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
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, from a topological viewpoint they are the same; the word homeomorphism comes from the Greek words ὅμοιος = similar or same and μορφή = shape, introduced to mathematics by Henri Poincaré in 1895. Speaking, a topological space is a geometric object, the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other. However, this description can be misleading; some continuous deformations are not homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms are not continuous deformations, such as the homeomorphism between a trefoil knot and a circle.
An often-repeated mathematical joke is that topologists can't tell the difference between a coffee cup and a donut, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's handle. A function f: X → Y between two topological spaces is a homeomorphism if it has the following properties: f is a bijection, f is continuous, the inverse function f − 1 is continuous. A homeomorphism is sometimes called a bicontinuous function. If such a function exists, X and Y are homeomorphic. A self-homeomorphism is a homeomorphism from a topological space onto itself. "Being homeomorphic" is an equivalence relation on topological spaces. Its equivalence classes are called homeomorphism classes; the open interval is homeomorphic to the real numbers R for any a < b.. The unit 2-disc D 2 and the unit square in R2 are homeomorphic. An example of a bicontinuous mapping from the square to the disc is, in polar coordinates, ↦.
The graph of a differentiable function is homeomorphic to the domain of the function. A differentiable parametrization of a curve is a homeomorphism between the domain of the parametrization and the curve. A chart of a manifold is an homeomorphism between an open subset of the manifold and an open subset of a Euclidean space; the stereographic projection is a homeomorphism between the unit sphere in R3 with a single point removed and the set of all points in R2. If G is a topological group, its inversion map. For any x ∈ G, the left translation y ↦ x y, the right translation y ↦ y x, the inner automorphism y ↦ x y x − 1 are homeomorphisms. Rm and Rn are not homeomorphic for m ≠ n; the Euclidean real line is not homeomorphic to the unit circle as a subspace of R2, since the unit circle is compact as a subspace of Euclidean R2 but the real line is not compact. The one-dimensional intervals and are not homeomorphic because no continuous bijection could be made; the third requirement, that f − 1 be continuous, is essential.
Consider for instance the function f: [ 0, 2 π ) → S 1 defined by f =. This function is bijective and continuous, but not a homeomorphism ( S
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. —— (
In mathematics, a normal surface is a surface inside a triangulated 3-manifold that intersects each tetrahedron so that each component of intersection is a triangle or a quad. A triangle cuts off a vertex of the tetrahedron. A normal surface may have many components of intersection, called normal disks, with one tetrahedron, but no two normal disks can be quads that separate different pairs of vertices since that would lead to the surface self-intersecting. Dually, a normal surface can be considered to be a surface that intersects each handle of a given handle structure on the 3-manifold in a prescribed manner similar to the above; the concept of normal surface can be generalized to arbitrary polyhedra. There are related notions of normal surface and spun normal surface; the concept of normal surface is due to Hellmuth Kneser, who utilized it in his proof of the prime decomposition theorem for 3-manifolds. Wolfgang Haken extended and refined the notion to create normal surface theory, at the basis of many of the algorithms in 3-manifold theory.
