User:JackSchmidt/AGT

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 =

Algebra

 * 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

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

Combinatorics

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

Numerical analysis

 * efficient matrix multiplication

Number theory

 * galois cohomology

Topology

 * fundamental group, homotopy groups

= Science =

Statistics

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

Biology

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

Chemistry

 * Crystallography

Computer science

 * (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
Hrm, dunno

Material science

 * quasicrystals and texture recognition

Physics

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

= Social science =

Economics

 * i think game theory uses some group theory

= Humanities =

Art

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

Music

 * tons of musicology