TORUS GROUPS

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TORUS GROUPS

• Wayne Lawton
• Department of Mathematics
• National University of Singapore
• matwml_at_nus.edu.sg
• http//www.math.nus.edu.sg/matwml

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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.

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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.

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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.

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Ancient Mathematics

• Monges 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.

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Ancient Mathematics
This theorem is also obvious when viewed in
three dimensions.
Pappus claims 13 that is was in Euclids
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
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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
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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 Geometry Norman E. Hurt. 2003
Mathematical Physics Of Quantum Wire And Devices
From Spectral Resonances To Anderson
Localization Norman E. Hurt. 2000
Quantum Chaos And Mesoscopic Systems
Mathematical Methods In The Quantum Signatures Of
Chaos Norman E. Hurt. 1997
Phase retrieval and zero crossings mathematical
methods in image reconstruction, Norman E. Hurt,
1987.
Geometric Quantization In Action Applications Of
Harmonic Analysis In Quantum Statistical Mechanic
Norman E. Hurt. 1983
but seen to be exceptional after Faltings in 1983
proved Mordells 1922 conjecture and Wiles in
1994 proved Fermats 1637 conjecture.
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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
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Modern Mathematics
Fouriers 1807 memoir on heat used sine and
cosine representation of functions
Eulers 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
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Modern Mathematics
Eulers formula gives a homomorphism
from the group of
onto the circle group
real numbers
whose kernel
is the group of integers
Therefore
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Modern Mathematics
category whose objects are locally
compact abelian topological groups, and morphisms
are continuous homomorphisms
Dual
defined by
Fourier transform of
is in
and gives isometry
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Modern Mathematics
dual
group
dual
group
compact
discrete
connected
torsion free
finite rank
finite dim
Weierstrass trig. polynomials
are dense in
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Modern Mathematics
torus group dim
(Harald) Bohr
Compactification
uniformly almost periodic
iff
Weierstrass? epicycle method of Claudius Ptolemy
(90-168), models planetary motion by of
circular motion
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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 mulatta International Journal
of Primatology
We should use geometric visualization and
symmetry.
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Research Review
A dynamical system
is expansive if
such that
there exists open
1971
compact, connected, abelian group
an expansive automorphism
?
and
is a solenoid group
(inverse or projective limit of torus groups)
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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
every
I obtained these results, and the solenoid
structure, using Pontryagin Duality
andproperties of equivariant maps.
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Research Review
Finitely Generated Conjecture If an
1972
is ergodic and
automorphism
entropy
? conclusion Result 2.
I tried to prove this using Kriegers result,
that implies that there exists a finite
measurable partition of G whose orbits under
generate and proved it implies
Lehmers Conjecture there exists
such that
if P is
a monic polynomial with integer coefficients.
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Lehmer-Pierce Sequences
1917 Pierce studied prime factors of seq.
that generalizes Mersennes seq.
1933 Lehmer proved
found primes
smallest known
1937 Lefschetz Fixed Point Theorem ?
?
1964 Arov
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Research Review
Mahler Measure
measurable
Jensens 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
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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 gt 1 L. Conj.
conclusion holds for P int. coef. and k terms
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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 gt 1 in 33
1983 Dobrowolski, Lawton, Schinzel proved the WLC
using algebraic geometry in 37
1983 I proved Boyds Conjecture in 38
using If P(z) is monic with k gt 1 terms, then
where
denotes Lebesque measure and
(Kron. dim gt 1 B. Conj easily ? WLC)
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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
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Research Review
March 2007 In 69 I submitted my proof of the
1997 Lagarius-Wang Conjecture 28
If
is a positively expansive
endomorphism and
is a real analytic
variety such that
then
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
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Future Research
Use methods developed in 69 toral
construction to lift (S,E),
hyperspace,
Hiraides result nonexistence of positively
expansive maps on compact connected manifolds
with boundaries, Lojasiewiczs 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.