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TORUS GROUPS. Wayne Lawton Department of Mathematics National University of Singapore Ancient Mathematics. Result 1. (Euclid, Elements, III, Prop. 20)

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torus groups


Wayne Lawton

Department of Mathematics

National University of Singapore

ancient mathematics

Ancient Mathematics

Result 1. (Euclid, Elements, III, Prop. 20)

In a circle the angle at the center is double the angle at the circumference, when the angles

have the same circumference at the base.

ancient mathematics1

Ancient Mathematics

Result 2. (Monge 1746-1818) Let there be three circles of different radii lyning completely outside of each other. Then the three points formed by the intersections of the external tangents of pairs of circles lie on a common line.

ancient mathematics2

Ancient Mathematics

Result 2. Extend the circles to spheres. Each pair of lines intersects at the vertex of the cone tangent to a pair of spheres. These vertices lie on the line where the two planes that are tangent to all three spheres intersect.

ancient mathematics3

Ancient Mathematics

Monge’s 3 Circles Theorem is equivalent to the Perspective Triangles Theorem attributed to Desargues (1591-1661): if lines through pairs of vertices meet at a point (here ) then their pairs of sides meet at points on a line.

ancient mathematics4
Ancient Mathematics

This theorem is also

obvious when viewed

in three dimensions.

Pappus claims [13] that

is was in Euclid’s lost

treatise on porisms.

It exemplifies the concept of DUALITY, in this case the fact that every assertion in projective geometry yields a logically equivalent assertion by interchanging the words ‘point’ and ‘line’


Ancient Mathematics

Appolonius (200BCE) parameterized the unit

circle with the rational stereographic map [4]

for Pythagorean triplets

~1900BCE Babylonia

~1000BCE China

This maps the set Q of rational numbers onto

all except one rational point in the unit circle

ancient mathematics5
Ancient Mathematics

Dense rational points is a property also shared by certain elliptic curves and useful for cryptography

Many Rational Points : Coding Theory And Algebraic GeometryNorman E. Hurt. 2003

Mathematical Physics Of Quantum Wire And Devices : From Spectral Resonances To Anderson LocalizationNorman E. Hurt. 2000

Quantum Chaos And Mesoscopic Systems : Mathematical Methods In The Quantum Signatures Of ChaosNorman E. Hurt. 1997

Phase retrieval and zero crossings : mathematical methods

in image reconstruction, Norman E. Hurt, 1987.

Geometric Quantization In ActionApplications Of Harmonic Analysis In Quantum Statistical MechanicNorman E. Hurt. 1983

but seen to be exceptional after Faltings in 1983 proved Mordell’s 1922 conjecture and Wiles in 1994 proved Fermat’s 1637 conjecture.


Modern Mathematics

emerges with a non-rational parameterization of the circle

Robert Coates 1714

Leonard Euler 1748

Richard Feynman 1963

“the most important formula in mathematics”


Modern Mathematics

Fourier’s 1807 memoir on heat used sine and cosine representation of functions

Euler’s formula facilitated modern Fourier analysis by providing complex exponential repesentations, but it took a long time to understand its geometric meaning

Caspar Wessel 1799

Jean-Robert Argand 1806

Carl Frederick Gauss 1832


Modern Mathematics

Euler’s formula gives a homomorphism

from the group of

real numbers

onto the circle group

whose kernel

is the group of integers



Modern Mathematics

category whose objects are locally

compact abelian topological groups, and morphisms are continuoushomomorphisms


defined by

Fourier transform of

is in

and gives isometry


Modern Mathematics








torsion free

finite rank

finite dim

Weierstrass: trig. polynomials

are dense in


Modern Mathematics

torus group dim =

(Harald) Bohr


uniformly almost periodic


Weierstrass epicycle method of Claudius Ptolemy (90-168), models planetary motion by + of circular motion


History Lessons

Charles Darwin, The Descent of Man, Ch11,p.2 “My object in this chapter is solely to show that there is no fundamental difference between man and the higher mammals in their mental faculties.”

Animals can geometrize and recognize symmetry

Rhesus monkeys use geometric and nongeometric information during a reorientation task, J. Exp. Psyc.

Preferences for Symmetry in Conspecific Facial Shape Among Macaca mulattaInternational Journal of Primatology

We should use geometric visualization and symmetry.


Research Review

A dynamical system

is expansive if

such that

there exists open


compact, connected, abelian group

an expansive automorphism


is a solenoid group

(inverse or projective limit of torus groups)


Research Review

Result 3. If

is expansive, then there

exists a finite subset

such that

is generated by the elements in the set

has finite entropy, then for

Result 4. If


I obtained these results, and the solenoid structure, using Pontryagin Duality andproperties of equivariant maps.


Research Review

Finitely Generated Conjecture: If an


is ergodic and



 conclusion Result 2.

I tried to prove this using Krieger’s result, that implies that there exists a finite measurable partition of G whose orbits under generate and proved it implies

Lehmer’s Conjecture: there exists

such that

if P is

a monic polynomial with integer coefficients.


Lehmer-Pierce Sequences

1917 Pierce studied prime factors of seq.

that generalizes Mersenne’s seq.

1933 Lehmer proved

found primes

smallest known

1937 Lefschetz Fixed Point Theorem 

1964 Arov


Research Review

Mahler Measure


Jensen’s formula  this extends M(P)

1920 Szeg

where Q is polynomial with Q(0) = 1.

1975 [31] I used this + prediction theory to compute M(P) as limit of rational sequence


Research Review

1976 I outlined a research strategy to attack the Lehmer Conjecture (LC) in [32]

that utilized facts: the toral hyperspace

with the Hausdorff topology is compact,

and conjectured

is continuous (later conjectured by Boyd),

Weak Lehmer Conjecture For k > 1 L. Conj. conclusion holds for P int. coef. and k terms


Research Review

1857 Kronecker P integer coef. and M(P)=1

 P is cyclotomic (all roots are roots of 1)

1977 I extended Kronecker dim > 1 in [33]

1983 Dobrowolski, Lawton, Schinzel proved the WLC using algebraic geometry in [37]

1983 I proved Boyd’s Conjecture in [38]

using: If P(z) is monic with k > 1 terms, then


denotes Lebesque measure and

(Kron. dim > 1 + B. Conj easily  WLC)


Research Review

My proof of this inequality is discussed by Schmidt [84] and by Everest and Ward [15].

It was used by Lind, Schmidt and Ward [72] to prove that ln M(P) is the entropy of a action and by Schinzel [83] to obtain inequalities for M(P) for

2003 Banff Workshop Boyd, Lind, Villegas and Deninger [7] explore M(P) in dynamical systems, K-theory, topology and analysis,

and Vincent Maillot announced “I can prove multidimensional Mahler measure of any polynomial can be expressed as a sum of periods of mixed motives”


Research Review

March 2007 In [69] I submitted my proof of the 1997 Lagarius-Wang Conjecture [28] :


is a positively expansive

endomorphism and

is a real analytic

variety such that


is a finite

union of translates of elements in

by elements in

that are period under

Remark 1. S = zero set of cyclotomic poly.

Remark 2. Possibly related to the dynamic Manin-Mumford Conjecture


Future Research

Use methods developed in [69]: toral


construction to lift (S,E),

Hiraide’s result : nonexistence of positively expansive maps on compact connected manifolds with boundaries, Lojasiewicz’s structure theorem for real analytic sets, and foliations for E, to examine the structure of more general algebraic mappings on real analytic sets, the dynamic Manin-Mumford conjecture, and LC.