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MATH/CHEM/COMP 2010

MATH/CHEM/COMP 2010. INTRINSIC FORMULA FOR FIVE POINTS ATIYAH DETERMINANT Dragutin Svrtan. Euclidean and Hyperbolic Geometry of point particles: A progress on the tantalizing Atiyah-Sutcliffe conjectures.

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MATH/CHEM/COMP 2010

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  1. MATH/CHEM/COMP 2010 INTRINSIC FORMULA FOR FIVE POINTS ATIYAH DETERMINANT Dragutin Svrtan

  2. Euclidean and Hyperbolic Geometry of point particles: A progress on the tantalizingAtiyah-Sutcliffe conjectures • Motivation:BERRY-ROBBINS PROBLEM(1997) coming from spin-statistics in particle physics • Cn(R^3):=configuration space of n ordered distinct points/particles in R^3 • PROBLEM: Does there exists a continuous equivariant map f_n:C_n(R^3)U(n)/T^n(=space of n orthogonal complex lines) ? • (leading to a connection between classical and quantum physics) • ATIYAH’s candidate map (2001) (via elementary construction, but not yet justified even for small n ) gave a renaissance to classical (euclidean and noneuclidean ) geometries and interconnects them with many other areas of modern mathematics.

  3. 3 POINTS INSIDE CIRCLE • Three points 1,2,3 inside circle (|z|=R) • 3 point-pairs on circle • p1 (12) (13) • p2 (21) (23) • p3 (31) (32) • point-pair u,v define quadratic with these roots • (z-u)*(z-v) • 3 point-pairs ---> 3 quadratics • p1, p2, p3 ---> p1, p2, p3 • THEOREM 1 (Atiyah 2001). For any triple 1,2,3 of distinct points inside circle the 3 quadratics are linearly independent • Remark: Atiyah gave a synthetic • proof which unfortunately does not generalize to more than 3 points

  4. SPECIAL CASE OF 3 COLLINEAR POINTS • (31)=(32)=(21) |---x----x------x--------| (12)=(13)=(23) u 1 2 3 v(≠u) p1 (z-u)^2 p2 (z-u)*(z-v) p3 (z-v)^2 clearly linearly independent || l*p2+ m*p3 always has v as root || but p1 has u,u as roots and u ≠ v THEOREM1  3-by-3 determinant of coefficient matrix 1 –v12-v13 v12*v13 det(M3) = det( 1 -v21-v23 v21*v23 ) is nonzero 1 -v31-v32 v31*v32

  5. NORMALIZED DETERMINANT D3_R • Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah’s geometric energy • det(M3) • D3:= -------------------------------------- • ( v12-v21)*(v13-v31)*(v23-v32) • D3=1 for collinear points • THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. • (TH.2 => TH.1) • R N LIMIT • Points on “circle at N” are directions in plane • TH.1 and TH.2 are also true for R =N .

  6. EXPLICIT FORMULAS FOR D3 • det(M3) • D3:= -------------------------------------- (original Atiyah’s definition) • ( v12-v21)*(v13-v31)*(v23-v32) • Extrinsic formula: • (v21 – v31) (v13 – v23) (v12 -v32) • D3= 1 + ---------------------------------------------- • (v12 - v21) (v13 - v31) (v23 - v32) • INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< π): • -------------------------------------------------------------------------------------------------------------- • D3 = ½*(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) – ½√(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !)

  7. Hilbert’s Arithmetic of Ends

  8. INTRINSIC FORMULA for D3 • INTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) • D3 = 1+exp(-p)* ∏ sinh(p-a)/sinh(a) • (=> TH2 Intrinsic proof) • EUCLIDEAN CASE: If we define 3-point function by • d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) • then • D3= ½*(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) • D3=1+ d3(a,b,c)/8*a*b*c

  9. SEVEN NEW ATIYAH-TYPE TRIANGLE’S ENERGIES • By switching simultaneously the directions on any edge of a set of edges of a triangle 123 we get 7 new Atiyah-type energies D3_ ε, ε=100,...,111 (with D3_ ε=D3 for ε =000) • E.g. • D3_001= 1-exp(-p+c)*sinh(p)*sh(p-a)* sh(p-b)/ ∏ sinh(a) • D3_110= 1-exp(p-c)*sinh(p)*sh(p-a)* sh(p-b)/ ∏ sinh(a) • D3_111= 1+exp(p)*∏ sinh(p-a)/sinh(a) • D3_111 = ½*(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) + ½*√(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) THEOREM2’(D.S): (i) D3_ εR 1, for ε=000 , 111. (ii) 0<D3_ ε#1, for ε≠000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3!

