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Encyclopedia > Areas of mathematics

The aim of this page is to list all areas of modern mathematics, with a brief explanation about their scope and links to other parts of this encyclopedia, set out in a systematic way. Euclid, detail from The School of Athens by Raphael. ...


The way research-level mathematics is internally organised is mostly determined by practitioners, and does change over time; this is in contrast with the apparently timeless syllabus divisions used in mathematics education, where calculus can seem to be much the same over a time scale of a century. Calculus itself does not appear as a major heading — most of the traditional material would be divided amongst topics under analysis. This illustrates, in part, the difficulty of communicating the principles of any large-scale organization. The research on most calculus topics was carried out in the eighteenth century, and has long been assimilated. The story of why fields exist as specialties involves, in most cases, quite a long intellectual history (and sometimes institutional history). A lecture on linear algebra Mathematics education is the study of practices and methods of both the teaching and learning of mathematics. ... Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ... (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... Assimilation, from Latin assimilatio meaning to render similar, is used to describe various phenomena: The process of assimilating new ideas into a schema (cognitive structure). ...


The American Mathematical Society's Mathematics Subject Classification (2000 edition) has been used as a starting point to ensure all areas are covered, and related areas are close together. However, the MSC aims to classify mathematical papers, not mathematics itself, so additional categories have been used. See also the list of mathematics lists. The American Mathematical Society (AMS) is dedicated to the interests of mathematical research and education, which it does with various publications and conferences as well as annual monetary awards to mathematicians. ... This article attempts to list all lists collecting articles about mathematics in Wikipedia. ...

Contents


Foundations / general

Euclid, detail from The School of Athens by Raphael. ... The word mathematics comes from the Greek μάθημα (máthema) which means science, knowledge, or learning; μαθηματικός (mathematikós) means fond of learning. Today, the term refers to a specific body of knowledge -- the rigorous, deductive study of quantity, structure, space and change. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... A lecture on linear algebra Mathematics education is the study of practices and methods of both the teaching and learning of mathematics. ... Recreational mathematics includes many mathematical games, and can be extended to cover such areas as logic and other puzzles of deductive reasoning. ...

Algebra

The study of structure starting with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by everyday numbers. Long standing questions about compass and straightedge construction were finally settled by Galois theory. The physically important concept of vectors, generalized to vector spaces is studied in linear algebra. Linear algebra lecture at Helsinki University of Technology This article is about the branch of mathematics; for other uses of the term see algebra (disambiguation). ... A number is an abstract entity that represents a count or measurement. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... Arithmetic is the current mathematics collaboration of the week! Please help improve it to featured article standard. ... Elementary algebra is the most basic form of algebra taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... This article presents the essential definitions. ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ...

Combinatorics (MSC 05) 
Studies finite collections of objects that satisfy specified criteria. In particular, it is concerned with "counting" the objects in those collections (enumerative combinatorics) and with deciding whether certain "optimal" objects exist (extremal combinatorics). It includes graph theory, used to describe inter-connected objects (a graph in this sense is a collection of connected points). See also the list of combinatorics topics, list of graph theory topics and glossary of graph theory. While these are the classical definitions, a combinatorial flavour is present in many parts of problem-solving.
Order theory (MSC 06) 
With any set of real numbers, it is possible to write them out in ascending order. Order Theory extends this idea to sets in general. It includes notions like lattices and ordered algebraic structures. See also the order theory glossary and the list of order topics.
General algebraic systems (MSC 08) 
Given a set, ways of combining or relating members of that set can be defined. If these obey certain rules, then a particular algebraic structure is formed. Universal algebra is the more formal study of these structures and systems.
Number theory (MSC 11) 
Number theory is traditionally concerned with the properties of integers. More recently, it has come to be concerned with wider classes of problems that have arisen naturally from the study of integers. It can be divided into elementary number theory (where the integers are studied without use of techniques from other mathematical fields); analytic number theory (where calculus and complex analysis are used as tools); algebraic number theory (which studies the algebraic numbers - the roots of polynomials with integer coefficients); Geometric number theory; combinatorial number theory and computational number theory. See also the list of number theory topics
Field theory and polynomials (MSC 12) 
Field theory studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined. A polynomial is an expression in which constants and variables are combined using only addition, subtraction, and multiplication.
Commutative rings and algebras (MSC 13) 
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. This means that if a and b are any elements of the ring, then a×b=b×a. Commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. It is foundational both for algebraic geometry and for algebraic number theory. The most prominent example for commutative rings are polynomial rings.

