glossary of areas of mathematics
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Glossary of areas of mathematicsFrom Wikipedia, the free encyclopedia
Contents
1 Glossary of areas of mathematics 11.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.10 J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.11 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.12 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.13 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.14 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.15 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.16 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.17 Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.18 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.19 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.20 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.21 U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.22 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.23 W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.24 X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.25 Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.26 Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.27 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 List of mathematical jargon 192.1 Philosophy of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Descriptive informalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Proof terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
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2.4 Proof techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Chapter 1
Glossary of areas of mathematics
This is a glossary of terms that are or have been considered areas of study in mathematics.
1.1 A
• Absolute differential calculus— the original name for tensor calculus developed around 1890.
• Absolute geometry — an extension of ordered geometry that is sometimes referred to as neutral geometrybecause its axiom system is neutral to the parallel postulate.
• Abstract algebra— the study of algebraic structures and their properties. Originally it was known as modernalgebra.
• Abstract analytic number theory — a branch mathematics that take ideas from classical analytic numbertheory and applies them to various other areas of mathematics.
• Abstract differential geometry — a form of differential geometry without the notion of smoothness fromcalculus. Instead it is built using sheaf theory and sheaf cohomology.
• Abstract harmonic analysis — a modern branch of harmonic analysis that extends upon the generalizedFourier transforms that can be defined on locally compact groups.
• Abstract homotopy theory
• Additive combinatorics — the part of arithmetic combinatorics devoted to the operations of addition andsubtraction.
• Additive number theory—a part of number theory that studies subsets of integers and their behaviour underaddition.
• Affine geometry — a branch of geometry that is centered on the study of geometric properties that remainunchanged by affine transformations. It can be described as a generalization of Euclidean geometry.
• Affine geometry of curves— the study of curves in affine space.
• Affine differential geometry — a type of differential geometry dedicated to differential invariants undervolume-preserving affine transformations.
• Ahlfors theory— a part of complex analysis being the geometric counterpart of Nevanlinna theory. It wasinvented by Lars Ahlfors
• Algebra—amajor part of pure mathematics centered on operations and relations. Beginning with elementaryalgebra, it introduces the concept of variables and how these can be manipulated towards problem solving;known as equation solving. Generalizations of operations and relations defined on sets have led to the ideaof an algebraic structure which are studied in abstract algebra. Other branches of algebra include universalalgebra, linear algebra and multilinear algebra.
1
2 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS
• Algebraic analysis—motivated by systems of linear partial differential equations, it is a branch of algebraicgeometry and algebraic topology that uses methods from sheaf theory and complex analysis, to study the prop-erties and generalizations of functions. It was started by Mikio Sato.
• Algebraic combinatorics— an area that employs methods of abstract algebra to problems of combinatorics.It also refers to the application of methods from combinatorics to problems in abstract algebra.
• Algebraic computation— see symbolic computation.
• Algebraic geometry— a branch that combines techniques from abstract algebra with the language and prob-lems of geometry. Fundamentally, it studies algebraic varieties.
• Algebraic graph theory— a branch of graph theory in which methods are taken from algebra and employedto problems about graphs. The methods are commonly taken from group theory and linear algebra.
• Algebraic K-theory — an important part of homological algebra concerned with defining and applying acertain sequence of functors from rings to abelian groups.
• Algebraic number theory— a part of algebraic geometry devoted to the study of the points of the algebraicvarieties whose coordinates belong to an algebraic number field. It is a major branch of number theory and isalso said to study algebraic structures related to algebraic integers.
• Algebraic statistics — the use of algebra to advance statistics, although the term is sometimes restricted tolabel the use of algebraic geometry and commutative algebra in statistics.
• Algebraic topology— a branch that uses tools from abstract algebra for topology to study topological spaces.
• Algorithmic number theory— also known as computational number theory, it is the study of algorithms forperforming number theoretic computations.
• Anabelian geometry — an area of study based on the theory proposed by Alexander Grothendieck in the1980s that describes the way a geometric object of an algebraic variety (such as an algebraic fundamentalgroup) can be mapped into another object, without it being an abelian group.
• Analysis— a rigorous branch of pure mathematics that had its beginnings in the formulation of infinitesimalcalculus. (Then it was known as infinitesimal analysis.) The classical forms of analysis are real analysis and itsextension complex analysis, whilst more modern forms are those such as functional analysis.
• Analytic combinatorics—part of enumerative combinatorics where methods of complex analysis are appliedto generating functions.
• Analytic geometry — usually this refer to the study of geometry using a coordinate system (also knownas Cartesian geometry). Alternatively it can refer to the geometry of analytic varieties. In this respect it isessentially equivalent to real and complex algebraic geometry.
• Analytic number theory—part of number theory using methods of analysis (as opposed to algebraic numbertheory)
• Appliedmathematics—acombination of various parts ofmathematics that concern a variety ofmathematicalmethods that can be applied to practical and theoretical problems. Typically the methods used are for science,engineering, finance, economics and logistics.
• Approximation theory — part of analysis that studies how well functions can be approximated by simplerones (such as polynomials or trigonometric polynomials)
• Arakelov geometry— also known as Arakelov theory
• Arakelov theory — an approach to Diophantine geometry used to study Diophantine equations in higherdimensions (using techniques from algebraic geometry). It is named after Suren Arakelov.
• Arithmetic—tomost people this refers to the branch known as elementary arithmetic dedicated to the usage ofaddition, subtraction, multiplication and division. However arithmetic also includes higher arithmetic referringto advanced results from number theory.
• Arithmetic algebraic geometry— see arithmetic geometry
1.2. B 3
• Arithmetic combinatorics—the study of the estimates from combinatorics that are associated with arithmeticoperations such as addition, subtraction, multiplication and division.
• Arithmetic dynamics
• Arithmetic geometry— the study of schemes of finite type over the spectrum of the ring of integers
• Arithmetic topology— a combination of algebraic number theory and topology studying analogies betweenprime ideals and knots
• Arithmetical algebraic geometry— an alternative name for arithmetic algebraic geometry
• Asymptotic combinatorics
• Asymptotic geometric analysis
• Asymptotic theory— the study of asymptotic expansions
• Auslander–Reiten theory— the study of the representation theory of Artinian rings
• Axiomatic geometry— see synthetic geometry.
• Axiomatic homology theory
• Axiomatic set theory— the study of systems of axioms in a context relevant to set theory and mathematicallogic.
1.2 B• Bifurcation theory— the study of changes in the qualitative or topological structure of a given family. It is apart of dynamical systems theory
• Birational geometry — a part of algebraic geometry that deals with the geometry (of an algebraic variety)that is dependent only on its function field.
• Bolyai-Lobachevskian geometry— see hyperbolic geometry.
1.3 C• C*-algebra theory
• Cartesian geometry— see analytic geometry
• Calculus—a branch usually associated with limits, functions, derivatives, integrals and infinite series. It formsthe basis of classical analysis, and historically was called the calculus of infinitesimals or infinitesimal calculus.Now it can refer to a system of calculation guided by symbolic manipulation.
• Calculus of infinitesimals — also known as infinitesimal calculus. It is a branch of calculus built upon theconcepts of infinitesimals.
• Calculus of moving surfaces—an extension of the theory of tensor calculus to include deforming manifolds.
