N J Wildberger


School of Maths UNSW Sydney 2052 Australia


Tel:61 (02) 9385 7098

Fax:61 (02) 9385 7123


Mathematical research interests

This page lists some of my more recent papers, including pdf downloads in some cases.

Articles on Geometry (Rational Trigonometry,
Universal Geometry, Chromogeometry)

  • N J Wildberger, Affine and projective universal geometry (pdf)

    This paper establishes the basics of universal geometry, a completely algebraic formulation of metrical geometry valid over a general field (not of characteristic two) and an arbitrary quadratic form. The fundamental laws of rational trigonometry are here shown to extend to the more general affine case, and also to a projective version, which has laws which are deformations of the affine case. This unifies both elliptic and hyperbolic geometries, in that the main trigonometry laws are identical in both.

    Euclidean versus non-Euclidean geometries are a manifestation of the distinction between the affine and the projective.

  • N J Wildberger, One dimensional metrical geometry (pdf)

    The basics of universal geometry are already visible in the one dimensional situation, the great blind spot of modern geometry. There is both an affine and a projective version. The affine version is interesting especially with regard to the quadruple quad formula, the relation between quadrances from four points, which anticipates the formula of Brahmagupta for cyclic quadrilaterals. The projective version depends on the one dimensional analog of a quadratic form. Chromogeometry already makes an appearance.

    The spread polynomials, which are rational equivalents of the Chebyshev polynomials of the first kind, contain already the seeds of two dimensional symmetry. They are valid over general fields.

  • N J Wildberger, Chromogeometry (pdf)

    My favourite discovery. A three fold symmetry in planar metrical geometry, that ends up transforming almost every aspect of the subject. Euclidean geometry meets two hyperbolic or relativistic geometries, and all three interact in a lovely way. This introductory paper illustrates applications to triangle geometry, in particular it `explains' in a new way the Euler line and introduces a new triangle of coloured orthocenters which controls all three Euler lines, along with circumcenters and nine point centers. There is much more to be said about this subject!
  • N J Wildberger, Neuberg cubics over finite fields (pdf)

    The Neuberg cubic is the most famous triangle cubic, organizing more than a hundred known triangle points and many lines. This paper uses the framework of Universal Geometry to extend triangle geometry to the finite field setting, and studies an example of a Neuberg cubic over the field of 23 elements. Many, but not all, of the usual real number properties hold. The paper also discusses tangent conics to elliptic curves in Weierstrasse form, showing them to be disjoint or identical if -3 is not a square.

Survey articles

  • N J Wildberger, Algebraic structures associated to group actions and sums of Hermitian matrices in Textos de Matematica, Serie B No28 Dept of Mathematics University of Coimbra 2001 (pdf)

    This is an introductory survey on hypergroups and their application to the Sums of Hermitian matrices problem, first solved by A H Dooley, J Repka and N J Wildberger. A hypergroup is an important generalization of a group, and in practice many are abelian, so this is a wide ranging generalization of the usual harmonic analysis on commutative groups. Important examples come about with random walks on symmetric spaces. Closely tied to work of Dooley and Wildberger on the wrapping map.

  • Hypergroups, symmetric spaces and wrapping maps in Probability Measures on Groups and related structures, Proceedings Oberwolfach 1994 (Editor H. Heyer) 406--425. (pdf)

    The wrapping map generalizes the Duflo isomorphism and results of Harish Chandra. It shows that for a compact Lie group ALL of the exponential map must be considered when pushing down a distribution from the Lie algebra to the Lie group. With an appropriate use of the `j(X)' function (square root of the Jacobian of the exponential map), a map from central distributions on the Lie algebra to central distributions on the Lie group can be defined which is an algebra homomorphism. This links to the theory of hypergroups and explains Kirillov theory in this context.

  • Characters, bi-modules and representations in Lie group harmonic analysis in Harmonic Analysis and Hypergroups Trends in Mathematics Eds. K. A. Ross et al Proceedings Delhi 1995 Birkhauser 1998. (pdf)

    A personal look at some issues in representation theory concerning constructions of representations, hypergroups and the role of bi-modules.

  • Finite commutative hypergroups and applications from group theory to conformal field in Applications of Hypergroups and Related Measure Algebras, Contemp. Math. 183 Proceedings Seattle 1993 (AMS) 413--434. (pdf)

    An overview of how finite commutative hypergroups arise in group theory, number theory, combinatorics, operator algebras and conformal field theory.


Papers On Lie theory/Representation theory

More papers will be added...especially the joint work with A. H. Dooley on the wrapping map and the `Wrapping theorem'...

