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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). The . FOUNDATIONS / GENERAL
ALGEBRA The study of structure starting with Number s, first the familiar Natural Number s and Integer s and their Arithmetic al 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 Field s, 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 Vector s, generalized to Vector Space s is studied in Linear Algebra . ; ) 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 . ; and ordered Algebraic Structure s. See also the Order Theory Glossary and the List Of Order Topics . ; General , 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. ; (where Calculus and Complex Analysis are used as tools); Algebraic Number Theory (which studies the algebraic numbers - the roots of Polynomial s with integer Coefficient s); Geometric Number Theory ; Combinatorial Number Theory and Computational Number Theory . See also the List Of Number Theory Topics ; . 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. ; , 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 Ring s.
(Also Transformation Group s, Abstract Harmonic Analysis ) ANALYSIS Analysis is primarily concerned with change. Rates Of Change , accumulated change, multiple things changing relative to (or independently of) one another, etc.
(Also: Probabilistic Potential Theory , Numerical Approximation , Representation Theory , Analysis on Manifold s) GEOMETRY Geometry (MSC 51) deals with spatial relationships, using fundamental qualities or Axiom s. 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 ; and Polyhedra . See also List Of Convexity Topics ;Discrete or 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 . ;. Covers such areas as Riemannian Geometry , Curvature and Differential Geometry Of Curves . See also the Glossary Of Differential Geometry And Topology . ; of two real Variable s, 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 . ;), Algebraic Topology , and the topology of Manifold s, defined below. ;s. Includes such notions as Open and Closed Set s, Compact Space s, Continuous Function s, Convergence , Separation Axiom s, Metric Space s, Dimension Theory . See also the Glossary Of General Topology and the List Of General Topology Topics . ;, Cohomology Theory , Homotopy Theory , and Homological Algebra , some of them examples of Functor s. Homotopy deals with Homotopy Group s (including the Fundamental Group ) as well as Simplicial Complexes and CW Complexes (also called ''cell complexes''). See also the List Of Algebraic Topology Topics . ;al 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 Manifold s. APPLIED MATHEMATICS Probability and statistics See also Glossary Of Probability And Statistics ;, and the List Of Probability Topics . ;. ;. Computational sciences ;). 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
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. ;, 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. ;, which is concerned with continuous matter. It deals with such notions as Stress , Strain and Elasticity . See also Continuum Mechanics . ;s in this sense includes not just Liquid s, but flowing Gas es, and even Solid s 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 .
Non-physical sciences
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