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Mathematics: Classification of Algebra

Mathematics: Algebra Classification

  • Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples (“models”) of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes “the theory of groups” as an object of study.  Subject area topics include:
    • Basic idea
    • Varieties
    • Basic construction
    • Some basic theorems
    • Motivations and applications
  • Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Subject area topics include:
    • Zeros of simultaneous polynomials
    • Affine varieties
    • Regular functions
    • The category of affine varieties
    • Projective space
    • The modern viewpoint
    • History
    • Applications
  • Functional analysis is the branch of mathematical analysis concerned with the study of the vector spaces in which limit processes can be defined, and the linear operators acting upon these spaces that are (in some way) compatible with these limits. Subject area topics include:
    • Normed linear spaces
    • Banach spaces
    • Hilbert spaces
    • Banach algebras
    • Normed algebras
    • Topological algebras
    • Topological groups
  • Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. The major difference between algebra and arithmetic is the inclusion of variables. While in arithmetic only numbers and their arithmetical operations (such as +, -, ×, ÷) occur, in algebra, one also uses symbols such as x and y, or a and b to denote variables. Subject area topics include:
    • Real number system
    • Constants
    • Variables
    • Mathematical expressions
    • Equations
    • Intermediary algebra
    • College-level algebra
  • Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and real or complex numbers, often now called elementary algebra. Subject area topics include:
    • Algebraic structures
    • Groups
    • Rings
    • Fields
    • Axiomatically
  • Linear algebra is a branch of mathematics concerned with the study of vectors, with families of vectors called vector spaces or linear spaces, and with functions that input one vector and output another, according to certain rules. These functions are called linear maps (or linear transformations or linear operators) and are often represented by matrices. Linear algebra is central to modern mathematics and its applications. Subject area topics include:
    • Vector spaces
    • Matrices
  • Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. Subject are topics include:
    • Enumerative
    • Matroids
    • Polytopes
    • Partially ordered sets
    • Finite geometries
  • Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization, the behavior of ideals, and field extensions. In this setting, the familiar features of the integers – such as unique factorization – need not hold. The virtue of the primary machinery employed – Galois theory, group cohomology, group representations, and L-functions – is that it allows one to deal with new phenomena and yet partially recover the behavior of the usual integers. Subject area topics include:
    • Basic notions
    • Major results

Source: Wikipedia

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