The notion of normal surfaces is due to Hyam Rubinstein. The notion of spun normal surface is due to Bill Thurston. Regina is software which enumerates normal and almost-normal surfaces in triangulated 3-manifolds, implementing Rubinstein's 3-sphere recognition algorithm, among other things. Hatcher, Notes on basic 3-manifold topology, available online Gordon, ed. Kent, The theory of normal surfaces, Hempel, 3-manifolds, American Mathematical Society, ISBN 0-8218-3695-1 Jaco, Lectures on three-manifold topology, American Mathematical Society, ISBN 0-8218-1693-4 R. H. Bing, The Geometric Topology of 3-Manifolds, American Mathematical Society Colloquium Publications Volume 40, Providence RI, ISBN 0-8218-1040-5. Hass, What is an normal surface?, arXiv:1208.0568, Bibcode:2012arXiv1208.0568H Tillmann, Normal surfaces in topologically finite 3-manifolds, arXiv:math/0406271, Bibcode:2004math......6271T
Hellmuth Kneser was a Baltic German mathematician, who made notable contributions to group theory and topology. His most famous result may be his theorem on the existence of a prime decomposition for 3-manifolds, his proof originated the concept of normal surface, a fundamental cornerstone of the theory of 3-manifolds. He was died in Tübingen, Germany, he was the father of the mathematician Martin Kneser. He assisted Wilhelm Süss in the founding of the Mathematical Research Institute of Oberwolfach and served as the director of the institute from 1958 to 1959. Kneser had formulated the problem of non-integer iteration of functions and proved the existence of the entire Abel function of the exponential. Kneser was a student of David Hilbert, he was an advisor including Reinhold Baer. Hellmuth Kneser was a member of the NSDAP and the SA. In July 1934 he wrote to Ludwig Bieberbach a short note supporting his anti-semitic views and stating: "May God grant German science a unitary and continued political position."
O'Connor, John J.. Hellmuth Kneser at the Mathematics Genealogy Project
In mathematics, topology, a fiber bundle is a space, locally a product space, but globally may have a different topological structure. The similarity between a space E and a product space B × F is defined using a continuous surjective map π: E → B that in small regions of E behaves just like a projection from corresponding regions of B × F to B; the map π, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, F the fiber. In the trivial case, E is just B × F, the map π is just the projection from the product space to the first factor; this is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles such as the tangent bundle of a manifold and more general vector bundles play an important role in differential geometry and differential topology, as do principal bundles.
Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps, the class of fiber bundles forms a category with respect to such mappings. A bundle map from the base space itself to E is called a section of E. Fiber bundles can be specialized in a number of ways, the most common of, requiring that the transitions between the local trivial patches lie in a certain topological group, known as the structure group, acting on the fiber F. In topology, the terms fiber and fiber space appeared for the first time in a paper by Herbert Seifert in 1933, but his definitions are limited to a special case; the main difference from the present day conception of a fiber space, was that for Seifert what is now called the base space of a fiber space E was not part of the structure, but derived from it as a quotient space of E. The first definition of fiber space was given by Hassler Whitney in 1935 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle.
The theory of fibered spaces, of which vector bundles, principal bundles, topological fibrations and fibered manifolds are a special case, is attributed to Seifert, Heinz Hopf, Jacques Feldbau, Norman Steenrod, Charles Ehresmann, Jean-Pierre Serre, others. Fiber bundles became their own object of study in the period 1935–1940; the first general definition appeared in the works of Whitney. Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a sphere bundle, a fiber bundle whose fiber is a sphere of arbitrary dimension. A fiber bundle is a structure, where E, B, F are topological spaces and π: E → B is a continuous surjection satisfying a local triviality condition outlined below; the space B is called the base space of the bundle, E the total space, F the fiber. The map π is called the projection map. We shall assume in. We require that for every x in E, there is an open neighborhood U ⊂ B of π such that there is a homeomorphism φ: π−1 → U × F in such a way that π agrees with the projection onto the first factor.
That is, the following diagram should commute: where proj1: U × F → U is the natural projection and φ: π−1 → U × F is a homeomorphism. The set of all is called a local trivialization of the bundle, thus for any p in B, the preimage π−1 is homeomorphic to F and is called the fiber over p. Every fiber bundle π: E → B is an open map, since projections of products are open maps; therefore B carries the quotient topology determined by the map π. A fiber bundle is denoted that, in analogy with a short exact sequence, indicates which space is the fiber, total space and base space, as well as the map from total to base space. A smooth fiber bundle is a fiber bundle in the category of smooth manifolds; that is, E, B, F are required to be smooth manifolds and all the functions above are required to be smooth maps. Let E = B × F and let π: E → B be the projection onto the first factor. E is a fiber bundle over B. Here E is not just locally a product but globally one. Any such fiber bundle is called a trivial bundle.