• Calculus of variations— the field dedicated to maximizing or minimizing functionals. It used to be calledfunctional calculus.
• Catastrophe theory— a branch of bifurcation theory from dynamical systems theory, and also a special caseof the more general singularity theory from geometry. It analyses the germs of the catastrophe geometries.
• Categorical logic—a branch of category theory adjacent to the mathematical logic. It is based on type theoryfor intuitionistic logics.
• Category theory — the study of the properties of particular mathematical concepts by formalising them ascollections of objects and arrows.
4 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS
• Chaos theory — the study of the behaviour of dynamical systems that are highly sensitive to their initialconditions.
• Character theory— a branch of group theory that studies the characters of group representations or modularrepresentations.
• Class field theory— a branch of algebraic number theory that studies abelian extensions of number fields.
• Classical differential geometry— also known as Euclidean differential geometry. see Euclidean differentialgeometry.
• Classical algebraic topology
• Classical analysis—usually refers to the more traditional topics of analysis such as real analysis and complexanalysis. It includes any work that does not use techniques from functional analysis and is sometimes calledhard analysis. However it may also refer to mathematical analysis done according to the principles of classicalmathematics.
• Classical analytic number theory
• Classical differential calculus
• Classical Diophantine geometry
• Classical Euclidean geometry— see Euclidean geometry
• Classical geometry—may refer to solid geometry or classical Euclidean geometry. See geometry
• Classical invariant theory — the form of invariant theory that deals with describing polynomial functionsthat are invariant under transformations from a given linear group.
• Classical mathematics— the standard approach to mathematics based on classical logic and ZFC set theory.
• Classical projective geometry
• Classical tensor calculus
• Clifford analysis— the study of Dirac operators and Dirac type operators from geometry and analysis usingclifford algebras.
• Clifford theory is a branch of representation theory spawned from Cliffords theorem.
• Cobordism theory
• Cohomology theory
• Combinatorial analysis
• Combinatorial commutative algebra—a discipline viewed as the intersection between commutative algebraand combinatorics. It frequently employsmethods from one to address problems arising in the other. Polyhedralgeometry also plays a significant role.
• Combinatorial design theory— a part of combinatorial mathematics that deals with the existence and con-struction of systems of fintie sets whose intersections have certain properties.
• Combinatorial game theory
• Combinatorial geometry— see discrete geometry
• Combinatorial group theory— the theory of free groups and the presentation of a group. It is closely relatedto geometric group theory and is applied in geometric topology.
• Combinatorial mathematics
• Combinatorial number theory
• Combinatorial set theory— also known as Infinitary combinatorics. see infinitary combinatorics
1.3. C 5
• Combinatorial theory
• Combinatorial topology— an old name for algebraic topology, when topological invariants of spaces wereregarded as derived from combinatorial decompositions.
• Combinatorics — a branch of discrete mathematics concerned with countable structures. Branches of itinclude enumerative combinatorics, combinatorial design theory, matroid theory, extremal combinatorics andalgebraic combinatorics, as well as many more.
• Commutative algebra— a branch of abstract algebra studying commutative rings.
• Complex algebra
• Complex algebraic geometry— the main stream of algebraic geometry devoted to the study of the complexpoints of algebraic varieties.
• Complex analysis— a part of analysis that deals with functions of a complex variable.
• Complex analytic dynamics— a subdivision of complex dynamics being the study of the dynamic systemsdefined by analytic functions.
• Complex analytic geometry— the application of complex numbers to plane geometry.
• Complex differential geometry— a branch of differential geometry that studies complex manifolds.
• Complex dynamics — the study of dynamical systems defined by iterated functions on complex numberspaces.
• Complex geometry—the study of complexmanifolds and functions of complex variables. It includes complexalgebraic geometry and complex analytic geometry.
• Complexity theory— the study of complex systems with the inclusion of the theory of complex systems.
• Computable analysis— the study of which parts of real analysis and functional analysis can be carried outin a computable manner. It is closely related to constructive analysis.
• Computable model theory— a branch of model theory dealing with the relevant questions computability.
• Computability theory—abranch ofmathematical logic originating in the 1930s with the study of computablefunctions and Turing degrees, but now includes the study of generalized computability and definability. Itoverlaps with proof theory and effective descriptive set theory.
• Computational algebraic geometry
• Computational complexity theory—a branch of mathematics and theoretical computer science that focuseson classifying computational problems according to their inherent difficulty, and relating those classes to eachother.
• Computational geometry
• Computational group theory— the study of groups by means of computers.
• Computational mathematics — the mathematical research in areas of science where computing plays anessential role.
• Computational number theory— also known as algorithmic number theory, it is the study of algorithms forperforming number theoretic computations.
• Computational real algebraic geometry
• Computational synthetic geometry
• Computational topology
• Computer algebra— see symbolic computation
• Conformal geometry— the study of conformal transformations on a space.
6 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS
• Constructive analysis—mathematical analysis done according to the principles of constructive mathematics.This differs from classical analysis.
• Constructive function theory—a branch of analysis that is closely related to approximation theory, studyingthe connection between the smoothness of a function and its degree of approximation
• Constructive mathematics—mathematics which tends to use intuitionistic logic. Essentially that is classicallogic but without the assumption that the law of the excluded middle is an axiom.
• Constructive quantum field theory — a branch of mathematical physics that is devoted to showing thatquantum theory is mathematically compatible with special relativity.
• Constructive set theory
• Contact geometry — a branch of differential geometry and topology, closely related to and considered theodd-dimensional counterpart of symplectic geometry. It is the study of a geometric structure called a contactstructure on a differentiable manifold.
• Convex analysis— the study of properties of convex functions and convex sets.
• Convex geometry— part of geometry devoted to the study of convex sets.
• Coordinate geometry— see analytic geometry
• CR geometry— a branch of differential geometry, being the study of CR manifolds.
1.4 D• Derived noncommutative algebraic geometry
• Descriptive set theory— a part of mathematical logic, more specifically a part of set theory dedicated to thestudy of Polish spaces.
• Differential algebraic geometry—the adaption of methods and concepts from algebraic geometry to systemsof algebraic differential equations.
• Differential calculus— a subfield of calculus concerned with derivatives or the rates that quantities change.It is one of two traditional divisions of calculus, the other being integral calculus.
• Differential Galois theory— the study of the Galois groups of differential fields.
• Differential geometry— a form of geometry that uses techniques from integral and differential calculus aswell as linear and multilinear algebra to study problems in geometry. Classically, these were problems ofEuclidean geometry, although now it has been expanded. It is generally concerned with geometric structureson differentiable manifolds. It is closely related to differential topology.
• Differential geometry of curves— the study of smooth curves in Euclidean space by using techniques fromdifferential geometry
• Differential geometry of surfaces — the study of smooth surfaces with various additional structures usingthe techniques of differential geometry.
• Differential topology— a branch of topology that deals with differentiable functions on differentiable mani-folds.
• Diffiety theory
• Diophantine geometry— in general the study of algebraic varieties over fields that are finitely generated overtheir prime fields.
• Discrepancy theory
• Discrete computational geometry
• Discrete differential geometry
1.5. E 7
• Discrete dynamics
• Discrete exterior calculus
• Discrete geometry
• Discrete mathematics
• Discrete Morse theory— a combinatorial adaption of Morse theory.