  • Quarks, diamonds and representations of sl(3) (pdf)

    A new model of the irreducible representations of the simple Lie algebra sl(3) which figures prominently in the quark/gluon theory. This model is constructed over the integers, and relies on remarkable polytopes in three dimensional space called diamonds. The integral points inside the diamond correspond to weight vectors, and these lie over the corresponding weights in the two dimensional weight space. The operators for the Lie algebra are determined by the geometry of chains of nodes inside the diamonds, and the operators for an entire basis is given, as in Gelfand Tsetlin. The coefficients are much easier than Gelfand Tsetlin to determine, however, and are integers! Further the diamonds share the property with Gelfand Tsetlin polytopes that they are both quantum and classical objects- the moment map projection of the volume is the Duistermaat-Heckman measure.

  • A combinatorial construction for simply laced Lie algebras (pdf)

    Simple Lie algebras are classified by Dynkin diagrams. This paper shows how to uniformly construct those simple Lie algebras associated to simply laced Dynkin diagrams (types A,D and E) using only combinatorics--no knowledge of Lie theory needed. The point is to start with the Dynkin diagrams, and then play the mutation and numbers games on these graphs, generating remarkable labelled lattices if one imposes some pleasant conditions. The Lie algebra is then exhibited explicitly as raising and lowering operators in the space of ideals of these lattices. This constructs the minuscule representations of the Lie algebra, and a complete basis of operators, corresponding to all roots (not just simple ones).

  • A combinatorial construction of G2 (pdf)

    An extension of the previous papers theme, this one shows how to push the theory further to construct the exceptional non simply laced Lie algebra G2 by exhibiting explicitly its 7 dimensional representation. Everything reduces to graphs and arrows to get raising and lowering operators that miraculously fit together.

  • Minuscule posets from neighbourly graph sequences (pdf)

    A companion paper to the last two, showing how to use graphs sequences and associated heaps to create the minuscule posets used for constructions of simple Lie algebras. By imposing a particular neighbourly condition on graph sequences, the ADE graphs figure in a classification. The representation theory of Lie algebras and combinatorics on graphs are closely linked, and this paper works towards this theme.

    Papers On Hypergroups

  • (with N. Obata) Generalized hypergroups and orthogonal polynomials (pdf)

    A theory of generalized discrete hypergroups is developed, with an eye to orthogonal polynomials. A boundedness condition is introduced which guarantees embedding into a C* algebra. Countable discrete hypergroups obey this condition, and Jacobi polynomials give another example. More general families of orthogonal polynomials are shown to satisfy the boundedness, ensuring a harmonic analysis even when the usual positivity of linearization coefficients is missing. Applications to a Gelfand theory for commutative generalized hypergroups is given.

  • Strong hypergroups of order three (pdf)

    A commutative hypergroup is strong if its dual (signed) hypergroup is in fact a hypergroup. This paper classifies strong hypergroups of order three. The moduli space exhibits a surprise- a small disconnected component. The Golden hypergroup obtained from a pentagon plays a distinguished role, and there are some rather subtle inequalities that determine boundary conditions. A number of pictures illustrate the parameter sets for different orders, including an early example of Jewett.

  • On the algebraic structure of Gaussian periods (pdf)

    This paper shows that there is a particular rigidity in the theory of cyclotomic constants, a subject which originated with Gauss. The algebraic structure of the gaussian periods obtained by the orbits of the unique subgroup of index n of the automorphisms of the field with p elements (p prime) when n divides p-1 is uniquely determined by a cyclic condition on the associated hypergroup. The proof uses harmonic analysis and some algebraic number theory .

    Papers On Finite Groups, Multisets

  • Finite group theory for a (future) computer (pdf)

    A heretical first course in finite group theory, ostensibly for a computer, but other readers might enjoy it too. Gives an introduction to order invariants, central invariants such as the class and character hypergroups, and subgroup invariants. By placing multisets firmly in the foreground (where they should have been all along), a firmer basis for counting and computing on finite groups emerges, and the powerful notion of a hypergroup is used to introduce character tables and quotient groups (finally in the correct way!)

  • A new look at multisets (pdf)

    For more than a century, mathematicians have been hypnotized by the allure of set theory. Unfortunately, the theory has at least two crucial failings. First of all, infinite set theory doesn't make proper logical sense. Secondly, the fundamental data structures in mathematics ought to be the same ones that are the most important in computer science, science and ordinary life---namely the multiset and the list. This paper begins the slow uphill struggle to put multiset theory in its rightful place. The notational innovations in this paper are particularly important. The tropical calculus also gets a look in, along with a general version of Hall's theorem.


    More papers will be added...