## N J Wildberger's mathematical Research

**Harmonic Analysis:**Lie groups, representation theory, noncommutative harmonic analysis, the wrapping map (with A H Dooley), diagonalization, combinatorics of the numbers and mutation games, construction of simple Lie algebras, Kirillov theory, positive definite functions on hypergroups (with W Bloom), diamond representations of sl(n) (with D Arnal and N Bel Baraka)

**Symplectic Geometry:**Moment maps of representations, Weyl’s sum of eigenvalues problem, solution of Thompson's conjecture, solution of Horn’s conjecture (with Dooley, Repka), symbol mappings for nilpotent Lie groups (with R Howe and G. Ratcliffe

**Number Theory:**Diophantine equations, Gaussian periods, Pell’s equation

**Algebra:**Spread polynomials, quasi-standard tableaux, root systems and mutation/numbers games for ADE graphs, circles in finite groups

**Hypergroups:**Duality for finite signed hypergroups, symmetric spaces, random walks and hypergroups, integrality, and fusion rule algebras and von Neumann algebras (with V S Sunder)

**Classical Geometry:**discovery of Rational trigonometry, Chromogeometry, incenter and orthocenter symmetries, triangle geometry, Universal Hyperbolic Geometry, Cayley-Klein geometries and conics

**Mathematical Physics:**Geometric quantization, coherent state representations of compact groups, *-products, diamond model for SU(3) symmetries and quarks

**Foundations:**Critique of real numbers and modern set theory, foundations of group theory, YouTube development of data structures in MathFoundations series

**History of Mathematics:**Plimpton 322 and ancient Babylonian trigonometry (with D Mansfield)

My main research areas are geometry (both Euclidean and hyperbolic), Harmonic analysis on Lie groups, Representation theory and combinatorics, and Hypergroups. I have secondary interests in the History of Mathematics, Number theory, Mathematical Physics and Foundations of mathematics. My Research Gate page has most of my 50 or so papers for download. My book is available at this website.

My involvement with each of these areas is summarized below, along with some links to selected papers.

My coauthors include 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 (Nagoya), R. Donnelly (Murray State)

Geometry

**Rational 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!)

It was laid out in my 2005 book: Divine Proportions: Rational Trigonometry to Universal Geometry which is available here, in physical or pdf form.

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.

Trisecting the equilateral triangle (pdf)---How to cut up an equilateral triangle and measure the result rationally.

Pythagoras, Euclid, 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 right way

**Universal 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.

See Affine and projective universal geometry (pdf)**One 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 dimensions.

See One dimensional metrical geometry (pdf)

**Chromogeometry**, 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.

See Chromogeometry (pdf)**Triangle 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 at Clark 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

**Spherical 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.

**Hyperbolic 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) **Moment 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.
**Moment 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.- Theorem: 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) - Construction 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 by Klyatchko. - (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 key 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.
- Extension 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 sl(n).

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 algebras.

Hypergroups

**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.

Duality and 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.

Strong 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 results.

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 is given.

Generalized hypergroups and orthogonal polynomials, N Obata and N J Wildberger, Nagoya Math J. Vol 142 (1996) 67-93.

Number theory

**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.

Row-reduction and 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 in cyclotomy.

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, largely due to the erroneous idea that one can do an infinite number of things and obtain a meaningful output. This sleight of hand is used almost every in modern mathematics to justify assertions that go beyond what our computers can do. It means that we have lost touch with the computational reality that ought to underpin the subject. Even our "examples" now have this ephemeral aspect: "just consider a pseudo-twisted trunk of functorial jets from the space of all semi folds to ... blah blah". You get the idea.

Much effort is required to (slowly) build a respectable logical framework for the subject, and we need to lean on our computer science friends --- and their machines --- to give us some ideas on how to proceed in a computationally realistic fashion.

In the MathFoundations YouTube series I set out to acknowledge and outline the difficulties, and explore ways that we can replace the current status quo with a more meaningful, concrete, computationally oriented mathematics. So far there are 4 playlists in this series, and more than 215 videos. This is my most exciting project!