• Distance geometry
• Domain theory
• Donaldson theory— the study of smooth 4-manifolds using gauge theory.
• Dynamical systems theory
1.5 E• Econometrics— the application of mathematical and statistical methods to economic data.
• Effective descriptive set theory — a branch of descriptive set theory dealing with set of real numbers thathave lightface definitions. It uses aspects of computability theory.
• Elementary algebra — a fundamental form of algebra extending on elementary arithmetic to include theconcept of variables.
• Elementary arithmetic— the simplified portion of arithmetic considered necessary for primary education.It includes the usage addition, subtraction, multiplication and division of the natural numbers. It also includesthe concept of fractions and negative numbers.
• Elementary mathematics—parts of mathematics frequently taught at the primary and secondary school lev-els. This includes elementary arithmetic, geometry, probability and statistics, elementary algebra and trigonometry.(calculus is not usually considered a part)
• Elementary group theory— the study of the basics of group theory
• Elimination theory— the classical name for algorithmic approaches to eliminating between polynomials ofseveral variables. It is a part of commutative algebra and algebraic geometry.
• Elliptic geometry — a type of non-Euclidean geometry (it violates Euclid's parallel postulate) and is basedon spherical geometry. It is constructed in elliptic space.
• Enumerative combinatorics — an area of combinatorics that deals with the number of ways that certainpatterns can be formed.
• Enumerative geometry— a branch of algebraic geometry concerned with counting the number of solutionsto geometric questions. This is usually done by means of intersection theory.
• Equivariant noncommutative algebraic geometry
• Ergodic Ramsey theory — a branch where problems are motivated by additive combinatorics and solvedusing ergodic theory.
• Ergodic theory— the study of dynamical systems with an invariant measure, and related problems.
• Euclidean geometry
• Euclidean differential geometry— also known as classical differential geometry. See differential geometry.
• Euler calculus
• Experimental mathematics
• Extraordinary cohomology theory
8 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS
• Extremal combinatorics— a branch of combinatorics, it is the study of the possible sizes of a collection offinite objects given certain restrictions.
• Extremal graph theory
1.6 F• Field theory— branch of abstract algebra studying fields.
• Finite geometry
• Finite model theory
• Finsler geometry— a branch of differential geometry whose main object of study is the Finsler manifold (ageneralisation of a Riemannian manifold).
• First order arithmetic
• Fourier analysis
• Fractional calculus — a branch of analysis that studies the possibility of taking real or complex powers ofthe differentiation operator.
• Fractional dynamics— investigates the behaviour of objects and systems that are described by differentiationand integration of fractional orders using methods of fractional calculus.
• Fredholm theory— part of spectral theory studying integral equations.
• Function theory — part of analysis devoted to properties of functions, especially functions of a complexvariable (see complex analysis).
• Functional analysis
• Functional calculus—historically the termwas used synonymously with calculus of variations, but now refersto a branch of functional analysis connected with spectral theory
• Fuzzy arithmetic
• Fuzzy geometry
• Fuzzy Galois theory
• Fuzzy mathematics— a branch of mathematics based on fuzzy set theory and fuzzy logic.
• Fuzzy measure theory
• Fuzzy qualitative trigonometry
• Fuzzy set theory— a form of set theory that studies fuzzy sets, that is sets that have degrees of membership.
• Fuzzy topology
1.7 G• Galois cohomology— an application of homological algebra, it is the study of group cohomology of Galoismodules.
• Galois theory—named after Évariste Galois, it is a branch of abstract algebra providing a connection betweenfield theory and group theory.
• Galois geometry— a branch of finite geometry concerned with algebraic and analytic geometry over a Galoisfield.
• Game theory
1.7. G 9
• Gauge theory
• General topology — also known as point-set topology, it is a branch of topology studying the properties oftopological spaces and structures defined on them. It differs from other branches of topology as the topologicalspaces do not have to be similar to manifolds.
• Generalized trigonometry—developments of trigonometric methods from the application to real numbers ofEuclidean geometry to any geometry or space. This includes spherical trigonometry, hyperbolic trigonometry,gyrotrigonometry, rational trigonometry, universal hyperbolic trigonometry, fuzzy qualitative trigonometry,operator trigonometry and lattice trigonometry.
• Geometric algebra— an alternative approach to classical, computational and relativistic geometry. It showsa natural correspondence between geometric entities and elements of algebra.
• Geometric analysis— a discipline that uses methods from differential geometry to study partial differentialequations as well as the applications to geometry.
• Geometric calculus
• Geometric combinatorics
• Geometric function theory— the study of geometric properties of analytic functions.
• Geometric homology theory
• Geometric invariant theory
• Geometric graph theory
• Geometric group theory
• Geometric measure theory
• Geometric topology— a branch of topology studying manifolds and mappings between them; in particularthe embedding of one manifold into another.
• Geometry—a branch of mathematics concerned with shape and the properties of space. Classically it arose aswhat is now known as solid geometry; this was concerning practical knowledge of length, area and volume. Itwas then put into an axiomatic form by Euclid, giving rise to what is now known as classical Euclidean geometry.The use of coordinates by René Descartes gave rise to Cartesian geometry enabling a more analytical approachto geometric entities. Since thenmany other branches have appeared including projective geometry, differentialgeometry, non-Euclidean geometry, Fractal geometry and algebraic geometry. Geometry also gave rise to themodern discipline of topology.
• Geometry of numbers— initiated by Hermann Minkowski, it is a branch of number theory studying convexbodies and integer vectors.
• Global analysis— the study of differential equations on manifolds and the relationship between differentialequations and topology.
• Global arithmetic dynamics
• Graph theory— a branch of discrete mathematics devoted to the study of graphs. It has many applicationsin physical, biological and social systems.
• Group-character theory— the part of character theory dedicated to the study of characters of group repre-sentations.
• Group representation theory
• Group theory
• Gyrotrigonometry—a form of trigonometry used in gyrovector space for hyperbolic geometry. (An analogyof the vector space in Euclidean geometry.)
10 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS
1.8 H• Hard analysis— see classical analysis
• Harmonic analysis— part of analysis concerned with the representations of functions in terms of waves. Itgeneralizes the notions of Fourier series and Fourier transforms from the Fourier analysis.
• High-dimensional topology
• Higher arithmetic
• Higher category theory
• Higher-dimensional algebra
• Hodge theory
• Holomorphic functional calculus— a branch of functional calculus starting with holomorphic functions.
• Homological algebra— the study of homology in general algebraic settings.
• Homology theory
• Homotopy theory
• Hyperbolic geometry — also known as Lobachevskian geometry or Bolyai-Lobachevskian geometry. It is anon-Euclidean geometry looking at hyperbolic space.
• hyperbolic trigonometry— the study of hyperbolic triangles in hyperbolic geometry, or hyperbolic functionsin Euclidean geometry. Other forms include gyrotrigonometry and universal hyperbolic trigonometry.
• Hypercomplex analysis
• Hyperfunction theory
1.9 I• Ideal theory— once the precursor name for what is now known as commutative algebra; it is the theory ofideals in commutative rings.
• Idempotent analysis
• Incidence geometry— the study of relations of incidence between various geometric objects, like curves andlines.
• Inconsistent mathematics— see paraconsistent mathematics.
