Part Ten X › Division II II › Section 10/22 10/22

Branches of Mathematics

Subject Outline

ASet theory
- 1Origins of set theory and the definitions of a set and a set element, or member
- 2Introduction to set theory
  - aFundamental set concepts
  - bEssential features of Cantorian set theory
- 3Axiomatic set theory: formal analyses of set theory based upon certain fundamental assumptions or undefined notions called axioms [see also 10/21.B.3.a.]
  - aPostulates of axiomatic set theory: the Zermelo-Fraenkel axioms, the von Neumann- Bernays-Godel axioms
  - bLimitations of axiomatic set theory: failure of attempts to prove the consistency of axiomatic set theory, Godel's theorem
  - cThe present status of axiomatic set theory: profound changes in axiomatic set theory as a result of recent discoveries
BAlgebra
- 1Arithmetic
  - aFundamental definitions and laws: the concepts of natural number and integer; the binary operations of addition and multiplication; the commutative and associative laws of addition; the commutative, associative, and distributive laws of multiplication
  - bTheory of divisors: extension of natural number concepts to non-integers, fractions resulting from the binary operation of division
  - cNumber systems and notation: use of the positional principle and the symbol zero to specify , magnitude in sequences of digits; number systems having different bases --e.g., binary, decimal, and sexagesimal systems [see also 10/23.A.1.]
  - dArithmetic calculation with decimals: binary operations with decimals; divisibility rules; calculation of square, cube, and higher roots
  - eLogarithms: formal definition of logarithms, use of logarithms to reduce the operations of multiplication and division to the simpler operations of addition and subtraction [see also 10/23.A.4.b.]
- 2Elementary and multivariate algebra
  - aAlgebra as an extension and generalization of arithmetic
  - bBasic algebraic properties of numbers
  - cPolynomials and rational functions
  - dSolution of equations: the principal problem of elementary algebra
- 3Linear and multilinear algebra
  - aLinear algebra
    - iVector spaces
    - iiMatrices
    - iiiLinear transformations and linear operators
    - ivLinear functionals and their relation to linear transformations
    - vInner products and inner product spaces: self-conjugate, or Hermitian, matrices; unitary and orthogonal matrices
    - viLinear operators in an inner product space: self-adjoint, or Hermitian, operators; unitary and orthogonal operators; the spectral theorem for normal operators
  - bMultilinear algebra
- 4Algebraic structures
  - aLattices
  - bGroups
  - cFields
  - dRings
  - eCategories
  - fHomological algebra
  - gUniversal algebra
CGeometry
- 1Euclidean geometry
  - aGeometry as an abstract doctrine: the axiomatization of the foundations of geometry; axioms of order, incidence, congruence, parallels, and continuity and results derived from them
  - bThe measure of polygons and polyhedra: the theories of equivalence and measure and their relation, Euclid's contribution and its modern extension and generalization
  - cTransformation geometry: reflection, rotation, and translation of geometric figures; homotheties and similitudes
  - dGeometric constructions: the equivalence between Euclidean constructions and existence theorems, gauge constructions, ruler and compass constructions, construction with compass only
  - eGeometry of more than three dimensions: the generalization of Euclidean geometry
  - fThe concept of convexity and convex sets
- 2Non-Euclidean geometry [see also 10/21.B.1.b.]
  - aDistinction between Euclidean and non-Euclidean geometry: hyperbolic geometry and elliptic geometry
  - bGeometric representations of the hyperbolic plane and hyperbolic space
  - cCoordinates in spherical and elliptical space: interpretations of four-dimensional Euclidean space
  - dCoordinates in the hyperbolic plane and hyperbolic trigonometry
  - eTransformations: hyperbolic geometry as characterized by its group of reflections
- 3Projective geometry
  - aThe procedure of projection as the foundation of projective geometry
  - bHomogeneous coordinates: location of points in space
  - cComplex geometry: introduction of complex numbers as homogeneous coordinates
  - dAbstract geometries: extension and generalization of projective geometry to space of any number of dimensions
- 4Analytic and trigonometric geometry
  - aPlane analytic geometry: fundamental procedures and concepts