(Also transformation groups, abstract harmonic analysis) Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ... Combinatorial enumeration is a subfield of enumeration that deals with the counting of objects whose symmetries do not exist or, if they exist, are combinatorial in nature. ... Extremal combinatorics is a field of combinatorics, which is itself a part of mathematics. ... A labeled graph with 6 vertices and 7 edges. ... This is a list of combinatorics topics, by Wikipedia page. ... This is a list of graph theory topics, by Wikipedia page. ... Graph theory is a growth area in mathematical research, and has a large specialized vocabulary. ... Problem solving forms part of thinking. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations. ... This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. ... This is a list of order topics, by Wikipedia page. ... In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations. ... Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Analytic number theory is the branch of number theory that uses methods from mathematical analysis. ... Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... In number theory, the geometry of numbers is a topic and method arising from the work of Hermann Minkowski, on the relationship between convex sets and lattices in n-dimensional space. ... In mathematics, computational number theory is a study of number theory with the aid of computer powers. ... This is a list of number theory topics, by Wikipedia page. ... Field theory is a branch of mathematics which studies the properties of fields. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... This article presents the essential definitions. ... This article presents the essential definitions. ... In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... Mathematical meaning A map or binary operation is said to be commutative when, for any x in A and any y in B . ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... In mathematics, multilinear algebra extends the methods of linear algebra. ... Matrix theory is a branch of mathematics which focuses on the study of matrices. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In abstract algebra, a nonassociative ring is a generalization of the concept of ring. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... In mathematics, K-theory is, firstly, an extraordinary cohomology theory which consists of topological K-theory. ... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... The symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...


Analysis

Analysis is primarily concerned with change. Rates of change, accumulated change, multiple things changing relative to (or independently of) one another, etc. Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ... In mathematics, the derivative is defined as the instantaneous rate of change of a function. ...

(Also: probabilistic potential theory, numerical approximation, representation theory, analysis on manifolds) In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, the derivative is defined as the instantaneous rate of change of a function. ... In calculus, the integral of a function is a generalization of area, mass, volume and total. ... Measure can mean: To perform a measurement. ... Integration may be any of the following: In the most general sense, integration may be any bringing together of things: the integration of two or more economies, cultures, religions (usually called syncretism), etc. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... In mathematics, approximation theory is concerned with how functions can be approximated with other, simpler, functions, and with characterising in a quantitative way the errors introduced thereby. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... Potential theory may be defined as the study of harmonic functions. ... The theory of functions of several complex variables is the branch of mathematics dealing with functions f(z1, z2, ... , zn) on the space Cn of n-tuples of complex numbers. ... In mathematics, there is a theory or theories of special functions, particular functions such as the trigonometric functions that have useful or attractive properties, and which occur in different applications often enough to warrant a name and attention of their own. ... In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ... In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, a series is often represented as the sum of a sequence of terms. ... In mathematics, approximation theory is concerned with how functions can be approximated with other, simpler, functions, and with characterising in a quantitative way the errors introduced thereby. ... Expansion can have several meanings, including: In physics: Expansion of space, thermal expansion In computer hardware: an Expansion card In computer programming: In-line expansion In computer gaming: an expansion pack See also: Wikipedia:Requests for expansion This is a disambiguation page — a navigational aid which lists pages that might... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ... In the mathematical subfield of numerical analysis, trigonometric interpolation is a special form of interpolation on the unit circle in the complex plane using trigonometric polynomials. ... Please do not move the orthogonal functions page to this page. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... In mathematics, an integral transform is any transform T of the following form: The input of this transform is a function f, and the output is another function Tf. ... In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. ... In mathematics, an integral transform is any transform T of the following form: The input of this transform is a function f, and the output is another function Tf. ... This page deals with mathematical distributions. ... In mathematics, operator theory is the branch of functional analysis which deals with bounded linear operators and their properties. ... Calculus of variations is a field of mathematics that deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... Optimal control theory is a mathematical field that is concerned with control policies that can be deduced using optimization algorithms. ... In mathematics, the term optimization refers to the study of problems that have the form Given: a function f : A R from some set A to the real numbers Sought: an element x0 in A such that f(x0) ≤ f(x) for all x in A (minimization) or such that... In the mathematical field of numerical ODEs, a geometric integrator is a numerical method that preserves geometric properties of the exact flow of a differential equation. ... ... In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. ... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... Look up Analysis in Wiktionary, the free dictionary An analysis is a critical evaluation, usually made by breaking a subject (either material or intellectual) down into its constituent parts, then describing the parts and their relationship to the whole. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...


