TORUS GROUPS Wayne Lawton Department of Mathematics National University of Singapore [email protected] http://www.math.nus.edu.sg/~m atwml
Jan 22, 2016
TORUS GROUPS
Wayne Lawton
Department of Mathematics
National University of Singapore
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,