Raoul Bott was a Hungarian-American mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, the Borel–Bott–Weil theorem. Bott was born in Budapest, the son of Margit Kovács and Rudolph Bott, his father was of Austrian descent, his mother was of Hungarian Jewish descent. Bott spent his working life in the United States, his family emigrated to Canada in 1938, subsequently he served in the Canadian Army in Europe during World War II. Bott went to college at McGill University in Montreal, where he studied electrical engineering, he earned a Ph. D. in mathematics from Carnegie Mellon University in Pittsburgh in 1949. His thesis, titled Electrical Network Theory, was written under the direction of Richard Duffin. Afterward, he began teaching at the University of Michigan in Ann Arbor. Bott continued his study at the Institute for Advanced Study in Princeton, he was a professor at Harvard University from 1959 to 1999.
In 2005 Bott died of cancer in San Diego. With Richard Duffin at Carnegie Mellon, Bott studied existence of electronic filters corresponding to given positive-real functions. In 1949 they proved a fundamental theorem of filter synthesis. Duffin and Bott extended earlier work by Otto Brune that requisite functions of complex frequency s could be realized by a passive network of inductors and capacitors; the proof, relying on induction on the sum of the degrees of the polynomials in the numerator and denominator of the rational function, was published in Journal of Applied Physics, volume 20, page 816. In his 2000 interview with Allyn Jackson of the American Mathematical Society, he explained that he sees "networks as discrete versions of harmonic theory", so his experience with network synthesis and electronic filter topology introduced him to algebraic topology. Bott met Arnold S. Shapiro at the IAS and they worked together, he studied the homotopy theory of Lie groups, using methods from Morse theory, leading to the Bott periodicity theorem.
In the course of this work, he introduced Morse–Bott functions, an important generalization of Morse functions. This led to his role as collaborator over many years with Michael Atiyah via the part played by periodicity in K-theory. Bott made important contributions towards the index theorem in formulating related fixed-point theorems, in particular the so-called'Woods Hole fixed-point theorem', a combination of the Riemann–Roch theorem and Lefschetz fixed-point theorem; the major Atiyah–Bott papers on what is now the Atiyah–Bott fixed-point theorem were written in the years up to 1968. In the 1980s, Atiyah and Bott investigated gauge theory, using the Yang–Mills equations on a Riemann surface to obtain topological information about the moduli spaces of stable bundles on Riemann surfaces. In 1983 he spoke to the Canadian Mathematical Society in a talk he called "A topologist marvels at Physics", he is well known in connection with the Borel–Bott–Weil theorem on representation theory of Lie groups via holomorphic sheaves and their cohomology groups.
He introduced Bott–Samelson varieties and the Bott residue formula for complex manifolds and the Bott cannibalistic class. In 1964, he was awarded the Oswald Veblen Prize in Geometry by the American Mathematical Society. In 1983, he was awarded the Jeffery–Williams Prize by the Canadian Mathematical Society. In 1987, he was awarded the National Medal of Science. In 2000, he received the Wolf Prize. In 2005, he was elected an Overseas Fellow of the Royal Society of London. Bott had 35 Ph. D. students, including Stephen Smale, Lawrence Conlon, Daniel Quillen, Peter Landweber, Robert MacPherson, Robert W. Brooks, Robin Forman, András Szenes, Kevin Corlette. 1995: Collected Papers. Vol. 4. Mathematics Related to Physics. Edited by Robert MacPherson. Contemporary Mathematicians. Birkhäuser Boston, xx+485 pp. ISBN 0-8176-3648-X MR1321890 1995: Collected Papers. Vol. 3. Foliations. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xxxii+610 pp. ISBN 0-8176-3647-1 MR1321886 1994: Collected Papers.
