUF, as A Course in Arithmetic, Springer-Verlag Représentations linéaires des groupes finis, Hermann, as Linear Represent
Homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are topological invariants, the homotopy groups are complex and difficult to compute; the n-dimensional unit sphere — called the n-sphere for brevity, denoted as Sn — generalizes the familiar circle and the ordinary sphere. The n-sphere may be defined geometrically as the set of points in a Euclidean space of dimension n + 1 located at a unit distance from the origin; the i-th homotopy group πi summarizes the different ways in which the i-dimensional sphere Si can be mapped continuously into the n-dimensional sphere Sn. This summary does not distinguish between two mappings if one can be continuously deformed to the other. An "addition" operation defined on these equivalence classes makes the set of equivalence classes into an abelian group.
The problem of determining πi falls into three regimes, depending on whether i is less than, equal to, or greater than n. For 0 < i < n, any mapping from Si to Sn is homotopic to a constant mapping, i.e. a mapping that maps all of Si to a single point of Sn. Therefore the homotopy group is the trivial group; when i = n, every map from Sn to itself has a degree that measures how many times the sphere is wrapped around itself. This degree identifies the homotopy group πn with the group of integers under addition. For example, every point on a circle can be mapped continuously onto a point of another circle; the most interesting and surprising results occur. The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere S3 around the usual sphere S2 in a non-trivial fashion, so is not equivalent to a one-point mapping; the question of computing the homotopy group πn+k for positive k turned out to be a central question in algebraic topology that has contributed to development of many of its fundamental techniques and has served as a stimulating focus of research.
One of the main discoveries is that the homotopy groups πn+k are independent of n for n ≥ k + 2. These are called the stable homotopy groups of spheres and have been computed for values of k up to 64; the stable homotopy groups form the coefficient ring of an extraordinary cohomology theory, called stable cohomotopy theory. The unstable homotopy groups are more erratic. Most modern computations use spectral sequences, a technique first applied to homotopy groups of spheres by Jean-Pierre Serre. Several important patterns much remains unknown and unexplained; the study of homotopy groups of spheres builds on a great deal of background material, here reviewed. Algebraic topology provides the larger context, itself built on topology and abstract algebra, with homotopy groups as a basic example. An ordinary sphere in three-dimensional space — the surface, not the solid ball — is just one example of what a sphere means in topology. Geometry defines a sphere rigidly, as a shape. Here are some alternatives.
Implicit surface: x02 + x12 + x22 = 1This is the set of points in 3-dimensional Euclidean space found one unit away from the origin. It is called the 2-sphere, S2, for reasons given below; the same idea applies for any dimension n. For example, the 1-sphere S1 is a circle. Disk with collapsed rim: written in topology as D2/S1This construction moves from geometry to pure topology; the disk D2 is the region contained by a circle, described by the inequality x02 + x12 ≤ 1, its rim is the circle S1, described by the equality x02 + x12 = 1. If a balloon is punctured and spread flat it produces a disk; the slash, pronounced "modulo", means to take the topological space on the left and in it join together as one all the points on the right. The region is 2-dimensional, why topology calls the resulting topological space a 2-sphere. Generalized, Dn/Sn−1 produces Sn. For example, D1 is a line segment, the construction joins its ends to make a circle. An equivalent description is that the boundary of an n-dimensional disk is glued to a point, producing a CW complex.
Suspension of equator: written in topology as ΣS1This construction, though simple, is of great theoretical importance. Take the circle S1 to be the equator, sweep each point on it to one point above, producing the northern hemisphere, to one point below, producing the southern hemisphere. For each positive integer n, the n-sphere x02 + x12 + ⋯ + xn2 = 1 has as equator the -sphere x02 + x12 + ⋯ + xn−12 = 1, the suspension ΣSn−1 produces Sn; some theory requires selecting a fixed point on the sphere. For some spaces the choice matters, but for a sphere all points are equivalent so the choice is a matter of convenience; the point, on the equator of all the spheres, works well for geometric spheres. The distinguishing feature of a topological space is its continuity structure, formalized in terms of open sets or neighborhoods
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though most classify up to homotopy equivalence. Although algebraic topology uses algebra to study topological problems, using topology to solve algebraic problems is sometimes possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Below are some of the main areas studied in algebraic topology: In mathematics, homotopy groups are used in algebraic topology to classify topological spaces; the first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. In algebraic topology and abstract algebra, homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group.
In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign'quantities' to the chains of homology theory. A manifold is a topological space. Examples include the plane, the sphere, the torus, which can all be realized in three dimensions, but the Klein bottle and real projective plane which cannot be realized in three dimensions, but can be realized in four dimensions. Results in algebraic topology focus on global, non-differentiable aspects of manifolds. Knot theory is the study of mathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone.
In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R 3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R 3 upon itself. A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments and their n-dimensional counterparts. Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory; the purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. A CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory; this class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones.
In the 1920s and 1930s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groups, which led to the change of name to algebraic topology. The combinatorial topology name is still sometimes used to emphasize an algorithmic approach based on decomposition of spaces. In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism of spaces; this allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure making these statement easier to prove. Two major ways in which this can be done are through fundamental groups, or more homotopy theory, through homology and cohomology groups; the fundamental groups give us basic information about the structure of a topological space, but they are nonabelian and can be difficult to work with. The fundamental group of a simplicial complex does have a finite presentation.
Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. Finitely generated abelian groups are classified and are easy to work with. In general, all constructions of algebraic topology are functorial. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms correspond — a continuous mapping of spaces induces a group homomorphism on the associated groups, these homomorphisms can be used to show non-existence of mappings. One of the first mathematicians to work with different types of cohomology was Georges de Rham. One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf co