MATH/CHEM/COMP 2011

1. MATH/CHEM/COMP 2011 HYPERBOLIC ANALOG OF THE EASTWOOD-NORBURY FORMULA FOR 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 • C_n(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(u12) (u13) • P2 (u21) (u23) • P3 (u31) (u32) • point-pair u12,u13 define quadratic with these roots • p1:= (Z-u12)*(Z-u13) • 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 { p1, p2, p3} are linearly independent • Remark: Atiyah – synthetic proof which does not generalize to more than 3 points

4. SPECIAL CASE OF 3 COLLINEAR POINTS • (u31)=(u32)=(u21) =-1|---x----x------x--------| (u12)=(u13)=(u23) =1 1 2 3 p1 (Z-1)^2 p2 (Z-1)*(Z+1) p3 (Z+1)^2 clearly linearly independent THEOREM1 : 3-by-3 determinant of the coefficient matrix 1 –u12-u13 u12*u13 det(M3) = det ( 1 -u21-u23 u21*u23 ) ≠ 0 1 -u31-u32 u31*u32

5. NORMALIZED DETERMINANT D3_R • Atiyah : normalized determinantD3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ... Atiyah’s geometric energy • det(M3) • D3:= ------------------------------------------ • ( u12-u21)*(u13-u31)*(u23-u32) • D3=1 only for collinear points • THM 2 (ATIYAH-synthetic proof): D3R1 . • (THM.2 => THM.1) • R N LIMIT GIVES THE EUCLIDEAN CASE • Points on “circle at N” are directions in plane • THM.1 and THM.2 are also true for R =N .

6. EXPLICIT FORMULAS FOR D3 EXTRINSIC FORMULA: (u21 – u31) (u13 – u23) (u12 -u32) D3= 1 + ---------------------------------------------- (u12 - u21) (u13 - u31) (u23 - u32) • INTRINSIC FORMULA: For hyperbolic triangles (0< A+B+C< π): • D3 = ½*(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) -1/2*Φ --------------------------------------------------------------------------------------------------- ------------- • where: Φ^2:= cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2 = ¼*(-1+cos^2(A)+cos^2(B)+cos^2(C)+2*cos(A)*cos(B)*cos(C))

7. Hilbert’s Arithmetic of Ends

8. INTRINSIC FORMULA for D3 • INTRINSIC FORMULA involving 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 functionby • 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)) • By cosine law: • D3=1+ (-a+b+c)*(a-b+c)*(a+b-c)/8*a*b*c

9. SEVEN NEW ATIYAH-TYPE TRIANGLE’S ENERGIES • We introduce 7 new Atiyah-type energies D3_ ε, ε=100,...,111 • (with D3_000=D3) • 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)) +1/2* Φ 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

13. 4 POINTS INSIDE CIRCLE • Four points 1,2,3,4 inside circle (|z|=R) • 4 point-triples on circle • P1 (u12) (u13) (u14) • P2 (u21) (u23) (u24) • P3 (u31) (u32) (u34) • P4 (u41) (u42) (u43) • point-tripleu12,u13,u14 defines cubic(polynomial) • p1:= (Z-u12)*(Z-u13)*(Z-u14) • =1*Z^3 -(u12-u13-u13)*Z^2+(u12*u13+ u12*u14+ u13*u14)*Z– u12*u13*u14 • 4 point-triples ---> 4 cubics • P1, P2, P3 ,P4 ---> { p1, p2, p3, p4}

14. NORMALIZED 4-points DETERMINANT D4 4-by-4 determinant of coefficient matrix of polynomials : ( 1 -u12-u13-u13 u12*u13+ u12*u14+ u13*u14 – u12*u13*u14) |M4| =det( 1 -u21-u23-u23 u21*u23 +u21*u24+u23*u24 – u21*u23*u24) ( 1 -u31-u32-u34 u31*u32 +u31*u34+u32*u34 – u31*u32*u34) ( 1 -u41-u42-u43 u41*u42 +u41*u43+u42*u43 – u41*u42*u43) Det(M4) D4:= ------------------------------------------------------------------------------------ (u12-u21)*(u13-u31)*(u14-u41)*(u23-u32)*(u24-u42)*(u34-u43) CONJECTURES : C1(Atiyah): D4 ≠0 (<--> p1, p2, p3, p4 lin. indep.) 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)