  10. Equations for Atiyah 3pt energies

  11. 4 POINTS INSIDE CIRCLE • Four points 1,2,3,4 inside circle (|z|=R) • 4 point-triples on circle • p1 (12) (13) (14) • p2 (21) (23) (24) • p3 (31) (32) (34) • p4 (41) (42) (43) • point-triple u,v,w define cubic(polynomial) with these roots • (z-u)*(z-v)*(z-w) • 4 point-triples ---> 4 cubics • p1, p2, p3 ,p4 ---> p1, p2, p3,p4

  12. NORMALIZED DETERMINANT D4=D4_R 4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14 – v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24 – v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34 – v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43 – v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 ≠0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4)

  13. Eastwood-Norbury formulas for euclidean D4 In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca’b’c’ - 4*d3(a.a’,b.b’,c.c’) +SUM +288*VOLUME^2, where SUM: = a’[(b’+c’)^2-a^2)]*d3(a,b,c)+... D4 /64abca’b’c’ = D4 (=>eucl. Conjecture 1, and “almost”(=60/64 ) of euclidean Conjecture2

  14. New proof of the Eastwood-Norbury formula

  15. Geometric interpretation of the "nonplanar"part in Eastwood-Norbury formula

  16. Remarks on Eastwood-Norbury REMARK1: With Urbiha (2006) many cases of euclidean C1-C3 (50 pages manuscript). Euclidean Atiyah_Sutcliffe Conjecture 3 is a “huge” inequality with 4500 terms of degree 12 in six variables (distances). REMARK2: We have (D.S. 2008) a “trigonometric variant” of the Eastwood_Norbury 16*Re(D4):=(1+C3_12+C2_34)*(1+C1_24+C4_13)+ (1+C2_13+C3_24)*(1+C4_12+C1_34)+ (1+C3_12+C1_34)*(1+C2_14+C4_23)+ (1+C1_23+C3_14)*(1+C2_34+C4_12)+ (1+C2_13+C1_24)*(1+C3_14+C4_23)+ (1+C1_23+C2_14)*(1+C3_24+C4_13)+ 2*(C14_23*C13_24 - C14_23*C12_34 +C13_24*C12_34)+ 72*normalized_VOLUME^2. where Ci_jk:=cos(ij,ik) and Cij,kl:=cos(ij,kl). OPEN PROBLEMS: Hyperbolic(Euclidean) version of Eastwood-Norbury formula for n R4 (n R5) points in terms of distances, or in terms of angles.

  17. EUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS • By using our positive parametrization we obtain a proof of the strongest Atiyah- Sutcliffe conjecture C3 for arbitrary 4 points in 3-dimensional Euclidean space. It is remarkable that the “huge” 4500-term polynomial (in r12,r13,r14,r23,r24,r34) |Re(D4)|^2 - D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) as a polynomial in our variables t1,t2,t3,t4,a12,b12 has all coefficients nonnegative.

  18. Atiyah – Sutcliffe 4 point determinant

  19. Verification of four point conjecture of Svrtan – Urbiha (implying Atiyah – Sutcliffe C3)

  20. POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS

  21. RELATIONS AND BASIC DISTANCES FOR 6 POINTS

  22. ĐOKOVIĆ’S RESULTS AND GENERALIZATIONS • In 2002 Đoković verified Atiyah’s conjecture (Conjecture 1) for almost collinear configurations and configurations with dihedral symmetry. • In 2006 (I.Urbiha ,D.S) we extended this to a variety of conjectures (with additional parameters) • proved a Đoković’s conjectural strengthening of Atiyah-Sutcliffe-Conjecture 2 for dihedral configurations and • Atiyah-Sutcliffe Conjecture 3 for 9 points on a line and one outside by extensive computer help.

  23. Remark • It turned out later that some of our generalizations are related to hyperbolic version for almost collinear configurations (with only 1 point aside a line). • Other generalizations are related to some (multi)-Schur symmetric function positivity.

  24. References • [1] Atiyah M, Sutcliffe P, The Geometry of Point Particles. arXiv: hep-th/0105179 (32 pages). Proc.R.Soc.Lond. A (2002) 458, 1089-115. • [2] Atiyah M, Sutcliffe P, Polyhedra in Physics, Chemistry and Geometry, arXiv: math-ph/03030701 (22 pages), “Milan J.Math.” 71:33-58 (2003) • [3].Eastwood M., Norbury P. A proof of Atiyah’s conjecture on configurations of four points in Euclidean three space, Geometry and Topology 5(2001) 885-893. • [4]. Svrtan D, Urbiha I, Atiyah-Sutcliffe Conjectures for almost Collinear Configurations and Some New Conjectures for Symmetric Functions, arXiv: math/0406386 (23 pages). • [5]. Svrtan D, Urbiha I,Verification and Strengthening of the Atiyah-Sutcliffe Conjectures for Several Types of Configurations, arXiv: math/0609174 (49 pages). • [6]. Atiyah M. An Unsolved Problem in Elementary Geometry , www.math.missouri.edu/archive/Miller-Lectures/atiyah/atiyah.html. • [7]. Atiyah M. An Unsolved Problem in Elementary Euclidean Geometry , http//c2.glocos.org/index.php/pedronunes/atiyah-uminho

  25. Thank you very much for your attention.

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