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14 results

  • 5 - Advanced Miscellany

  • Peter J. Cameron, University of St Andrews, Pierre-Philippe Dechant, University of Leeds, Yang-Hui He, London Institute for Mathematical Sciences, John McKay, Concordia University, Montréal
  • Book:
    ADE
    Published online:
    01 August 2025
    Print publication:
    07 August 2025, pp 122-164

  • Summary

    We treat some more advanced topics: monstrous (and other) moonshine, Monster and E_8, Niemeier lattices, the triangle property, generalized line graphs, quiver representations, cluster algebras, von Neumann algebras, catastrophes, Calabi–Yau, elliptic fibrations.

  • 3 - ADE Classifications

  • Peter J. Cameron, University of St Andrews, Pierre-Philippe Dechant, University of Leeds, Yang-Hui He, London Institute for Mathematical Sciences, John McKay, Concordia University, Montréal
  • Book:
    ADE
    Published online:
    01 August 2025
    Print publication:
    07 August 2025, pp 40-92

  • Summary

    We discuss some areas where the ADE classification arises: polytopes, tessellations, root systems, Coxeter groups, spectra of graphs, binary polyhedral groups, reflections, Clifford algebras, Lie groups and algebras.

ADE
  • Patterns in Mathematics
  • Peter J. Cameron, Pierre-Philippe Dechant, Yang-Hui He, John McKay
  • Published online:
    01 August 2025
    Print publication:
    07 August 2025
  • The ADE diagrams, shown on the cover, constitute one of the most universal and mysterious patterns in all of mathematics. John McKay's remarkable insights unveiled a connection between the 'double covers' of the groups of regular polyhedra, known since ancient Greek times, and the exceptional Lie algebras, recognised since the late nineteenth century. The correspondence involves the ADE diagrams being interpreted in different ways: as quivers associated with the groups and as Dynkin diagrams of root systems of Lie algebras. The ADE diagrams arise in many areas of mathematics, including topics in algebraic geometry, string theory, spectral theory of graphs and cluster algebras. Accessible to students, this book explains these connections with exercises and examples throughout. An excellent introduction for students and researchers wishing to learn more about this unifying principle of mathematics, it also presents standard undergraduate material from a novel perspective.