My main research areas
are Geometry, Harmonic analysis on Lie groups, Representation
theory and combinatorics, and Hypergroups. I have secondary
interests in Number theory, Mathematical Physics and
Foundations of mathematics.
My involvement with each
of these areas is summarized below, and links to relevant
papers are given.
My coauthors are D Arnal
(Dijon), N Bel-Baraka (Dijon), W Bloom (Murdoch University),
A H Dooley (UNSW), H Grundling (UNSW), F Loose (Tuebingen),
J Repka (Toronto), R Srinivasan (Bangalore), V S Sunder
(Institute of Mathematical Sciences, Chennai), N Obata
Trigonometry, a simpler and more direct approach
to trigonometry which uses rational arithmetic and
polynomial formulas instead of transcendental trigonometric
functions, with many new applications. This theory
opens up dramatic new opportunities for revitalizing
modern mathematics teaching---even in high schools,
offers engineers simpler and more accurate tools for
solving practical geometric problems, allows more
efficient application of computers to robotics, connects
naturally with Einstein's special theory of relativity,
and shows how to finally organize Euclidean geometry
into a logically coherent subject in which waffling
is unnecessary. (Yes, it is possible!)
See Divine Proportions: Rational
Trigonometry to Universal Geometry
Survivor: the trigonometry
challenge (pdf)---a light hearted
look at why rational trigonometry is superior to classical
trigonometry when it comes to specific geometric problems
involving medians, bisectors, altitudes and areas.
the equilateral triangle (pdf)---How to cut up
an equilateral triangle and measure the result rationally.
Archimedes and a new trigonometry ---shows how
rational trigonometry follows naturally from the geometry
of the ancient Greeks, once you look at it in the
Geometry, a far reaching generalization of
classical Euclidean planar geometry, valid over a
general field, and extending to arbitrary quadratic
forms. By formulating geometry in a purely algebraic
setting, with distance and angle replaced by appropriate
quadratic algebraic quantities (quadrance and spread),
we may investigate those aspects of geometry that
are truly universal.
The usual division of geometry into Euclidean versus
non-Euclidean is much better understood in terms of
the more fundamental affine versus projective division.
Affine means it applies in a linear space, projective
means it applies to a projective space, ie framework
of lines through the origin in a linear space. Spherical
or elliptic and hyperbolic geometries are best thought
of as projective theories. The projective version
of universal geometry has laws which are deformations
of the affine version.
Affine and projective
universal geometry (pdf)
dimensional geometry, a chapter in Universal
Geometry, in which the metrical structure of one dimensional
space, both affine and projective, yields already
many of the main directions of geometry in higher
metrical geometry (pdf)
a surprising three-fold symmetry which brings together
Euclidean and non-Euclidean geometry, and transcends
Klein's Erlangen program. There are three quadratic
forms in the plane which interact with each other.
One is the usual Euclidean form, the other two are
relativistic. Many features of plane geometry are
seen in a new light when the standard Euclidean picture
is supplemented by the other two geometries. In particular
we get new views on triangle geometry, conics, quadrilaterals,
cubics and much more.
geometry is one of the oldest and richest
branches of geometry, and investigates triangle points
like the orthocenter, circumcenter, Fermat points,
isodynamic points and others (more than 1000 are listed
Kimberling's site), triangle lines, such as the
Euler line, Lemoine line, Brocard line, and triangle
cubics such as the Neuberg cubic. I am interested
in extending this business to general fields, in particular
to finite fields.
Finite fields form an interesting laboratory for exploration,
as generally calculations are simpler. The study of
elliptic curves in particular could benefit, since
triangle geometry gives metrical definitions of
cubics, and as a bonus come with many (more than
a hundred, see
Bernard Gibert's site) predetermined points. There
ought to be more interaction between the geometry
of a triangle and the arithmetic of the elliptic curve!
However for finite fields, the usual tri-linear coordinates
will need to be rethought. Universal geometry is the
ideal toolbox for formulating and proving results
in these directions.
Also I am interested in tangent conics, an idea promoted
in my book Divine Proportions. I have shown that for
an elliptic curve in Weierstrasse form over a field
in which -3 is not a square, the tangent conics are
either identical or disjoint.
See Neuberg cubics over
finite fields (pdf)
and Projective Trigonometries, I am currently
writing up a new theory of rational spherical trigonometry,
which shows how to extend the ideas of `Divine Proportions'
to the spherical case. This establishes new metrical
structures on projective space of remarkable simplicity
and elegance, thus furthering the case that algebraic
geometry should start to think more metrically. I
am also working on an exposition of ZOME geometry.
Trigonometries, I am also currently writing
up a new theory of hyperbolic geometry. This new version
generalizes the old theory to arbitrary fields, features
duality as a key organizing point, uses the Beltrami-Klein
model, both inside and outside the unit circle, so
that the entire plane is used, and provides much simpler
rational definitions of all the main concepts. Differential
geometry is not needed, hyperbolic trig functions
are also unnecessary.
