Mathematical research interests
This page lists some of my more recent
papers, including pdf downloads in
Articles on Geometry (Rational
Universal Geometry, Chromogeometry)
- N J Wildberger, Affine and projective universal
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
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
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
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.
- 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
- 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
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.
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
More papers will be added...especially the joint work
with A. H. Dooley on the wrapping map and the `Wrapping