TORUS GROUPS. Wayne Lawton Department of Mathematics National University of Singapore [email protected] http://www.math.nus.edu.sg/~matwml. Ancient Mathematics. Result 1. (Euclid, Elements, III, Prop. 20)
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.
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.
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.
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.
This theorem is also
obvious when viewed
in three dimensions.
Pappus claims  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’
Appolonius (200BCE) parameterized the unit
circle with the rational stereographic map 
for Pythagorean triplets
This maps the set Q of rational numbers onto
all except one rational point in the unit circle
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.
emerges with a non-rational parameterization of the circle
Robert Coates 1714
Leonard Euler 1748
Richard Feynman 1963
“the most important formula in 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
Euler’s formula gives a homomorphism
from the group of
onto the circle group
is the group of integers
category whose objects are locally
compact abelian topological groups, and morphisms are continuoushomomorphisms
Fourier transform of
and gives isometry
Weierstrass: trig. polynomials
are dense in
torus group dim =
uniformly almost periodic
Weierstrass epicycle method of Claudius Ptolemy (90-168), models planetary motion by + of circular motion
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.
A dynamical system
is expansive if
there exists open
compact, connected, abelian group
an expansive automorphism
is a solenoid group
(inverse or projective limit of torus groups)
Result 3. If
is expansive, then there
exists a finite subset
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.
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
if P is
a monic polynomial with integer coefficients.
1917 Pierce studied prime factors of seq.
that generalizes Mersenne’s seq.
1933 Lehmer proved
1937 Lefschetz Fixed Point Theorem
Jensen’s formula this extends M(P)
where Q is polynomial with Q(0) = 1.
1975  I used this + prediction theory to compute M(P) as limit of rational sequence
1976 I outlined a research strategy to attack the Lehmer Conjecture (LC) in 
that utilized facts: the toral hyperspace
with the Hausdorff topology is compact,
is continuous (later conjectured by Boyd),
Weak Lehmer Conjecture For k > 1 L. Conj. conclusion holds for P int. coef. and k terms
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 
1983 Dobrowolski, Lawton, Schinzel proved the WLC using algebraic geometry in 
1983 I proved Boyd’s Conjecture in 
using: If P(z) is monic with k > 1 terms, then
denotes Lebesque measure and
(Kron. dim > 1 + B. Conj easily WLC)
My proof of this inequality is discussed by Schmidt  and by Everest and Ward .
It was used by Lind, Schmidt and Ward  to prove that ln M(P) is the entropy of a action and by Schinzel  to obtain inequalities for M(P) for
2003 Banff Workshop Boyd, Lind, Villegas and Deninger  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”
March 2007 In  I submitted my proof of the 1997 Lagarius-Wang Conjecture  :
is a positively expansive
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
Use methods developed in : 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.