Harmonic analysis on Lie groups
- Weyl symbol
calculus for nilpotent Lie groups: Using Weyl calculus
of differential operators (related to Kohn -Nirenberg,
but more symmetrical) and a particular ruling of an
arbitrary coadjoint orbit of a simply connected nilpotent
Lie group, Kirillov representations of the Lie algebra
are given by the operators whose symbols are the corresponding
Hamiltonian functions obtained by restriction to the
dual of the Lie algebra. This allows a geometric approach
to the definition and study of the Fourier transform
for such groups.
( PhD thesis: Yale University 1984)
map of a Lie group representation. The moment
map of symplectic geometry can be applied to the projective
space of a finite dimensional unitary representation
of a Lie group to yield a map into the dual of the
Lie algebra which intertwines with the coadjoint action.
An explicit formula allows the extension to infinite
dimensional representations. The image of this moment
map I call the moment set of the
representation. It turns out that this procedure can
be expanded to yield essentially the dual process
of geometric quantization, and helps explain Kirillov's
theory of coadjoint orbits and irreducible representations.
- Theorem: The
moment set of an irreducible unitary representation
of a nilpotent Lie group is the closed convex hull
of the associated Kirillov co-adjoint orbit.
sets of finite dimensional irreducible representations
of compact Lie groups. Determined explicity the moment
sets for all irred reps of a compact simple Lie group,
and showed that they are mostly all convex, with some
special exceptions when the highest weight is close
to a wall of the postive Weyl chamber. This result
obtained independently by Arnal and Ludwig.
For compact Lie groups, there is a neighborhood of
zero in the Lie algebra such that the exponential
of the sum of two adjoint orbits is exacly the same
as the product of the exponentials of each orbit in
the group. In the case of U(n) this settled a conjecture
of Thomson (Rouviere discovered another way
to settle this conjecture independently)
of a geometric Fourier transform
for a compact Lie group, involving a particular function
on the integral part of the cotangent bundle of the
group which is a kernel for the Fourier transform
taking functions on the group to functions on the
integral coadjoint orbits. This involved constructing
a star product (as in Flato, Fronsdale et al) on an
integral coadjoint orbit, and developing a symbol
calculus. In the case of SU(2), this kernel is closely
related to the Cayley transform, as well as interesting
formulas from classical spherical trigonometry.
- Another proof
of Schur Horn/Kostant theorem about projection of
coadjoint orbits for compact Lie groups onto maximal
torus, using modern version of approximation algorithm
going back to Lagrange
- (with J Repka
and A H Dooley) Resolution of Weyl's famous problem
on sums of Hermitian matrices. We
showed how to explicitly describe not only the set,
but more precisely the probability measure
obtained by convolving two invariant measures
on two co-adjoint orbits for a general compact semisimple
Lie group. This result uses symplectic geometry (moment
map results of Guillemin and Sternberg, Atiyah) and
the Kirillov form of the Weyl character formula. The
result is in terms of origami operations on
measures on the maximal torus: starting with basic
tenet measures, fold over until everything is in the
positive Weyl chamber, taking care to maintain carrect
signs. This work predates resolution of same problem
- (with A H
Dooley) Discovery of the wrapping map
and the wrapping theorem: A global
understanding of the role between the hypergroup of
adjoint orbits and conjugacy classes on a complact
Lie group. This generalizes results of Duflo, Harish-Chandra
and Kashiwara-Vergne. In my view, the wrapping theorem
is the key structural fact about
a compact semisimple Lie group. Once you understand
it, the representation theory falls into place--including
why Kirillov orbit theory works, Vergne's Poisson-Plancherel
formula. It also suggests a kew problem of harmonic
analysis in the non-compact setting: to develop a
suitable wrapping map that works with central noncompact
distributions. The Duflo isomorphism is the infinitesimal
version, so it should exist...
- (with A H
Dooley) Using the wrapping map to greatly enlarge
the possibilities for Kirillov-type character formulas
for compact semisimple Lie groups. Introduction to
the idea of a modulator.
- (with A H
Dooley) Extension of the wrapping map and theorem
to semidirect products of compact groups by vector
spaces. This requires a suitable use of almost periodic
functions, together with the former theory.
of Weyl calculus to general (locally compact) abelian
groups, involving a corresponding star product,
and suitable oscillator representation of an associated
symplectic group. The Cayley transform also appears.
As an application, the usual n by n matrices used
to represent operators are replaced by symbols, giving
new descriptions of the structure of the Lie algebra
Representation theory and combinatorics
- Combinatorial construction of
the simply laced simply Lie algebras except for E8.