• Infinitary combinatorics— an expansion of ideas in combinatorics to account for infinite sets.
• Infinitesimal analysis— once a synonym for infinitesimal calculus
• Infinitesimal calculus— see calculus of infinitesimals
• Information geometry
• Integral calculus
• Integral geometry
• Intersection theory— a branch of algebraic geometry and algebraic topology
• Intuitionistic type theory
• Invariant theory— studies how group actions on algebraic varieties affect functions.
• Inversive geometry— the study of invariants preserved by a type of transformation known as inversion
1.10. J 11
• Inversive plane geometry— inversive geometry that is limited to two dimensions• Inversive ring geometry• Itō calculus• Iwasawa theory
1.10 J
1.11 K• K-theory— originated as the study of a ring generated by vector bundles over a topological space or scheme.In algebraic topology it is an extraordinary cohomology theory known as topological K-theory. In algebra andalgebraic geometry it is referred to as algebraic K-theory. In physics, K-theory has appeared in type II stringtheory. (In particular twisted K-theory.)
• K-homology• Kähler geometry — a branch of differential geometry, more specifically a union of Riemannian geometry,complex differential geometry and symplectic geometry. It is the study of Kähler manifolds. (named afterErich Kähler)
• KK-theory• Klein geometry• Knot theory— part of topology dealing with knots• Kummer theory
1.12 L• L-theory• Large deviations theory— part of probability theory studying events of small probability (tail events).• Large sample theory— also known as asymptotic theory• Lattice theory— the study of lattices, being important in order theory and universal algebra• Lattice trigonometry• Lie algebra theory• Lie group theory• Lie sphere geometry• Lie theory• Line geometry• Linear algebra – a branch of algebra studying linear spaces and linear maps. It has applications in fields suchas abstract algebra and functional analysis; it can be represented in analytic geometry and it is generalized inoperator theory and in module theory. Sometimes matrix theory is considered a branch, although linear algebrais restricted to only finite dimensions. Extensions of the methods used belong to multilinear algebra.
• Linear functional analysis• Local algebra— a term sometimes applied to the theory of local rings.• Local arithmetic dynamics— also known as p-adic dynamics or nonarchimedean dynamics.• Local class field theory• Low-dimensional topology
12 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS
1.13 M
• Malliavin calculus
• Mathematical logic
• Mathematical optimization
• Mathematical physics— a part of mathematics that develops mathematical methods motivated by problemsin physics.
• Mathematical sciences— refers to academic disciplines that are mathematical in nature, but are not consid-ered proper subfields of mathematics. Examples include statistics, cryptography, game theory and actuarialscience.
• Matrix algebra
• Matrix calculus
• Matrix theory
• Matroid theory
• Measure theory
• Metric geometry
• Microlocal analysis
• Model theory
• Modern algebra— see abstract algebra
• Modern algebraic geometry— the form of algebraic geometry given by Alexander Grothendieck and Jean-Pierre Serre drawing on sheaf theory.
• Modern invariant theory— the form of invariant theory that analyses the decomposition of representationsinto irreducibles.
• Modular representation theory
• Module theory
• Molecular geometry
• Morse theory — a part of differential topology, it analyzes the topological space of a manifold by studyingdifferentiable functions on that manifold.
• Motivic cohomology
• Multilinear algebra— an extension of linear algebra building upon concepts of p-vectors and multivectorswith Grassmann algebra.
• Multivariable calculus
• Multiplicative number theory—a subfield of analytic number theory that deals with prime numbers, factorizationand divisors.
• Multiple-scale analysis
1.14. N 13
1.14 N• Neutral geometry— see absolute geometry
• Nevanlinna theory— part of complex analysis studying the value distribution of meromorphic functions. Itis named after Rolf Nevanlinna
• Nielsen theory—an area of mathematical research with its origins in fixed point topology, developed by JakobNielsen
• Non-abelian class field theory
• Non-classical analysis
• Non-Euclidean geometry
• Non-standard analysis
• Non-standard calculus
• Nonarchimedean dynamics— also known as p-adic analysis or local arithmetic dynamics
• Noncommutative algebraic geometry — a direction in noncommutative geometry studying the geometricproperties of formal duals of non-commutative algebraic objects.
• Noncommutative geometry
• Noncommutative harmonic analysis— see representation theory
• Noncommutative topology
• Nonlinear analysis
• Nonlinear functional analysis
• Number theory— a branch of pure mathematics primarily devoted to the study of the integers. Originally itwas known as arithmetic or higher arithmetic.
• Numerical analysis
• Numerical geometry
• Numerical linear algebra
1.15 O• Operad theory— a type of abstract algebra concerned with prototypical algebras.
• Operator geometry
• Operator K-theory
• Operator theory— part of functional analysis studying operators.
• Operator trigonometry
• Optimal control theory— a generalization of the calculus of variations.
• Orbifold theory
• Order theory— a branch that investigates the intuitive notion of order using binary relations.
• Ordered geometry— a form of geometry omitting the notion of measurement but featuring the concept ofintermediacy. It is a fundamental geometry forming a common framework for affine geometry, Euclideangeometry, absolute geometry and hyperbolic geometry.
• Oriented elliptic geometry
• Oriented spherical geometry
14 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS
1.16 P• p-adic analysis— a branch of number theory that deals with the analysis of functions of p-adic numbers.
• p-adic dynamics— an application of p-adic analysis looking at p-adic differential equations.
• p-adic Hodge theory
• Parabolic geometry
• Paraconsistent mathematics — sometimes called inconsistent mathematics, it is an attempt to develop theclassical infrastructure of mathematics based on a foundation of paraconsistent logic instead of classical logic.
• Partition theory
• Perturbation theory
• Picard–Vessiot theory
• Plane geometry
• Point-set topology— see general topology
• Pointless topology
• Poisson geometry
• Polyhedral combinatorics— a branch within combinatorics and discrete geometry that studies the problemsof describing convex polytopes.
• Polyhedral geometry
• Possibility theory
• Potential theory
• Precalculus
• Predicative mathematics
• Probability theory
• Probabilistic combinatorics
• Probabilistic graph theory
• Probabilistic number theory
• Projective geometry — a form of geometry that studies geometric properties that are invariant under aprojective transformation.
• Projective differential geometry
• Proof theory
• Pseudo-Riemannian geometry — generalizes Riemannian geometry to the study of pseudo-Riemannianmanifolds.
• Pure mathematics— the part of mathematics that studies entirely abstract concepts.
1.17 Q• Quantum calculus— a form of calculus without the notion of limits. There are 2 forms known as q-calculusand h-calculus
• Quantum geometry— the generalization of concepts of geometry used to describe the physical phenomenaof quantum physics
• Quaternionic analysis
1.18. R 15
1.18 R
• Ramsey theory— the study of the conditions in which order must appear. It is named after Frank P. Ramsey.
• Rational geometry
• Rational trigonometry— a reformulation of trigonometry in terms of spread and quadrance instead of angleand length.
• Real algebra— the study of the part of algebra relevant to real algebraic geometry.
• Real algebraic geometry— the part of algebraic geometry that studies real points of the algebraic varieties.
• Real analysis—abranch ofmathematical analysis; in particular hard analysis, that is the study of real numbersand functions of Real values. It provides a rigorous formulation of the calculus of real numbers in terms ofcontinuity and smoothness, whilst the theory is extended to the complex numbers in complex analysis.