Vol. 2. Differential Operators. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xxxiv+802 pp. ISBN 0-8176-3646-3 MR1290361 1994: Collected Papers. Vol. 1. Topology and Lie Groups. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xii+584 pp. ISBN 0-8176-3613-7 MR1280032 1982: Differential Forms in Algebraic Topology. Graduate Texts in Mathematics #82. Springer-Verlag, New York-Berlin. Xiv+331 pp. ISBN 0-387-90613-4 doi:10.1007/978-1-4757-3951-0 MR0658304 1969: Lectures on K. Mathematics Lecture Note New York-Amsterdam x +203 pp. MR0258020 Raoul Bott at the Mathematics Genealogy Project Commemorative website at Harvard Math Department "The Life and Works of Raoul Bott", by Loring Tu. "Raoul Bott, an Innovator in Mathematics, Dies at 82", The New York Times, January 8, 2006
Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, did foundational work in manifolds, immersions, characteristic classes, geometric integration theory. Hassler Whitney was born on March 23, 1907, in New York City, where his father Edward Baldwin Whitney was the First District New York Supreme Court judge, his mother, A. Josepha Newcomb Whitney, was an artist and active in politics, his paternal grandfather was William Dwight Whitney, professor of Ancient Languages at Yale University and Sanskrit scholar. Whitney was the great grandson of Connecticut Governor and US Senator Roger Sherman Baldwin, the great-great-grandson of American founding father Roger Sherman, his maternal grandparents were astronomer and mathematician Simon Newcomb, a Steeves descendant, Mary Hassler Newcomb, granddaughter of the first superintendent of the Coast Survey Ferdinand Rudolph Hassler. His great uncle Josiah Whitney was the first to survey Mount Whitney, he married three times: his first wife was Margaret R. Howell, married on the 30 May 1930.
They had three children, James Newcomb and Marian. After his first divorce, on January 16, 1955 he married Mary Barnett Garfield, he and Mary had Sarah Newcomb and Emily Baldwin. Whitney divorced his second wife and married Barbara Floyd Osterman on 8 February 1986. Whitney and his first wife Margaret made an innovative decision in 1939 that influenced the history of modern architecture in New England, when they commissioned the architect Edwin B. Goodell, Jr. to design a new residence for their family in Weston, Massachusetts. They purchased a rocky hillside site on a historic road, next door to another International Style house by Goodell from several years earlier, designed for Richard and Caroline Field. Distinctively featuring flat roofs, flush wood siding, corner windows—all of which were unusual architectural elements at the time—the Whitney House was a creative response to its site, in that it placed the main living spaces one floor above ground level, with large banks of windows opening to the south sun and to views of the beautiful property.
The Whitney House survives today, along with the Field House, more than 75 years following its original construction. Throughout his life he pursued two particular hobbies with excitement: mountain-climbing. An accomplished player of the violin and the viola, Whitney played with the Princeton Musical Amateurs, he would run outside, 6 to 12 miles every other day. As an undergraduate, with his cousin Bradley Gilman, Whitney made the first ascent of the Whitney–Gilman ridge on Cannon Mountain, New Hampshire in 1929, it was the hardest and most famous rock climb in the East. He was a member of the Swiss Alpine Society and the Yale Mountaineering Society and climbed most of the mountain peaks in Switzerland. Three years after his third marriage, on 10 May 1989, Whitney died in Princeton, after suffering a stroke. In accordance with his wish, Hassler Whitney ashes rest atop mountain Dents Blanches in Switzerland where Oscar Burlet, another mathematician and member of the Swiss Alpine Club, placed them on August 20, 1989.
Whitney attended Yale University where he received baccalaureate degrees in physics and in music in 1928 and in 1929. In 1932, he earned a PhD in mathematics at Harvard University, his doctoral dissertation was The Coloring of Graphs, written under the supervision of George David Birkhoff. At Harvard, Birkhoff got him a job as Instructor of Mathematics for the years 1930–31, an Assistant Professorship for the years 1934–35. On he held the following working positions: NRC Fellow, Mathematics, 1931–33, he was a member of the National Academy of Science. In 1947 he was elected member of the American Philosophical Society. In 1969 he was awarded the Lester R. Ford Award for the paper in two parts "The mathematics of Physical quantities". In 1976 he was awarded the National Medal of Science. In 1980 he was elected honorary member of the London Mathematical Society. In 1983 he received the Wolf Prize from the Wolf Foundation, in 1985, he was awarded the Steele Prize from the American Mathematical Society.
Whitney's earliest work, from 1930 to 1933, was on graph theory. Many of his contributions were to the graph-coloring, the ultimate computer-assisted solution to the four-color problem relied on some of his results, his work in graph theory culminated in a 1933 paper, where he laid the foundations for matroids, a fundamental notion in modern combinatorics and representation the
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
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More each point of an n-dimensional manifold has a neighbourhood, homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold. One-dimensional manifolds include circles, but not figure eights. Two-dimensional manifolds are called surfaces. Examples include the plane, the sphere, the torus, which can all be embedded in three dimensional real space, but the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space. Although a manifold locally resembles Euclidean space, meaning that every point has a neighbourhood homeomorphic to an open subset of Euclidean space, globally it may not: manifolds in general are not homeomorphic to Euclidean space. For example, the surface of the sphere is not homeomorphic to the Euclidean plane, because it has the global topological property of compactness that Euclidean space lacks, but in a region it can be charted by means of map projections of the region into the Euclidean plane.
