Mathematics

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

Contractible space

In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space. A contractible space is one with the homotopy type of a point, it follows. Therefore any space with a nontrivial homotopy group cannot be contractible. Since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial. For a topological space X the following are all equivalent: X is contractible. X is homotopy equivalent to a one-point space. X deformation retracts onto a point. For any space Y, any two maps f,g: Y → X are homotopic. For any space Y, any map f: Y → X is null-homotopic; the cone on a space X is always contractible. Therefore any space can be embedded in a contractible one. Furthermore, X is contractible if and only if there exists a retraction from the cone of X to X; every contractible space is path connected and connected.

Moreover, since all the higher homotopy groups vanish, every contractible space is n-connected for all n ≥ 0. A topological space is locally contractible if every point has a local base of contractible neighborhoods. Contractible spaces are not locally contractible nor vice versa. For example, the comb space is contractible but not locally contractible. Locally contractible spaces are locally n-connected for all n ≥ 0. In particular, they are locally connected, locally path connected, locally connected. Any Euclidean space is contractible; the Whitehead manifold is contractible. Spheres of any finite dimension are not contractible; the unit sphere in an infinite-dimensional Hilbert space is contractible. The house with two rooms is standard example of a space, contractible, but not intuitively so; the Dunce hat is not collapsible. The cone on a Hawaiian earring is contractible, but not locally contractible or locally connected. All manifolds and CW complexes are in general not contractible; the Warsaw circle is obtained by "closing up" the topologist's sine curve by an arc connecting and.

It is a one-dimensional continuum whose homotopy groups are all trivial, but it is not contractible

Topology

In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting and bending, but not tearing or gluing. An n-dimensional topological space is a space with certain properties of connectedness and compactness; the space discrete. It can be closed. Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space and transformation; such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems; the term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics. Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries.

Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realised the importance of the edges of a polyhedron; this led to his polyhedron formula, V − E + F = 2. Some authorities regard this analysis as the first theorem. Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term "Topologie" in Vorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print; the English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". The term "topologist" in the sense of a specialist in topology was used in 1905 in the magazine Spectator.

Their work was corrected and extended by Henri Poincaré. In 1895, he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a special case of a general topological space, with any given topological space giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space. A topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski. Modern topology depends on the ideas of set theory, developed by Georg Cantor in the part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series.

For further developments, see point-set topology and algebraic topology. Topology can be formally defined as "the study of qualitative properties of certain objects that are invariant under a certain kind of transformation those properties that are invariant under a certain kind of invertible transformation." Topology is used to refer to a structure imposed upon a set X, a structure that characterizes the set X as a topological space by taking proper care of properties such as convergence and continuity, upon transformation. Topological spaces show up in every branch of mathematics; this has made topology one of the great unifying ideas of mathematics. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects and both separate the plane into two parts, the part inside and the part outside.

In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg that would cross each of its seven bridges once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks; this Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory. The hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is convincing to most people though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of t

Ronald Brown (mathematician)

Ronald Brown is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor University, he has authored many books and more than 160 journal articles. Born on 4 January 1935 in London, Brown attended Oxford University, obtaining a B. A. in 1956 and a D. Phil. in 1962. Brown began his teaching career during his doctorate work, serving as an assistant lecturer at the University of Liverpool before assuming the position as Lecturer. In 1964, he took a position at the University of Hull, serving first as a Senior Lecturer and as a Reader before becoming a Professor of pure mathematics at Bangor University a part of the University of Wales, in 1970. Brown served as Professor of Pure Mathematics for 30 years. In 1999, Brown took a half-time research professorship until he became Professor Emeritus in 2001, he was elected as a Fellow of the Learned Society of Wales in 2016. Brown has served as an editor or on the editorial board for a number of print and electronic journals.