• Real analytic geometry
• Real K-theory
• Recreational mathematics— the area dedicated to mathematical puzzles and mathematical games.
• Recursion theory— see computability theory
• Representation theory— a subfield of abstract algebra; it studies algebraic structures by representing theirelements as linear transformations of vector spaces. It also studies modules over these algebraic structures,providing a way of reducing problems in abstract algebra to problems in linear algebra.
• Representation theory of algebraic groups
• Representation theory of algebras
• Representation theory of diffeomorphism groups
• Representation theory of finite groups
• Representation theory of groups
• Representation theory of Hopf algebras
• Representation theory of Lie algebras
• Representation theory of Lie groups
• Representation theory of the Galilean group
• Representation theory of the Lorentz group
• Representation theory of the Poincaré group
• Representation theory of the symmetric group
• Ribbon theory— a branch of topology studying ribbons.
• Riemannian geometry—a branch of differential geometry that is more specifically, the study of Riemannianmanifolds. It is named after Bernhard Riemann and it featuresmany generalizations of concepts fromEuclideangeometry, analysis and calculus.
• Rough set theory — the a form of set theory based on rough sets.
16 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS
1.19 S
• Scheme theory — the study of schemes introduced by Alexander Grothendieck. It allows the use of sheaftheory to study algebraic varieties and is considered the central part of modern algebraic geometry.
• Secondary calculus
• Semialgebraic geometry—apart of algebraic geometry; more specifically a branch of real algebraic geometrythat studies semialgebraic sets.
• Set-theoretic topology
• Set theory
• Sheaf theory
• Sheaf cohomology
• Sieve theory
• Single operator theory— deals with the properties and classifications of single operators.
• Singularity theory— a branch, notably of geometry; that studies the failure of manifold structure.
• Smooth infinitesimal analysis — a rigorous reformation of infinitesimal calculus employing methods ofcategory theory. As a theory, it is a subset of synthetic differential geometry.
• Solid geometry
• Spatial geometry
• Spectral geometry— a field that concerns the relationships between geometric structures of manifolds andspectra of canonically defined differential operators.
• Spectral graph theory— the study of properties of a graph using methods from matrix theory.
• Spectral theory—part of operator theory extending the concepts of eigenvalues and eigenvectors from linearalgebra and matrix theory.
• Spectral theory of ordinary differential equations— part of spectral theory concerned with the spectrumand eigenfunction expansion associated with linear ordinary differential equations.
• Spectrum continuation analysis— generalizes the concept of a Fourier series to non-periodic functions.
• Spherical geometry— a branch of non-Euclidean geometry, studying the 2-dimensional surface of a sphere.
• Spherical trigonometry— a branch of spherical geometry that studies polygons on the surface of a sphere.Usually the polygons are triangles.
• Statistics—although the termmay refer to the more general study of statistics, the term is used in mathematicsto refer to the mathematical study of statistics and related fields. This includes probability theory.
• Stochastic calculus
• Stochastic calculus of variations
• Stochastic geometry— the study of random patterns of points
• Stratified Morse theory
• Super category theory
• Super linear algebra
• Surgery theory — a part of geometric topology referring to methods used to produce one manifold fromanother (in a controlled way.)
1.20. T 17
• Symbolic computation— also known as algebraic computation and computer algebra. It refers to the tech-niques used to manipulate mathematical expressions and equations in symbolic form as opposed to manipulat-ing them by the numerical quantities represented by them.
• Symbolic dynamics
• Symmetric function theory
• Symplectic geometry — a branch of differential geometry and topology whose main object of study is thesymplectic manifold.
• Symplectic topology
• Synthetic differential geometry— a reformulation of differential geometry in the language of topos theoryand in the context of an intuitionistic logic.
• Synthetic geometry — also known as axiomatic geometry, it is a branch of geometry that uses axioms andlogical arguments to draw conclusions as opposed to analytic and algebraic methods.
• Systolic geometry—abranch of differential geometry studying systolic invariants ofmanifolds and polyhedra.
• Systolic hyperbolic geometry— the study of systoles in hyperbolic geometry.
1.20 T• Tensor analysis—the study of tensors, which play a role in subjects such as differential geometry, mathematicalphysics, algebraic topology, multilinear algebra, homological algebra and representation theory.
• Tensor calculus— an older term for tensor analysis.
• Tensor theory— an alternative name for tensor analysis.
• Theoretical physics—abranch primarily of the science physics that usesmathematical models and abstractionof physics to rationalize and predict phenomena.
• Time-scale calculus
• Topology
• Topological combinatorics— the application of methods from algebraic topology to solve problems in com-binatorics.
• Topological degree theory
• Topological fixed point theory
• Topological graph theory
• Topological K-theory
• Topos theory
• Toric geometry
• Transcendental number theory— a branch of number theory that revolves around the transcendental num-bers.
• Transfinite order theory
• Transformation geometry
• Trigonometry— the study of triangles and the relationships between the length of their sides, and the anglesbetween them. It is essential to many parts of applied mathematics.
• Tropical analysis— see idempotent analysis
18 CHAPTER 1. GLOSSARY OF AREAS OF MATHEMATICS
• Tropical geometry
• Twisted K-theory — a variation on K-theory, spanning abstract algebra, algebraic topology and operatortheory.
• Type theory
1.21 U• Umbral calculus— the study of Sheffer sequences
• Uncertainty theory—a new branch of mathematics based on normality, monotonicity, self-duality, countablesubadditivity, and product measure axioms.
• Unitary representation theory
• Universal algebra— a field studying the formalization of algebraic structures itself.
• Universal hyperbolic trigonometry— an approach to hyperbolic trigonometry based on rational geometry.
1.22 V• Valuation theory
• Variational analysis
• Vector algebra—apart of linear algebra concernedwith the operations of vector addition and scalarmultiplication,although it may also refer to vector operations of vector calculus, including the dot and cross product. In thiscase it can be contrasted with geometric algebra which generalizes into higher dimensions.
• Vector analysis— also known as vector calculus, see vector calculus.
• Vector calculus— a branch of multivariable calculus concerned with differentiation and integration of vectorfields. Primarily it is concerned with 3-dimensional Euclidean space.
1.23 W• Wavelet
• Windowed Fourier transform
• Window function
1.24 X
1.25 Y
1.26 Z
1.27 See also
Glossary of engineering
Chapter 2
List of mathematical jargon
The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount ofjargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon oftenappears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much ofthis is common English, but with a specific non-obvious meaning when used in a mathematical sense.Some phrases, like “in general”, appear below in more than one section.
2.1 Philosophy of mathematicsabstract nonsense Also general abstract nonsense or generalized abstract nonsense, a tongue-in-cheek reference to
category theory, using which one can employ arguments that establish a (possibly concrete) result withoutreference to any specifics of the present problem.
[The paper of Eilenberg and Mac Lane (1942)] introduced the very abstract idea of a 'category' —a subject then called 'general abstract nonsense'!— Saunders Mac Lane (1997)
[Grothendieck] raised algebraic geometry to a new level of abstraction...if certain mathematicianscould console themselves for a time with the hope that all these complicated structures were 'abstract non-sense'...the later papers of Grothendieck and others showed that classical problems...which had resistedefforts of several generations of talented mathematicians, could be solved in terms of...complicated con-cepts.— Michael Monastyrsky (2001)
canonical A reference to a standard or choice-free presentation of some mathematical object. The term canonical isalso used more informally, meaning roughly “standard” or “classic”. For example, one might say that Euclid'sproof is the “canonical proof” of the infinitude of primes.