Geometry

Geometry (MSC 51) deals with spatial relationships, using fundamental qualities or axioms. Such axioms can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions. See also List of geometry topics Table of Geometry, from the 1728 Cyclopaedia. ... An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths. ... This is list of geometry topics, by Wikipedia page. ...

Convex geometry and discrete geometry (MSC 52)
Includes the study of objects such as polytopes and polyhedra. See also List of convexity topics
Discrete or combinatorial geometry (MSC 52)
The study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation. It includes the study of shapes such as the Platonic solids and the notion of tessellation.
Differential geometry (MSC 53)
The study of geometry using calculus, and is very closely related to differential topology. Covers such areas as Riemannian geometry, curvature and differential geometry of curves. See also the glossary of differential geometry and topology.
Algebraic geometry (MSC 14)
Given a polynomial of two real variables, then the points on a plane where that function is zero will form a curve. An algebraic curve extends this notion to polynomials over a field in a given number of variables. Algebraic geometry may be viewed the study of these curves. See also the list of algebraic geometry topics and list of algebraic surfaces.
Topology
Deals with the properties of a figure that do not change when the figure is continuously deformed. The main areas are point set topology (or general topology), algebraic topology, and the topology of manifolds, defined below.
General topology (MSC 54)
Also called point set topology. Properties of topological spaces. Includes such notions as open and closed sets, compact spaces, continuous functions, convergence, separation axioms, metric spaces, dimension theory. See also the glossary of general topology and the list of general topology topics.
Algebraic topology (MSC 55)
Properties of algebraic objects associated with a topological space and how these algebraic objects capture properties of such spaces. Contains areas like homology theory, cohomology theory, homotopy theory, and homological algebra, some of them examples of functors. Homotopy deals with homotopy groups (including the fundamental group) as well as simplicial complexes and CW complexes (also called cell complexes). See also the list of algebraic topology topics.
Manifolds (MSC 57)
A manifold can be thought of as an n-dimensional generalization of a surface in the usual 3-dimensional Euclidean space. The study of manifolds includes differential topology, which looks at the properties of differentiable functions defined over a manifold. See also complex manifolds.

Convex Geometry is the branch of geometry studying convex bodies: compact, convex sets in Euclidean space. ... Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuity. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... This is a list of convexity topics, by Wikipedia page. ... Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuity. ... The word discrete comes from the Latin word discretus which means separate. ... Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with counting the objects in those collections (enumerative combinatorics) and with deciding whether certain optimal objects exist (extremal combinatorics). ... A Platonic solid is a convex polyhedron whose faces all use the same regular polygon and such that the same number of faces meet at all its vertices. ... A tessellated plane. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ... Curvature refers to a number of loosely related concepts in different areas of geometry. ... In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ... This is a glossary of terms specific to differential geometry and differential topology. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... In computer science and mathematics, a variable (sometimes called a pronumeral) is a symbol denoting a quantity or symbolic representation. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... This article presents the essential definitions. ... This is a list of algebraic geometry topics, by Wikipedia page. ... This is a list of named (classes of) algebraic surfaces. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ... In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... This is a list of general topology topics, by Wikipedia page. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ... In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ... This is a list of algebraic topology topics, by Wikipedia page. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... 2-dimensional renderings (ie. ... An open surface with X-, Y-, and Z-contours shown. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ...

Applied mathematics

Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ...

Probability and statistics

See also glossary of probability and statistics Terms in statistics and probability theory : Concerned fields Probability theory Algebra of random variables (linear algebra) Statistics Measure theory Estimation theory Probability interpretations: Bayesian probability (or personal probability) Frequency probability Eclectic probability Glossary Atomic event : another name for elementary event. ...

Probability theory (MSC 60) 
The study of how likely a given event is to occur. See also Category:probability theory, and the list of probability topics.
Stochastic processes (MSC 60G/H) 
Considers with aggregate effect of a random function, either over time (a time series) or physical space (a random field). See also List of stochastic processes topics, and Category:Stochastic processes.
Statistics (MSC 62)
Analysis of data, and how representative it is. See also the list of statistical topics.

Probability theory is the mathematical study of probability. ... This is a list of probability topics, by Wikipedia page. ... In the mathematics of probability, a stochastic process is a random function. ... In statistics and signal processing, a time series is a sequence of data points, measured typically at successive times, spaced apart at uniform time intervals. ... In probability theory, let S = {X1, ..., Xn}, with the Xi in {0,1,...,G-1}, be a set of random variables on the sample space Ω={0,1,...,G-1}n, a probability measure π is a random field if . There exist several types of random fields, such as Markov... In the mathematics of probability, a stochastic process can be thought of as a random function. ... A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... Please add any Wikipedia articles related to statistics that are not already on this list. ...