He began in 1968 with the Chapman & Hall Mathematics Series, contributing through 1986. In 1975, he joined the editorial advisory board of the London Mathematical Society, remaining through 1994. Two years he joined the editorial board of Applied Categorical Structures, continuing through 2007. From 1995 and 1999 he has been active with the electronic journals Theory and Applications of Categories and Homology and Applications, which he helped found. Since 2006, he has been involved with Journal of Related Structures, his mathematical research interests range from algebraic topology and groupoids, to homology theory, category theory, mathematical biology, mathematical physics and higher-dimensional algebra. Brown has authored or edited a number of books and over 150 academic papers published in academic journals or collections, his first published paper was "Ten topologies for X × Y", published in the Quarterly Journal of Mathematics in 1963 Since his publications have appeared in many journals, including but not limited to the Journal of Algebra, Proceedings of the American Mathematical Society, Mathematische Zeitschrift, College Mathematics Journal, American Mathematical Monthly.

He is known for several recent co-authored papers on Categorical ontology. Among his several books and standard topology and algebraic topology textbooks are: Elements of Modern Topology, Low-Dimensional Topology, Topology: a geometric account of general topology, homotopy types, the fundamental groupoid and Groupoids and Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids, his recent fundamental results that extend the classical Van Kampen theorem to higher homotopy in higher dimensions are of substantial interest for solving several problems in algebraic topology, both old and new. Moreover, developments in algebraic topology have had wider implications, as for example in algebraic geometry and in algebraic number theory; such higher-dimensional theorems are about homotopy invariants of structured spaces, those for filtered spaces or n-cubes of spaces. An example is the fact that the relative Hurewicz theorem is a consequence of HHSvKT, this suggested a triadic Hurewicz theorem.

Higher-dimensional algebra Higher category theory Seifert–van Kampen theorem Groupoids Algebraic topology Nonabelian algebraic topology R-algebroids Double groupoids Homology Alexander Grothendieck arXiv Ronald Brown at the Mathematics Genealogy Project "Ronald Brown's Biography and publications". "Ronald Brown's Home Page". "MathOverflow user page". Higher-Dimensional Algebra citations list Editorial Board of Journal of Homotopy and Related Structures nLab Abstract Mathematics Website Editorial Board of Homology and Applications The Origins of `Pursuing Stacks' by Alexander Grothendieck Homology and Applications Theory and Applications of Categories

Line segment

In geometry, a line segment is a part of a line, bounded by two distinct end points, contains every point on the line between its endpoints. A closed line segment includes both endpoints. Examples of line segments include the sides of a square. More when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices, or otherwise a diagonal; when the end points both lie on a curve such as a circle, a line segment is called a chord. If V is a vector space over R or C, L is a subset of V L is a line segment if L can be parameterized as L = for some vectors u, v ∈ V, in which case the vectors u and u + v are called the end points of L. Sometimes one needs to distinguish between "open" and "closed" line segments. One defines a closed line segment as above, an open line segment as a subset L that can be parametrized as L = for some vectors u, v ∈ V. Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points.

In geometry, it is sometimes defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R 2 the line segment with endpoints A = and C = is the following collection of points:. A line segment is a non-empty set. If V is a topological vector space a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More than above, the concept of a line segment can be defined in an ordered geometry. A pair of line segments can be any one of the following: intersecting, skew, or none of these; the last possibility is a way that line segments differ from lines: if two nonparallel lines are in the same Euclidean plane they must cross each other, but that need not be true of segments. In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line.

Segments play an important role in other theories. For example, a set is convex if the segment that joins any two points of the set is contained in the set; this is important because it transforms some of the analysis of convex sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and substitute other segments into another statement to make segments congruent. A line segment can be viewed as a degenerate case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints, the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant. A complete orbit of this ellipse traverses the line segment twice; as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, line segments appear in numerous other locations relative to other geometric shapes.