There are two canonical proofs that are always used to show non-mathematicians what a mathemat-ical proof is like:
• —The proof that there are infinitely many prime numbers.• —The proof of the irrationality of the square root of two.
— Freek Wiedijk (2006, p.2)
deep A result is called “deep” if its proof requires concepts and methods that are advanced beyond the conceptsneeded to formulate the result. The prime number theorem, proved with techniques from complex analysis,
19
20 CHAPTER 2. LIST OF MATHEMATICAL JARGON
was thought to be a deep result until elementary proofs were found. The fact that π is irrational is a deep resultbecause it requires considerable development of real analysis to prove, even though it can be stated in terms ofsimple number theory and geometry.
elegant Also beautiful; an aesthetic term referring to the ability of an idea to provide insight into mathematics,whether by unifying disparate fields, introducing a new perspective on a single field, or providing a techniqueof proof which is either particularly simple, or captures the intuition or imagination as to why the result it provesis true. Gian-Carlo Rota distinguished between elegance of presentation and beauty of concept, saying that forexample, some topics could be written about elegantly although the mathematical content is not beautiful, andsome theorems or proofs are beautiful but may be written about inelegantly.
The beauty of a mathematical theory is independent of the aesthetic qualities...of the theory’s rig-orous expositions. Some beautiful theories may never be given a presentation which matches theirbeauty....Instances can also be found of mediocre theories of questionable beauty which are given bril-liant, exciting expositions....[Category theory] is rich in beautiful and insightful definitions and poor inelegant proofs....[The theorems] remain clumsy and dull....[Expositions of projective geometry] vied forone another in elegance of presentation and in cleverness of proof....In retrospect, one wonders what allthe fuss was about.Mathematicians may say that a theorem is beautiful when they really mean to say that it is enlightening.We acknowledge a theorem’s beauty when we see how the theorem 'fits’ in its place....We say that a proofis beautiful when such a proof finally gives away the secret of the theorem....— Gian-Carlo Rota (1977, pp.173–174, pp.181–182)
elementary A proof or result is called “elementary” if it requires only basic concepts and methods, in contrast to so-called deep results. The concept of “elementary proof” is used specifically in number theory, where it usuallyrefers to a proof that does not use methods from complex analysis.
folklore A result is called “folklore” if it is non-obvious, has not been published, and yet is generally known amongthe specialists in a field. Usually, it is unknown who first obtained the result. If the result is important, it mayeventually find its way into the textbooks, whereupon it ceases to be folklore.
Many of the results mentioned in this paper should be considered “folklore” in that they merely for-mally state ideas that are well-known to researchers in the area, but may not be obvious to beginners andto the best of my knowledge do not appear elsewhere in print.— Russell Impagliazzo (1995)
natural Similar to “canonical” but more specific, this term makes reference to a description (almost exclusively inthe context of transformations) which holds independently of any choices. Though long used informally, thisterm has found a formal definition in category theory.
pathological An object behaves pathologically (or, somewhat more broadly used, in a degenerated way) if it failsto conform to the generic behavior of such objects, fails to satisfy certain regularity properties (depending oncontext), or simply disobeys mathematical intuition. These can be and often are contradictory requirements.Sometimes the term is more pointed, referring to an object which is specifically and artificially exhibited as acounterexample to these properties.
Since half a century we have seen arise a crowd of bizarre functions which seem to try to resembleas little as possible the honest functions which serve some purpose....Nay more, from the logical pointof view, it is these strange functions which are the most general....to-day they are invented expressly toput at fault the reasonings of our fathers....— Henri Poincaré (1913)
[The Dirichlet function] took on an enormous importance...as giving an incentive for the creation ofnew types of function whose properties departed completely from what intuitively seemed admissible. Acelebrated example of such a so-called 'pathological' function...is the one provided byWeierstrass....This
2.2. DESCRIPTIVE INFORMALITIES 21
function is continuous but not differentiable.— J. Sousa Pinto (2004)
Note for that latter quote that as the differentiable functions are meagre in the space of continuous functions, asBanach found out in 1931, differentiable functions are colloquially speaking a rare exception among the con-tinuous ones. Thus it can hardly be defended any-more to call non-differentiable continuous functions patho-logical.
rigor (rigour) Mathematics strives to establish its results using indisputable logic rather than informal descriptiveargument. Rigor is the use of such logic in a proof.
well-behaved An object is well-behaved (in contrast with being pathological) if it does satisfy the prevailing regularityproperties, or sometimes if it conforms to intuition (but intuition often suggests the opposite behavior as well).
2.2 Descriptive informalities
Although ultimately every mathematical argument must meet a high standard of precision, mathematicians use de-scriptive but informal statements to discuss recurring themes or concepts with unwieldy formal statements. Note thatmany of the terms are completely rigorous in context.
almost all A shorthand term for “all except for a set of measure zero", when there is a measure to speak of. Forexample, “almost all real numbers are transcendental" because the algebraic real numbers form a countablesubset of the real numbers with measure zero. One can also speak of “almost all” integers having a propertyto mean “all but finitely many”, despite the integers not admitting a measure for which this agrees with theprevious usage. For example, “almost all prime numbers are odd”. There is a more complicated meaning forintegers as well, discussed in the main article. Finally, this term is sometimes used synonymously with generic,below.
arbitrarily large Notions which arise mostly in the context of limits, referring to the recurrence of a phenomenonas the limit is approached. A statement such as that predicate P is satisfied by arbitrarily large values, can beexpressed in more formal notation by ∀x : ∃y ≥ x : P(y). See also frequently. The statement that quantity f(x)depending on x “can be made” arbitrarily large, corresponds to ∀y : ∃x : f(x) ≥ y.
arbitrary A shorthand for the universal quantifier. An arbitrary choice is one which is made unrestrictedly, oralternatively, a statement holds of an arbitrary element of a set if it holds of any element of that set. Also muchin general-language use among mathematicians: “Of course, this problem can be arbitrarily complicated”.
eventually, definitely In the context of limits, this is shorthand for sufficiently large arguments; the relevant argu-ment(s) are implicit in the context. As an example, one could say that “The function log(log(x)) eventuallybecomes larger than 100"; in this context, “eventually” means “for sufficiently large x".
factor through A term in category theory referring to composition of morphisms. If we have three objects A, B,and C and a map f : A → C which is written as a composition f = h ◦ g with g : A → B and h : B → C ,then f is said to factor through any (and all) of B , g , and h .
finite Next to the usual meaning of “not infinite”, in another more restrictive meaning that one may encounter, avalue being said to be “finite” also excludes infinitesimal values and the value 0. For example, if the varianceof a random variable is said to be finite, this implies it is a positive real number.
frequently In the context of limits, this is shorthand for arbitrarily large arguments and its relatives; as with eventually,the intended variant is implicit. As an example, one could say that “The function sin(x) is frequently zero”,where “frequently” means “for arbitrarily large x".
generic This term has similar connotations as almost all but is used particularly for concepts outside the purview ofmeasure theory. A property holds “generically” on a set if the set satisfies some (context-dependent) notionof density, or perhaps if its complement satisfies some (context-dependent) notion of smallness. For example,a property which holds on a dense Gδ (intersection of countably many open sets) is said to hold generically.In algebraic geometry, one says that a property of points on an algebraic variety that holds on a dense Zariskiopen set is true generically; however, it is usually not said that a property which holds merely on a dense set(which is not Zariski open) is generic in this situation.