Computational sciences

Numerical analysis, (MSC 65)
Many problems in mathematrics cannot in general be solved exactly (e.g. the quintic equation). Numerical analysis is the study of algorithms to provide an aproximate solution to problems to a given degree of accuracy. Includes numerical differentiation, numerical integration and numerical methods. See also List of numerical analysis topics

Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ... Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. ... Flowcharts are often used to represent algorithms. ... Numerical differentiation is a technique of numerical analysis to produce an estimate of the derivative of a mathematical function or function subroutine using values from the function and perhaps other knowledge about the function. ... Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ... This is a list of numerical analysis topics, by Wikipedia page. ... Computer science is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...

Physical sciences

Mechanics
Addresses what happens when a real physical object is subjected to forces. This divides naturally into the study of rigid solids, deformable solids, and fluids, detailed below.
Particle mechanics (MSC 70)
In mathematics, a particle is a point-like, perfectly rigid, solid object. Particle mechanics deals with the results of subjecting particles to forces. It includes celestial mechanics — the study of the motion of celestial objects.
Mechanics of deformable solids (MSC 74) 
Most real-world objects are not point-like nor perfectly rigid. More importantly, objects change shape when subjected to forces. This subject has a very strong overlap with continuum mechanics, which is concerned with continuous matter. It deals with such notions as stress, strain and elasticity. See also continuum mechanics.
Fluid mechanics (MSC 76)
Fluids in this sense includes not just liquids, but flowing gases, and even solids under certain situations. (For example, dry sand can behave like a fluid). It includes such notions as viscosity, turbulent flow and laminar flow (its opposite). See also fluid dynamics.

Mechanics can be seen as the prime, and even as the original, discipline of physics. ... Mechanics refers to: a craft relating to machinery (from the Latin mechanicus, from the Greek mechanikos, meaning one skilled in machines), or a range of disciplines in science and engineering. ... Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. ... Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ... Figure 1  Stress tensor In physics, stress is a measure of the internal distribution of force per unit area within a body that balances and reacts to the loads applied to it. ... Look up strain in Wiktionary, the free dictionary. ... Elasticity has meanings in two different fields: In physics and mechanical engineering, the theory of elasticity describes how a solid object moves and deforms in response to external stress. ... Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ... The hydrogeology is study about of water-bearing formation. ... A subset of the phases of matter, fluids include liquids, gases, plasmas and, to some extent, plastic solids. ... A liquid will assume the shape of its container. ... A gas is one of the ninety three main phases of matter (after solid and liquid,it could possibly arouse you if you inhale it and followed by plasma), that subsequently appear as a solid material is subjected to increasingly higher temperatures. ... In jewelry, a solid gold piece is the alternative to gold-filled or gold-plated jewelry. ... Patterns in the sand Sand is an example of a class of materials called granular matter. ... The pitch drop experiment at the University of Queensland. ... Turbulent flow around an obstacle; the flow further away is laminar Laminar and turbulent water flow over the hull of a submarine Turbulence creating a vortex on an airplane wing In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by low-momentum diffusion, high momentum convection, and... Laminar flow (bottom) and turbulent flow (top) over a submarine hull. ... Fluid dynamics is the subdiscipline of fluid mechanics that studies fluids (liquids and gases) in motion. ... Table of Opticks, 1728 Cyclopaedia Optics (appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. ... Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... Thermodynamics (from the Greek thermos meaning heat and dynamis meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Heat transfer is the study of the energy transfer via either conduction, convection, or radiation. ... Fig. ... Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In physics, the term relativity is used in several, related contexts: Galileo first developed the principle of relativity, which is the postulate that the laws of physics are the same for all observers. ... This article covers the physics of gravitation. ... A simple introduction to this subject is provided in Special relativity for beginners Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. ... Radio telescopes are among many different tools used by astronomers Astronomy (Greek: αστρονομία = άστρον + νόμος, astronomia = astron + nomos, literally, law of the stars) is the science of celestial objects and phenomena that originate outside the Earths atmosphere, such as stars, planets, comets, auroras, galaxies, and the cosmic background radiation. ... Spiral Galaxy ESO 269-57 // Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties (luminosity, density, temperature and chemical composition) of astronomical objects such as stars, galaxies, and the interstellar medium, as well as their interactions. ... Geophysics, the study of the earth by quantitative physical methods, especially by seismic reflection and refraction, geodesy, gravity, magnetic, electrical, electromagnetic, and radioactivity methods. ...

Non-physical sciences


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