15. Eastwood-Norbury formulas for euclidean D4 In 2001 EASTWOOD -NORBURY, by tricky use of MAPLE ( n=4 points in E^3) : ----------------------------------------------------------Re(D4)=64abca’b’c’ - 4*d3(a*a’,b*b’,c*c’) + Σ*+ 288*Vol^2 -------------------------------------- Σ*:= a’[(b’+c’)^2-a^2)]d3(a,b,c)+...(11 terms) Recall: d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) D4= D4 /64abca’b’c’ =>eucl. C1, => “almost”(=60/64 of) C2

16. New proof of the Eastwood-Norbury formula

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

22. 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). In 2008 we have discovered: TRIGONOMETRIC (euclidean) Eastwood_Norbury formula: 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. Here: 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.

23. TRIGONOMETRIC (hyperbolic-planar case) Eastwood_Norbury formula: 16*Re(D4_hyp):= (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) +(Φ1+ Φ2+ Φ3+ Φ4)/4 +( Φ12_13_24*c14,23+...)(12 terms) +1/2*sqrt(Φ1* Φ2* Φ3* Φ4) Here: Cij,kl:=cos(ij,kl)=2*cij_kl-1 cij_kl:=(u_ij-u_lk)*(u_kl-u_ji)/(u_ij-u_ji)*(u_kl-u_lk) (“ Cross ratio”)

26. EUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (IN R^3) • By using our positive parametrization we prove the strongest Atiyah- Sutcliffe conjecture C3 for arbitrary 4 points in 3-dim Eucl. 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 t1,t2,t3,t4,a12,b12 has all coefficients nonnegative.

27. Atiyah – Sutcliffe 4 point determinant

28. Verification of 4 point conjecture of Svrtan – Urbiha (→ Atiyah – Sutcliffe C3)

29. NEW DEVELOPMENTS • In 2011 M.Mazur and B.V.Petrenko restated the original Eastwood Norbury formula in trigonometric form which besides face angles of a tetrahedron uses also angles of the so called Crelle triangle(associated to the tetrahedron). Our formula in [5] does not involve Crelles angles, but uses “skew” angles . • C2 for convex (planar) quadrilaterals and • C3 for cyclic quadrilaterals (we have proved it already in [5]) and • stated 3 conjectures which are consequences of some of our conjectures in [5] .(Hence we have a proof of all 3)

30. POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS

31. RELATIONS AND BASIC DISTANCES FOR 6 POINTS

33. ĐOKOVIĆ’S RESULTS AND GENERALIZATIONS • In 2002. Đoković verified C1 for • almost collinear configurations and configurations with dihedral symmetry. • In 2006 (I.Urbiha ,D.S) have: (i) extended this to a variety of conjectures (with parameters) including Schur positivity conjectures for some symmetric functions , (ii) proved a Đoković’s conjectural strengthening of C2 for dihedral configurations and (iii) proved C3 for 9 points on a line and 1 outside, by computer trickery. • Recently Mazur and Petrenko (2011) proved C2 for regular polygons by first establishing an amazing result : lim(ln(D_n)/n^2) = 7*ζ (3)/2*π^2-ln(2)/2 ( = 0.007970... )

34. 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.

35. 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 • [8] M.Mazur and B.V.Petrenko :On the conjectures of Atiyah and Sutcliffe arXiv:1102.4662v1 • [9]Atiyah M. Edinburgh Lectures on Geometry,Analysis and Physics,arXiv:1009.4827v1.

36. Thank you very much for your attention.