22 CHAPTER 2. LIST OF MATHEMATICAL JARGON
in general In a descriptive context, this phrase introduces a simple characterization of a broad class of objects, withan eye towards identifying a unifying principle. This term introduces an “elegant” description which holds for"arbitrary" objects. Exceptions to this description may be mentioned explicitly, as "pathological" cases.
Norbert A’Campo of the University of Basel once asked Grothendieck about something related tothe Platonic solids. Grothendieck advised caution. The Platonic solids are so beautiful and so excep-tional, he said, that one cannot assume such exceptional beauty will hold in more general situations.— Allyn Jackson (2004, p.1197)
left-hand side, right-hand side (LHS, RHS) Most often, these refer simply to the left-hand or the right-hand sideof an equation; for example, x = y + 1 has x on the LHS and y + 1 on the RHS. Occasionally, these are usedin the sense of lvalue and rvalue: an RHS is primitive, and an LHS is derivative.
nice A mathematical object is colloquially called nice or sufficiently nice if it satisfies hypotheses or properties,sometimes unspecified or even unknown, that are especially desirable in a given context. It is an informalantonym for pathological. For example, one might conjecture that a differential operator ought to satisfy acertain boundedness condition “for nice test functions,” or one might state that some interesting topologicalinvariant should be computable “for nice spaces X.”
onto A function (which in mathematics is generally defined as mapping the elements of one set A to elements ofanother B) is called “A onto B” (instead of “A to B”) only if it is surjective; it may even be said that “f is onto”(i. e. surjective). Not translatable (without circumlocutions) to languages other than English.
proper If, for some notion of substructure, objects are substructures of themselves (that is, the relationship is re-flexive), then the qualification proper requires the objects to be different. For example, a proper subset of a setS is a subset of S that is different from S, and a proper divisor of a number n is a divisor of n that is differentfrom n. This overloaded word is also non-jargon for a proper morphism.
regular A function is called regular if it satisfies satisfactory continuity and differentiability properties, which areoften context-dependent. These properties might include possessing a specified number of derivatives, withthe function and its derivatives exhibiting some nice property, such as Hölder continuity. Informally, this termis sometimes used synonymously with smooth, below. These imprecise uses of the word regular are not to beconfused with the notion of a regular topological space, which is rigorously defined.
resp. (Respectively) A convention to shorten parallel expositions. “A (resp. B) [has some relationship to] X (resp.Y)" means that A [has some relationship to] X and also that B [has (the same) relationship to] Y. For example,squares (resp. triangles) have 4 sides (resp. 3 sides); or compact (resp. Lindelöf) spaces are ones where everyopen cover has a finite (resp. countable) open subcover.
sharp Often, a mathematical theorem will establish constraints on the behavior of some object; for example, afunction will be shown to have an upper or lower bound. The constraint is sharp (sometimes optimal) if itcannot be made more restrictive without failing in some cases. For example, for arbitrary nonnegative realnumbers x, the exponential function ex, where e = 2.7182818..., gives an upper bound on the values of thequadratic function x2. This is not sharp; the gap between the functions is everywhere at least 1. Among theexponential functions of the form αx, setting α = e2/e = 2.0870652... results in a sharp upper bound; the slightlysmaller choice α = 2 fails to produce an upper bound, since then α3 = 8 < 32. In applied fields the word “tight”is often used with the same meaning.[1]
smooth Smoothness is a concept which mathematics has endowed with many meanings, from simple differentiabilityto infinite differentiability to analyticity, and still others which are more complicated. Each such usage attemptsto invoke the physically intuitive notion of smoothness.
strong, stronger A theorem is said to be strong if it deduces restrictive results from general hypotheses. One cel-ebrated example is Donaldson’s theorem, which puts tight restraints on what would otherwise appear to be alarge class of manifolds. This (informal) usage reflects the opinion of the mathematical community: not onlyshould such a theorem be strong in the descriptive sense (below) but it should also be definitive in its area. Atheorem, result, or condition is further called stronger than another one if a proof of the second can be easilyobtained from the first. An example is the sequence of theorems: Fermat’s little theorem, Euler’s theorem,Lagrange’s theorem, each of which is stronger than the last; another is that a sharp upper bound (see above) isa stronger result than a non-sharp one. Finally, the adjective strong or the adverb strongly may be added to a
2.3. PROOF TERMINOLOGY 23
mathematical notion to indicate a related stronger notion; for example, a strong antichain is an antichain sat-isfying certain additional conditions, and likewise a strongly regular graph is a regular graph meeting strongerconditions. When used in this way, the stronger notion (such as “strong antichain”) is a technical term witha precisely defined meaning; the nature of the extra conditions cannot be derived from the definition of theweaker notion (such as “antichain”).
sufficiently large, suitably small, sufficiently close In the context of limits, these terms refer to some (unspecified,even unknown) point at which a phenomenon prevails as the limit is approached. A statement such as thatpredicate P holds for sufficiently large values, can be expressed in more formal notation by ∃x : ∀y ≥ x : P(y).See also eventually.
upstairs, downstairs A descriptive term referring to notation in which two objects are written one above the other;the upper one is upstairs and the lower, downstairs. For example, in a fiber bundle, the total space is oftensaid to be upstairs, with the base space downstairs. In a fraction, the numerator is occasionally referred to asupstairs and the denominator downstairs, as in “bringing a term upstairs”.
up to, modulo, mod out by An extension to mathematical discourse of the notions of modular arithmetic. A state-ment is true up to a condition if the establishment of that condition is the only impediment to the truth ofthe statement. Also used when working with members of equivalence classes, esp. in category theory, wherethe equivalence relation is (categorical) isomorphism; for example, “The tensor product in a weak monoidalcategory is associative and unital up to a natural isomorphism.”
vanish To assume the value 0. For example, “The function sin(x) vanishes for those values of x that are integermultiples of π.” This can also apply to limits: see Vanish at infinity.
weak, weaker The converse of strong.
well-defined Accurately and precisely described or specified.
2.3 Proof terminology
The formal language of proof draws repeatedly from a small pool of ideas, many of which are invoked through variouslexical shorthands in practice.
aliter An obsolescent term which is used to announce to the reader an alternative method, or proof of a result. In aproof, it therefore flags a piece of reasoning that is superfluous from a logical point of view, but has some otherinterest.
by way of contradiction (BWOC), or “for, if not, ...” The rhetorical prelude to a proof by contradiction, preced-ing the negation of the statement to be proved. Also, starting a proof or a sub-proof with Assume... indicatesthat a proof by contradiction will be employed.
if and only if (iff) An abbreviation for logical equivalence of statements.
in general In the context of proofs, this phrase is often seen in induction arguments when passing from the basecase to the “induction step”, and similarly, in the definition of sequences whose first few terms are exhibited asexamples of the formula giving every term of the sequence.
necessary and sufficient A minor variant on “if and only if"; “A is necessary (sufficient) for B” means “A if (onlyif) B”. For example, “For a field K to be algebraically closed it is necessary and sufficient that it have no finitefield extensions" means "K is algebraically closed if and only if it has no finite extensions”. Often used in lists,as in “The following conditions are necessary and sufficient for a field to be algebraically closed...”.
need to show (NTS), required to prove (RTP), wish to show, want to show (WTS)Proofs sometimes proceed by enumerating several conditions whose satisfaction will together imply the desiredtheorem; thus, one needs to show just these statements.
one and only one A statement of the uniqueness of an object; the object exists, and furthermore, no other suchobject exists.
