TORUS GROUPS

1. TORUS GROUPS Wayne Lawton Department of Mathematics National University of Singapore matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml

2. 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.

3. 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.

4. 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.

5. 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.

6. 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’

7. 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

8. 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.

9. 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”

10. 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

11. 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 Therefore

12. Modern Mathematics category whose objects are locally compact abelian topological groups, and morphisms are continuoushomomorphisms Dual defined by Fourier transform of is in and gives isometry

13. Modern Mathematics dual group dual group compact discrete connected torsion free finite rank finite dim Weierstrass: trig. polynomials are dense in

14. 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

15. 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.

16. 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)

17. 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.

18. Research Review Finitely Generated Conjecture: If an 1972 is ergodic and automorphism entropy  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.

19. 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

20. Research Review Mahler Measure measurable 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

21. 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

22. 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 where denotes Lebesque measure and (Kron. dim > 1 + B. Conj easily  WLC)

23. 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”

24. 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

25. Future Research Use methods developed in [69]: toral hyperspace, 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.