  - bTrigonometry
  - cCoordinates and transformation of coordinates
  - dProjective and solid analytic geometry: extensions of analytic geometry to the projective plane and to three or more dimensions
  - eSpecial curves: named curves that have been studied with regard to problems in mathematics or the physical sciences; e.g., the folium of Descartes, the lemniscate of Bernoulli, the cardioid, the cycloid, the catenary, the brachistochrone
- 5Combinatorial geometry [see E.1.c., below]
- 6Differential geometry [see also F.2.e., below]
- 7Algebraic geometry [see also F.3., below]
DAnalysis
- 1Real analysis
  - aOrigins and concepts of real analysis
  - bNumber systems and their properties
  - cFunctions and differential calculus
  - dMeasure and integral calculus
- 2Complex analysis
  - aTheory of analytic functions of one complex variable
  - bTheory of analytic functions of several complex variables
  - cPotential theory
- 3Differential equations
  - aOrdinary differential equations
  - bPartial differential equations
  - cSpecial functions that arise as solutions to differential equations; e.g., the hypergeometric function, Legendre polynomials, spherical harmonics, Bessel functions
  - dDynamical systems on manifolds
- 4Functional analysis
  - aGeneral features of functional analysis
  - bCalculus of variations
  - cGeneralized functions: the theory of distributions
- 5Fourier analysis
  - aThe theory of series
  - bFourier series
  - cHarmonic analysis and integral transforms
  - dRepresentations of groups and algebras: Fourier analysis on non-Abelian groups
- 6Theory of probability
  - aHeuristic introduction to probability: the need for a mathematically precise definition of probability
  - bProbability on finite dimensional spaces
  - cProbability on infinite dimensional spaces
- 7Vector and tensor analysis
  - aScalars, vectors, tensors, and the physical quantities that give rise to them
  - bVector algebra and analysis
  - cTensor algebra and analysis
ECombinatorics and number theory
- 1Combinatorics and combinatorial geometry
  - aThe nature and scope of combinatorics: the definition of combinatorics as the branch of mathematics concerned with arrangements, operations, and selections within a finite or a discrete system
  - bMethods, results, and unsolved problems of combinatorial theory, exclusive of geometric considerations
  - cCombinatorial geometry
- 2Number theory
  - aElementary number theory: properties of the whole numbers, or integers [see B.1.a., above]
  - bAlgebraic number theory: properties of algebraic numbers
  - cAnalytic number theory
  - dGeometric number theory
  - eProbabilistic number theory
FTopology
- 1General topology
  - aDefinition and basic concepts of topology; the subject matter and applications of topology as exemplified by certain simple topological problems and their solutions
  - bTopological spaces: methods for constructing topological spaces; Euclidean ^-dimensional space, Hilbert space, Cartesian-product space, and other examples of topological spaces
  - cTopological properties
  - dTopological problems of current interest; e.g., the planar fixed-point problem, the polyhedral Schoenflies problem
- 2Topological groups and differential topology
  - aInteraction between analysis and topology
  - bThe theorems of Tikhonov and Ascoli: embedding of a topological space as a subspace of a compact space
  - cContinuous groups
  - dAnalysis on manifolds: topological implications of problems in global analysis
  - eDifferential topology [see also C.6., above]
- 3Algebraic topology [see also C.7., above]
  - aThe nature and scope of algebraic topology and its context within general topology, the basic concepts of topological spaces and maps
  - bInvariants: unchanging quantities that play a central role in the classification of spaces and maps
  - cHomotopy theory: homotopy classes and the concept of homotopy-equivalent spaces
  - dHomology and cohomology theory: definition of a simplex, axiomatic homology theory
  - eHomotopy groups: stability and suspension
  - fDefinition and properties of fibres, fibre bundles, and fibrings
  - gSheaf cohomology
  - hSpectral sequences: Serre, Rothenberg-Steenrod, and Eilenberg-Moore spectral sequences
  - iFurther developments in homotopy theory. Eilenberg-MacLane spaces, the methods of killing homotopy groups, Serre's C-theory
  - jGeneralized homology and cohomology theory: K-theory, the spectral sequence of G.W. Whitehead-Atiyah-Hirzebruch