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User:JackSchmidt/AGT

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The mathematical field of abstract algebra known as group theory is important to many disciplines, because of the wide range of applications of group theory.

Mathematics

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Algebra

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  • Galois theory of equations
  • Steinberg and fundamental groups of rings
  • Group rings as important examples in ring theory
    • Easy counterexample for "noncomm domains have skew fields of fractions"
    • "If G is torsion free, is kG a domain?" has generated tons of ring theory
    • basically just check passman
  • Maximal orders, and the Albert-Brauer-Hasse-Noether theorem more or less come down to crossed algebras, a simple application of groups to algebras

Analysis

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  • Lie groups
    • Classical elliptic integrals, etc.
  • Harmonic analysis
  • Haar measure type arguments
  • Homogenous spaces

Combinatorics

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  • burnside-cauchy-frobenius
  • transitive graphs
  • dense codes
  • analysis of block designs
  • finite geometry

Numerical analysis

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  • efficient matrix multiplication

Number theory

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  • galois cohomology

Topology

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  • fundamental group, homotopy groups

Science

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Statistics

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  • dense block designs, analysis of block designs
  • examples of rapidly mixing markov chains

Biology

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  • check biostats literature, algebraic statistics mostly uses abelian groups and commutative algebra, but probably some real groups too

Chemistry

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  • Crystallography

Computer science

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  • (coding theory again)
  • efficient network design
  • Crypto
    • Group theoretic analysis of block ciphers
    • Counting arguments for stream ciphers
    • Generalized Diffie Hellman problems (solve the word or conjugacy problem in some infinite nonabelian group)

Earth science

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Hrm, dunno

Material science

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  • quasicrystals and texture recognition

Physics

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  • Symmetry principles in general
  • Quantum groups, quantum mechanics
  • Heisenberg groups

Social science

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Economics

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  • i think game theory uses some group theory

Humanities

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Art

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  • symmetry based art, old pottery, fabrics, and modern escher style

Music

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  • tons of musicology