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Terry Tao
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math.GR Group Theory

  • Group theory provides methods for understanding the Rubik's cube, and for generating algorithms for solving the cube remarkably quickly from any state the cube may be in.
  • Groups find various applications in chemistry, eg. in the study of crystal structures and spectroscopy.
  • Cryptography - various hard algorithmic problems about groups are used to design crypto-systems.
  • Groups of symmetries are used to reduce the dimension of parameter spaces in engineering models to make model verification more tractable.
  • Potentially fast matrix multiplication; see this MO question.
  • Card tricks that don't work by sleight of hand, but via the arrangements of the cards. e.g. Sim Sala Bim, see this site for a description. If you think about it, the symmetric group explains the trick and shows you how you to extend it past three piles of seven cards, but to N piles of M cards.