When a region appears in two neighbouring charts, the two representations do not coincide and a transformation is needed to pass from one to the other, called a transition map. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described and understood in terms of the simpler local topological properties of Euclidean space. Manifolds arise as solution sets of systems of equations and as graphs of functions. Manifolds can be equipped with additional structure. One important class of manifolds is the class of differentiable manifolds. A Riemannian metric on a manifold allows angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. A surface is a two dimensional manifold, meaning that it locally resembles the Euclidean plane near each point. For example, the surface of a globe can be described by a collection of maps, which together form an atlas of the globe.
Although no individual map is sufficient to cover the entire surface of the globe, any place in the globe will be in at least one of the charts. Many places will appear in more than one chart. For example, a map of North America will include parts of South America and the Arctic circle; these regions of the globe will be described in full in separate charts, which in turn will contain parts of North America. There is a relation between adjacent charts, called a transition map that allows them to be patched together to cover the whole of the globe. Describing the coordinate charts on surfaces explicitly requires knowledge of functions of two variables, because these patching functions must map a region in the plane to another region of the plane. However, one-dimensional examples of manifolds can be described with functions of a single variable only. Manifolds have applications in computer-graphics and augmented-reality given the need to associate pictures to coordinates. In an augmented reality setting, a picture can be seen as something associated with a coordinate and by using sensors for detecting movements and rotation one can have knowledge of how the picture is oriented and placed in space.
After a line, the circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Consider, for instance, the top part of the unit circle, x2 + y2 = 1, where the y-coordinate is positive. Any point of this arc can be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous, invertible, mapping from the upper arc to the open interval: χ t o p = x; such functions along with the open regions they map are called charts. There are charts for the bottom and right parts of the circle: χ b o t t o m = x χ l e f t = y χ r i g h t = y. Together, these parts cover the four charts form an atlas for the circle; the top and right charts, χ t o
Leonhard Euler was a Swiss mathematician, astronomer and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while making pioneering contributions to several branches such as topology and analytic number theory. He introduced much of the modern mathematical terminology and notation for mathematical analysis, such as the notion of a mathematical function, he is known for his work in mechanics, fluid dynamics, optics and music theory. Euler was one of the most eminent mathematicians of the 18th century and is held to be one of the greatest in history, he is widely considered to be the most prolific mathematician of all time. His collected works fill more than anybody in the field, he spent most of his adult life in Saint Petersburg, in Berlin the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, Marguerite née Brucker, a pastor's daughter.
He had two younger sisters: Anna Maria and Maria Magdalena, a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Paul Euler was a friend of the Bernoulli family. Euler's formal education started in Basel. In 1720, aged thirteen, he enrolled at the University of Basel, in 1723, he received a Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who discovered his new pupil's incredible talent for mathematics. At that time Euler's main studies included theology and Hebrew at his father's urging in order to become a pastor, but Bernoulli convinced his father that Leonhard was destined to become a great mathematician. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono. At that time, he was unsuccessfully attempting to obtain a position at the University of Basel.
In 1727, he first entered the Paris Academy Prize Problem competition. Pierre Bouguer, who became known as "the father of naval architecture", won and Euler took second place. Euler won this annual prize twelve times. Around this time Johann Bernoulli's two sons and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia, when Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727, he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration.
Euler settled into life in Saint Petersburg. He took on an additional job as a medic in the Russian Navy; the Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made attractive to foreign scholars like Euler; the academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Few students were enrolled in the academy in order to lessen the faculty's teaching burden, the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions; the Academy's benefactress, Catherine I, who had continued the progressive policies of her late husband, died on the day of Euler's arrival. The Russian nobility gained power upon the ascension of the twelve-year-old Peter II; the nobility was suspicious of the academy's foreign scientists, thus cut funding and caused other difficulties for Euler and his colleagues.
Conditions improved after the death of Peter II, Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731. Two years Daniel Bernoulli, fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department. On 7 January 1734, he married Katharina Gsell, a daughter of Georg Gsell, a painter from the Academy Gymnasium; the young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood. Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia, he lived for 25 years in Berlin. In Berlin, he published the two works for which he would become most renowned: the Introductio in analysin infinitorum, a text on functions published in 1748, the Institutiones calculi differentialis, published in 1755 on differential calculus.