The minuscule representations are constructed purely
using combinatorial constructions, involving the mutation
and numbers games, spaces built from ideals of certain
lattices associated to these games, and raising and
lowering operators defined using a parity function
and root layers. The result is an explicit Chevalley
basis with computable integer structure constants.
- Combinatorial construction of
G2, by extending the previous work to the non-simply
laced case. The 7 dimensional representation of this
14 dimensional Lie algebra is constructed, resulting
in an explicit basis and integral structure constants.
- Neighbourly graph sequences are
defined and maximal two neighbourly sequences are
defined, resulting in a classification involving the
ubiquitous ADE Dynkin diagrams. This gives another
approach to minuscule posets purely from graph theory
(without reference to the mutation and numbers games),
which are used above in constructions of simple Lie
- Duality for finite commutative hypergroups:
Hypergroups are lovely generalizations of
groups, where the product of two elements is not a
single element, but rather a probablility distribution
of elements. There are many examples, coming from
group theory, von Neumann algebras, mathematical physics
and special functions and more... In the finite commutative
case, there is a duality theory if one extends to
signed hypergroups, generalizing Pontryagin duality
for finite groups. The beauty of this is that the
commmutative hypergroups situation is much richer,
and relates to many non-commutative group situations.
This is closely related to work of Kawada on C- algebras.
- Entropy: A second law of thermodynamics
applies to multiplication in a finite commutative
hypergroup, once one defines entropy appropriately.
Applications to characters of finite groups, as well
as to the fusion rule algebras of mathematical physics.
Entropy for finite commutative hypergroups and fusion
rule algebras, N J Wildberger, J. London Math
Soc. (2) 56 (1997) 275-291
- Strong hypergroups of order three: A classification
of hypergroups with three elements whose duals are
also hypergroups (as opposed to just being signed
hypergroups). Surprisingly the moduli space has a
small disconnected component, closely related to the
Golden hypergroup. Many rather mysterious inequalities
that require explaining are involved here.
hypergroups of order three, N J Wildberger, J.
Pure and Appl. Alg. 174 (2002) 95-115
- Actions of finite hypergroups:
(with V S Sunder) The notion of action is introduced,
parallel to that of a representation but capturing
better the probabilistic nature of a hypergroup. Maximal
actions are related to association schemes, and a
geometric criteria is given for maximal *-actions
ofa Hermitian hypergroup. Following a suggestion of
E. Bannai, we find a three element hypergroup with
infinitely many irreducible *-actions. Some basic
inequalities in this paper generalize my earlier entropy
Actions of finite hypergroups, V S Sunder and N J
Wildberger, J. Algebraic Combin. 18, (2003) 135-151
- Generalized hypergroups and orthogonal polynomials:
(with N Obata) We generalize a discrete hypergroup
to include examples coming from linearization coefficients
for orthogonal polynomials, allowing coefficients
which are not necessarily positive, and determine
when such an object can be embedded in a C* algebra.
In the commutative setting a Gelfand theory is derived
and harmonic analysis (Fourier transform etc) set
up. Applications to maximum values of orthogonal polynomials
hypergroups and orthogonal polynomials, N Obata
and N J Wildberger, Nagoya Math J. Vol 142 (1996)
- Power method for solving
Diophantine equations: While most work on
Diophantine equations concentrates upon equations
of low degree with a few variables, this method tackles
the general situation, giving an explicit and surprisingly
powerful tool to solve Diophantine equations either
over the rationals or a finite field of prime order.
Although not always succesful (that would be too much
to ask!) it works especially well with `random' equations
with many variables and terms.
Michael Leeming, a student at UNSW, has written a
lovely implementation of this method in an applet
which you can try.
invariants of Diophantine equations
N J Wildberger, Proc. Indian Acad. Sci. (Math. Sci.),
Vol. 104, No. 3, August 1994, 549-555.
- Cyclotomic constants:
Gauss discovered that associated to quadratic residues
and non-residues over a finite field of p elements
were certain integers, which count how many times
a residue plus a residue is a residue, or a non-residue
etc. This can be generalized by looking at orbits
of the subgroup of the cyclic group of automorphisms
of index n, and the determination of the associated
`cyclotomic constants' is one of the main problems
I showed that there is an `integral hypergroup' called
an assembly in this situation, and that this assembly
obeys an algebraic rigidity that determines it. In
effect, the algebra in the situation controls the
number theory. The proof of the theorem (which is
too specialized to state here) involves harmonic analysis
on hypergroups as well as some algebraic number theory.
Foundations of mathematics
The foundations of modern mathematics
are something of a mess. With the exception of combinatorics
and some parts of algebra, almost nothing makes precise
sense (including most of my own work). Much effort
is required to (slowly) build a respectable logical
framework for the subject.
I outline some of the difficulties that ought to be
seriously acknowledged about both elementary foundational
mathematics and modern set theory and analysis.