Some frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, the internal angle bisectors. In each case there are various equalities relating these segment lengths to others as well as various inequalities. Other segment

Sphere

A sphere is a round geometrical object in three-dimensional space, the surface of a round ball. Like a circle in a two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point, but in a three-dimensional space; this distance r is the radius of the ball, made up from all points with a distance less than r from the given point, the center of the mathematical ball. These are referred to as the radius and center of the sphere, respectively; the longest straight line segment through the ball, connecting two points of the sphere, passes through the center and its length is thus twice the radius. While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics the above distinction is made between a sphere, a two-dimensional closed surface, embedded in a three-dimensional Euclidean space, a ball, a three-dimensional shape that includes the sphere and everything inside the sphere, or, more just the points inside, but not on the sphere.

The distinction between ball and sphere has not always been maintained and older mathematical references talk about a sphere as a solid. This is analogous to the situation in the plane, where the terms "circle" and "disk" can be confounded. In analytic geometry, a sphere with center and radius r is the locus of all points such that 2 + 2 + 2 = r 2. Let a, b, c, d, e be real numbers with a ≠ 0 and put x 0 = − b a, y 0 = − c a, z 0 = − d a, ρ = b 2 + c 2 + d 2 − a e a 2; the equation f = a + 2 + e = 0 has no real points as solutions if ρ < 0 and is called the equation of an imaginary sphere. If ρ = 0 the only solution of f = 0 is the point P 0 = and the equation is said to be the equation of a point sphere. In the case ρ > 0, f = 0 is an equation of a sphere whose center is P 0 and whose radius is ρ. If a in the above equation is zero f = 0 is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius; the points on the sphere with radius r > 0 and center can be parameterized via x = x 0 + r sin θ cos φ y = y 0 + r sin θ sin φ z = z 0 + r cos θ The parameter θ {

Allen Hatcher

Allen Edward Hatcher is an American topologist. Hatcher received his Ph. D. under the supervision of Hans Samelson at Stanford University in 1971. He went on to become a professor at the University of California, Los Angeles. Since 1983 he has been a professor at Cornell University, he has worked in geometric topology, both in high dimensions, relating pseudoisotopy to algebraic K-theory, in low dimensions: surfaces and 3-manifolds, such as proving the Smale conjecture for the 3-sphere. Among his most recognized results in 3-manifolds concern the classification of incompressible surfaces in certain 3-manifolds and their boundary slopes. William Floyd and Hatcher classified all the incompressible surfaces in punctured-torus bundles over the circle. William Thurston and Hatcher classified; as corollaries, this gave more examples of non-Haken, non-Seifert fibered, irreducible 3-manifolds and extended the techniques and line of investigation started in Thurston's Princeton lecture notes. Hatcher showed that irreducible, boundary-irreducible 3-manifolds with toral boundary have at most "half" of all possible boundary slopes resulting from essential surfaces.

In the case of one torus boundary, one can conclude that the number of slopes given by essential surfaces is finite. Hatcher has made contributions to the so-called theory of essential laminations in 3-manifolds, he invented the notion of "end-incompressibility" and several of his students, such as Mark Brittenham, Charles Delman, Rachel Roberts, have made important contributions to the theory. Hatcher and Thurston exhibited an algorithm to produce a presentation of the mapping class group of a closed, orientable surface, their work relied on moves that relate any two systems. Allen Hatcher and William Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19, no. 3, 221—237. Allen Hatcher, On the boundary curves of incompressible surfaces, Pacific Journal of Mathematics 99, no. 2, 373—377. William Floyd and Allen Hatcher, Incompressible surfaces in punctured-torus bundles and its Applications 13, no. 3, 263—282. Allen Hatcher and William Thurston, Incompressible surfaces in 2 -bridge knot complements, Inventiones Mathematicae 79, no.

2, 225—246. Allen Hatcher, A proof of the Smale conjecture, D i f f ≃ O, Annals of Mathematics 117, no. 3, 553—607. Hatcher, Algebraic topology. Cambridge University Press, Cambridge, 2002. Xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0 Vector Bundles and K-Theory Spectral Sequences in Algebraic Topology Basic Topology of 3-Manifolds Hatcher's official page Hatcher's personal homepage Allen Hatcher at the Mathematics Genealogy Project