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. Milnor is a distinguished professor at Stony Brook University and one of the four mathematicians to have won the Fields Medal, the Wolf Prize, the Abel Prize. Milnor was born on February 1931 in Orange, New Jersey, his father was J. Willard Milnor and his mother was Emily Cox Milnor; as an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and proved the Fary–Milnor theorem. He continued on to graduate school at Princeton under the direction of Ralph Fox and submitted his dissertation, entitled "Isotopy of Links", which concerned link groups and their associated link structure, in 1954. Upon completing his doctorate he went on to work at Princeton, he was a professor at the Institute for Advanced Study from 1970 to 1990. His students have included Tadatoshi Akiba, Jon Folkman, John Mather, Laurent C. Siebenmann, Michael Spivak, his wife, Dusa McDuff, is a professor of mathematics at Barnard College.
One of his published works is his proof in 1956 of the existence of 7-dimensional spheres with nonstandard differential structure. With Michel Kervaire, he showed that the 7-sphere has 15 differentiable structures. An n-sphere with nonstandard differential structure is called an exotic sphere, a term coined by Milnor. Egbert Brieskorn found simple algebraic equations for 28 complex hypersurfaces in complex 5-space such that their intersection with a small sphere of dimension 9 around a singular point is diffeomorphic to these exotic spheres. Subsequently Milnor worked on the topology of isolated singular points of complex hypersurfaces in general, developing the theory of the Milnor fibration whose fiber has the homotopy type of a bouquet of μ spheres where μ is known as the Milnor number. Milnor's 1968 book on his theory inspired the growth of a huge and rich research area which continues to mature to this day. In 1961 Milnor disproved the Hauptvermutung by illustrating two simplicial complexes which are homeomorphic but combinatorially distinct.
In 1984 Milnor introduced a definition of attractor. The objects generalize standard attractors, include so-called unstable attractors and are now known as Milnor attractors. Milnor's current interest is dynamics holomorphic dynamics, his work in dynamics is summarized by Peter Makienko in his review of Topological Methods in Modern Mathematics: It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from the beginning, looking at the simplest nontrivial families of maps; the first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston. The case of a unimodal map, that is, one with a single critical point, turns out to be rich; this work may be compared with Poincaré's work on circle diffeomorphisms, which 100 years before had inaugurated the qualitative theory of dynamical systems.
Milnor's work has opened several new directions in this field, has given us many basic concepts, challenging problems and nice theorems. He was an editor of the Annals of Mathematics for a number of years after 1962, he has written a number of books. In 1962 Milnor was awarded the Fields Medal for his work in differential topology, he went on to win the National Medal of Science, the Lester R. Ford Award in 1970 and again in 1984, the Leroy P Steele Prize for "Seminal Contribution to Research", the Wolf Prize in Mathematics, the Leroy P Steele Prize for Mathematical Exposition, the Leroy P Steele Prize for Lifetime Achievement "... for a paper of fundamental and lasting importance, On manifolds homeomorphic to the 7-sphere, Annals of Mathematics 64, 399–405". In 1991 a symposium was held at Stony Brook University in celebration of his 60th birthday. Milnor was awarded the 2011 Abel Prize, for his "pioneering discoveries in topology and algebra." Reacting to the award, Milnor told the New Scientist "It feels good," adding that "ne is always surprised by a call at 6 o'clock in the morning."
In 2013 he became a fellow of the American Mathematical Society, for "contributions to differential topology, geometric topology, algebraic topology and dynamical systems". Milnor, John W.. Morse theory. Annals of Mathematics Studies, No. 51. Notes by M. Spivak and R. Wells. Princeton, NJ: Princeton University Press. ISBN 0-691-08008-9. ——. Lectures on the h-cobordism theorem. Notes by L. Siebenmann and J. Sondow. Princeton, NJ: Princeton University Press. ISBN 0-691-07996-X. OCLC 58324. ——. Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton, NJ: Princeton University Press. ISBN 0-691-08065-8. ——. Introduction to algebraic K-theory. Annals of Mathematics Studies, No. 72. Princeton, NJ: Princeton University Press. ISBN 978-0-691-08101-4. Husemoller, Dale. Symmetric bilinear forms. New York, NY: Springer-Verlag. ISBN 978-0-387-06009-5. Milnor, John W.. Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton, NJ: Princeton University Press. ISBN 0-691-08122-0. Milnor, John W..
Topology from the differentiable viewpoint. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-04833-9. —— (
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