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140 results

Page 1 of 7

  • 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.

  • 3 - Laplacian eigenvalues and optimality

  • Edited by R. A. Bailey, University of St Andrews, Scotland, Peter J. Cameron, University of St Andrews, Scotland, Yaokun Wu, Shanghai Jiao Tong University, China
  • Book:
    Groups and Graphs, Designs and Dynamics
    Published online:
    11 May 2024
    Print publication:
    30 May 2024, pp 176-265

  • Summary

    Eigenvalues of the Laplacian matrix of a graph have been widely used in studying connectivity and expansion properties of networks, and also in analyzing random walks on a graph. Independently, statisticians introduced various optimality criteria in experimental design, the goal being to obtain more accurate estimates of quantities of interest in an experiment. It turns out that the most popular of these optimality criteria for block designs are determined by the Laplacian eigenvalues of the concurrence graph, or of the Levi graph, of the design. The most important optimality criteria, called A (average), D (determinant) and E (extreme), are related to the conductance of the graph as an electrical network, the number of spanning trees, and the isoperimetric properties of the graphs, respectively. The number of spanning trees is also an evaluation of the Tutte polynomial of the graph, and is the subject of the Merino–Welsh conjecture relating it to acyclic and totally cyclic orientations, of interest in their own right. This chapter ties these ideas together, building on the work in [4] and [5].

Page 1 of 7