24 CHAPTER 2. LIST OF MATHEMATICAL JARGON
Q.E.D. (Quod erat demonstrandum): A Latin abbreviation, meaning “which was to be demonstrated”, historicallyplaced at the end of proofs, but less common currently, having been supplanted by the Halmos end-of-proofmark.
sufficiently nice A condition on objects in the scope of the discussion, to be specified later, that will guarantee thatsome stated property holds for them. When working out a theorem, the use of this expression in the statementof the theorem indicates that the conditions involved may be not yet known to the speaker, and that the intentis to collect the conditions that will be found to be needed in order for the proof of the theorem to go through.
the following are equivalent (TFAE) Often several equivalent conditions (especially for a definition, such as normalsubgroup) are equally useful in practice; one introduces a theorem stating an equivalence of more than twostatements with TFAE.
transport of structure It is often the case that two objects are shown to be equivalent in some way, and that oneof them is endowed with additional structure. Using the equivalence, we may define such a structure on thesecond object as well, via transport of structure. For example, any two vector spaces of the same dimensionare isomorphic; if one of them is given an inner product and if we fix a particular isomorphism, then we maydefine an inner product on the other space by factoring through the isomorphism.
Let V be a finite-dimensional vector space over k....Let (ei)₁ ≤ i ≤ n be a basis for V....There is anisomorphism of the polynomial algebra k[Tij]₁ ≤ i,j ≤ n onto the algebra Symk(V ⊗ V*)....It extendsto an isomorphism of k[GLn] to the localized algebra Symk(V ⊗ V*)D, where D = det(ei ⊗ ej*)....Wewrite k[GL(V)] for this last algebra. By transport of structure, we obtain a linear algebraic groupGL(V)isomorphic to GLn.— Igor Shafarevich (1991, p.12)
without (any) loss of generality (WLOG, WOLOG, WALOG), we may assume (WMA)Sometimes a proposition can be more easily proved with additional assumptions on the objects it concerns. Ifthe proposition as stated follows from this modified one with a simple and minimal explanation (for example, ifthe remaining special cases are identical but for notation), then the modified assumptions are introduced withthis phrase and the altered proposition is proved.
2.4 Proof techniques
Mathematicians have several phrases to describe proofs or proof techniques. These are often used as hints for fillingin tedious details.
angle chasing Used to describe a geometrical proof that involves finding relationships between the various anglesin a diagram.[2]
back-of-the-envelope calculation An informal computation omitting much rigor without sacrificing correctness.Often this computation is “proof of concept” and treats only an accessible special case.
by inspection A rhetorical shortcut made by authors who invite the reader to verify, at a glance, the correctnessof a proposed expression or deduction. If an expression can be evaluated by straightforward application ofsimple techniques and without recourse to extended calculation or general theory, then it can be evaluated byinspection. It is also applied to solving equations; for example to find roots of a quadratic equation by inspectionis to 'notice' them, or mentally check them. 'By inspection' can play a kind of gestalt role: the answer or solutionsimply clicks into place.
clearly, can be easily shown A term which shortcuts around calculation the mathematician perceives to be tediousor routine, accessible to any member of the audience with the necessary expertise in the field; Laplace usedobvious (French: évident).
complete intuition commonly reserved for jokes (puns on complete induction).
2.5. NOTES 25
diagram chasing [3] Given a commutative diagram of objects and morphisms between them, if one wishes to provesome property of the morphisms (such as injectivity) which can be stated in terms of elements, then the proofcan proceed by tracing the path of elements of various objects around the diagram as successive morphismsare applied to it. That is, one chases elements around the diagram, or does a diagram chase.
handwaving A non-technique of proof mostly employed in lectures, where formal argument is not strictly necessary.It proceeds by omission of details or even significant ingredients, and is merely a plausibility argument.
in general In a context not requiring rigor, this phrase often appears as a labor-saving device when the technicaldetails of a complete argument would outweigh the conceptual benefits. The author gives a proof in a simpleenough case that the computations are reasonable, and then indicates that “in general” the proof is similar.
index battle for proofs involving object with plurious indices which can be solved by going to the bottom (if anyonewishes to take up the effort). Similar to diagram chasing.
trivial Similar to clearly. A concept is trivial if it holds by definition, is immediate corollary to a known statement,or is a simple special case of a more general concept.
2.5 Notes[1] Boyd, Stephen (2004). Convex Optimization. Cambridge University Press. ISBN 0521833787.
[2] Roe, John (1993), Elementary Geometry, Oxford science publications, p. 119, ISBN 0-19-853456-6
[3] Numerous examples can be found in (Mac Lane 1998), for example on p. 100.
2.6 References• Eilenberg, Samuel; Mac Lane, Saunders (1942), “Natural Isomorphisms in Group Theory”, Proc. Natl. Acad.Sci. USA 28: 537–543, doi:10.1073/pnas.28.12.537.
• Impagliazzo, Russell (1995), “A personal view of average-case complexity”, Proc. Tenth Annual Structure inComplexity Theory Conference (SCT'95), pp. 134–147, doi:10.1109/SCT.1995.514853.
• Jackson, Allyn (2004), “Comme Appelé du Néant — As If Summoned from the Void: The Life of AlexandreGrothendieck”, AMS Notices 51 (9,10) (Parts I and II).
• Mac Lane, Saunders (1997), “The PNAS way back then” (PDF), Proc. Natl. Acad. Sci. USA 94 (12): 5983–5985, doi:10.1073/pnas.94.12.5983.
• Mac Lane, Saunders (1998), Categories for the Working Mathematician, Springer.
• Monastyrsky, Michael (2001), “Some Trends in Modern Mathematics and the Fields Medal” (PDF), Can.Math. Soc. Notes 33 (2 and 3).
• Pinto, J. Sousa (2004), Hoskins, R.F., ed., Infinitesimal methods for mathematical analysis, Horwood Publish-ing, p. 246, ISBN 978-1-898563-99-0.
• Poincare, Henri (1913), Halsted, Bruce, ed., The Foundations of Science, The Science Press, p. 435.
• Rota, Gian-Carlo (1977), “The phenomenology ofmathematical beauty”, Synthese 111 (2): 171–182, doi:10.1023/A:1004930722234,ISSN 0039-7857.
• Shafarevich, Igor (1991), Kandall, G.A., ed., Algebraic Geometry IV, Springer.
• Wiedijk, Freek, ed. (2006), The Seventeen Provers of the World, Birkhäuser, ISBN 3-540-30704-4.
26 CHAPTER 2. LIST OF MATHEMATICAL JARGON
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