TORUS GROUPS

Wayne Lawton

Department of Mathematics

National University of Singapore

matwml@nus.edu.sg

http://www.math.nus.edu.sg/~matwml

Ancient MathematicsResult 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 MathematicsResult 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 MathematicsResult 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 MathematicsMonge’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.

A

B

Ca

b

c

Ancient Mathematics

This theorem is alsoobvious when viewed in three dimensions.

Pappus claims [13] thatis was in Euclid’s losttreatise 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 [4]

Ancient Mathematics

22

2

1

2,

1

1

x

x

x

xx 222 cba

for Pythagorean triplets

ca

bThis maps the set Q of rational numbers onto

all except one rational point in the unit circle

~1900BCE Babylonia

~1000BCE China

Geometric Quantization In Action Applications Of Harmonic Analysis In Quantum Statistical Mechanic Norman E. Hurt. 1983

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

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

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

Phase retrieval and zero crossings : mathematical methods in image reconstruction, Norman E. Hurt, 1987.

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

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

ixxix )sinln(cosLeonard Euler 1748 xixix sincos)exp( 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

R)2exp( ixx

whose kernel

onto the circle group

}1||:{ zCzT

Therefore

Z

real numbers

is the group of integers

ZRT /

Modern Mathematics

category whose objects are locallyA

Dual ),(ˆ TGHomG

compact abelian topological groups, and

morphisms are continuous homomorphisms

HHGHom ˆ),,(,)(ˆ )ˆ,ˆ(ˆ),( GHHomHGHom

GFF )(ˆ

Fourier transform of )(1 GLF defined by

is in )ˆ(GC and gives isometry

)ˆ()( 22 GLGL

Modern Mathematics

connected

G

TZR R

dG dG

torsion freefinite dim finite rank

group dual

G

compact discrete

group dual

nZZ / nZZ /

}ˆ,:{)( GCccGP kkk kktrig Weierstrass: trig. polynomials

are dense in )(GC

Modern Mathematics

RT)2exp()))(((,: irxrxTR R

)()( RRB uniformly almost periodic

iff

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

torus group dim =

2

(Harald) BohrCompactification

)(RCf ))((, RBCFFf

History Lessons

Animals can geometrize and recognize symmetry

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.”

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

Rhesus monkeys use geometric and nongeometric

information during a reorientation task, J. Exp. Psyc.

We should use geometric visualization and symmetry.

Research Review

A dynamical system ),( X is expansive if

Gdim

there exists open

compact, connected, abelian group

XxxxOO j

n

j jZk

k

:),()(1

such that

1971 GGG : an expansive automorphism

and G is a solenoid group

(inverse or projective limit of torus groups)

XOO n ,...,1

Research Review

Result 3. If ),( G is expansive, then there

,G

exists a finite subset is generated by the elements in the set

},:)(ˆ{ ZnSn

such that

Result 4. If

GS ˆ

has finite entropy, then forevery

G

}:)(ˆ{rank Znn I obtained these results, and the solenoid structure, using Pontryagin Duality andproperties of equivariant maps.

2121 ,:;2,1),,( XXjX jj

Research Review

Finitely Generated Conjecture: If an1972GG :automorphism

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 impliesG

conclusion Result 2.is ergodic and

entropy )(

Lehmer’s Conjecture: there exists 0such that )1,1(|}|,1max{)(

0)(

P

PM

a monic polynomial with integer coefficients.

if P is

Lehmer-Pierce Sequences

,:),det()( dd TTCCzIzP

1917 Pierce studied prime factors of seq. that generalizes Mersenne’s seq.

dkTHTHC dkdkk 0),()(:

0)(1)(

P

nn P 12 n

1933 Lehmer proved )(|)(|lim /1 PMP nn

found primes 733,032,251,514,233,3)1( 3127 zz

1937 Lefschetz Fixed Point Theorem

d

k

nk

knn

n CCIP0

)(Trace)1()det()()1(

d

k kn nnz

nkn

zCIPz1

)1(

1det)(exp)(

1964 Arov )(ln)(Entropy PMC

smallest known ...1763.1)1( 34567910 zzzzzzzzM

Research Review

Mahler Measure CGF :

Jensen’s formula this extends M(P)

||||inf)( 2 FQFMT

measurable

G

FFM ||lnexp)(

o1920 Szeg

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

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

Research Review

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

and conjectured

with the Hausdorff topology is compact,jHHHH jj largefor

}connectedclosed,{)( dd TGTH

)()(: ddP THTH

)(),|()( dGP TTPPMG

is continuous (later conjectured by Boyd),

that utilized facts: the toral hyperspace

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

Research Review

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

kk VcVzPTz /11})|)(|:({

where denotes Lebesque measure andkk

kc

kkkcc /126/11112 )()(,/24

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]

(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

nZ

),...,( 1 nzzP2003 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] :

nTS

nn TTE :If is a positively expansive

endomorphism and is a real analytic

variety such that ,)(SES then S is a finite union of translates of elements in )( nTHby elements in

nTthat are period under

.ERemark 1. S = zero set of cyclotomic poly. Remark 2. Possibly related to the dynamic Manin-Mumford Conjecture

Future Research

Use methods developed in [69]: toralconstruction 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.

taleehyperspace,