Any fiber bundle over a contractible CW-complex is trivial. The simplest example of a nontrivial bundle E is the Möbius strip, it has the circle that runs lengthwise along the center of the strip as a base B and a line segment for the fiber F, so the Möbius strip is a bundle of the line segment over the circle. A neighborhood U of π ∈ B is an arc; the preimage π − 1 in the picture is a slice of the strip one long. A homeomorphism exists that maps the preimage of U to a slice of a cylinder: curved, but not twisted; this pair locally trivializes the strip. The corresponding trivial bundle B × F would be a cylinder, but the Möbius strip has an overall "twist". Note that this twist is visible only globally.
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer; this is made more precise in the definition below. A topological space X is a 3-manifold if it is a second-countable Hausdorff space and if every point in X has a neighbourhood, homeomorphic to Euclidean 3-space; the topological, piecewise-linear, smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, so there is a prevalence of specialized techniques that do not generalize to dimensions greater than three; this special role has led to the discovery of close connections to a diversity of other fields, such as knot theory, geometric group theory, hyperbolic geometry, number theory, Teichmüller theory, topological quantum field theory, gauge theory, Floer homology, partial differential equations.
3-manifold theory is considered a part of low-dimensional topology or geometric topology. A key idea in the theory is to study a 3-manifold by considering special surfaces embedded in it. One can choose the surface to be nicely placed in the 3-manifold, which leads to the idea of an incompressible surface and the theory of Haken manifolds, or one can choose the complementary pieces to be as nice as possible, leading to structures such as Heegaard splittings, which are useful in the non-Haken case. Thurston's contributions to the theory allow one to consider, in many cases, the additional structure given by a particular Thurston model geometry; the most prevalent geometry is hyperbolic geometry. Using a geometry in addition to special surfaces is fruitful; the fundamental groups of 3-manifolds reflect the geometric and topological information belonging to a 3-manifold. Thus, there is an interplay between group topological methods. Euclidean 3-space is the most important example of a 3-manifold, as all others are defined in relation to it.
This is just the standard 3-dimensional vector space over the real numbers. A 3-sphere is a higher-dimensional analogue of a sphere, it consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere is a two-dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions. Real projective 3-space, or RP3, is the topological space of lines passing through the origin 0 in R4, it is a compact, smooth manifold of dimension 3, is a special case Gr of a Grassmannian space. RP3 is SO, hence admits a group structure; the 3-dimensional torus is the product of 3 circles. That is: T 3 = S 1 × S 1 × S 1; the 3-torus, T3 can be described as a quotient of R3 under integral shifts in any coordinate. That is, the 3-torus is R3 modulo the action of the integer lattice Z3. Equivalently, the 3-torus is obtained from the 3-dimensional cube by gluing the opposite faces together.
A 3-torus in this sense is an example of a 3-dimensional compact manifold. It is an example of a compact abelian Lie group; this follows from the fact. Group multiplication on the torus is defined by coordinate-wise multiplication. Hyperbolic space is a homogeneous space, it is the model of hyperbolic geometry. It is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, models of elliptic geometry that have a constant positive curvature; when embedded to a Euclidean space, every point of a hyperbolic space is a saddle point. Another distinctive property is the amount of space covered by the 3-ball in hyperbolic 3-space: it increases exponentially with respect to the radius of the ball, rather than polynomially; the Poincaré homology sphere is a particular example of a homology sphere. Being a spherical 3-manifold, it is the only homology 3-sphere with a finite fundamental group, its fundamental group is known as the binary icosahedral group and has order 120.
This shows. In 2003, lack of structure on the largest scales in the cosmic microwave background as observed for one year by the WMAP spacecraft led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere. In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft. However, there is no strong support for the correctness of the model, as yet. In mathematics, Seifert–Weber space is a closed hyperbolic 3-manifold, it is known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space