MATH/CHEM/COMP 2011

HYPERBOLIC ANALOG OF THE EASTWOOD-NORBURY FORMULA

FOR ATIYAH DETERMINANT

Dragutin Svrtan

Euclidean and Hyperbolic Geometry of point particles: A progress on the tantalizing

Atiyah-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 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

3

2

1

(31)

(13)

(12)

(21)

(23)

(32)

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

NORMALIZED DETERMINANT D3_R

• Atiyah : 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:= ------------------------------------------ • ( 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 .

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)) D3 = ½*(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) -1/2*-1/2*ΦΦ --------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------- --------------------------• wherewhere: :

ΦΦ^2:=^2:= cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/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))

Hilbert’s Arithmetic of Ends

INTRINSIC FORMULA for D3• INTRINSIC FORMULA involving side lengths a,b,c (p=(a+b+c)/2

semiperimeter)

• D3 = 1+expD3 = 1+exp(-p(-p)* )* ∏∏ sinh(p-a)/sinh(a) sinh(p-a)/sinh(a)• (=> TH2 (=> TH2 Intrinsic proofIntrinsic proof)) • EUCLIDEAN CASEEUCLIDEAN CASE: : If we define If we define 3-point function3-point function byby • d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-cd3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c))• thenthen

• D3= D3= ½*(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))½*(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))• By cosine law:By cosine law:

• D3=1+ (-a+b+c)*(a-b+c)*(a+b-c)/8*a*b*cD3=1+ (-a+b+c)*(a-b+c)*(a+b-c)/8*a*b*c

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= 1-exp(-p+c(-p+c)*)*sinh(p)*sh(p-a)* sh(p-b)/ sinh(p)*sh(p-a)* sh(p-b)/ ∏∏ sinh(a) sinh(a)

• D3_110= = 1-exp(1-exp(p-cp-c)*sinh(p)*sh(p-a)* sh(p-b)/ )*sinh(p)*sh(p-a)* sh(p-b)/ ∏∏ sinh(a) sinh(a)

• D3_111= = 1+exp(1+exp(pp)*)*∏∏ sinh(p-a)/sinh(a) sinh(p-a)/sinh(a)

• D3_111 D3_111 = ½*(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) = ½*(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) +1/2*+1/2* Φ 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!

Equations for Atiyah 3pt energies

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-triple u12,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}

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)

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

a' b'c'

b

ca

a'((b'+c')^2-a^2)*d3(a,b,c)

1

2

3

4

New proof of the Eastwood-Norbury formula

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

Remarks on Eastwood-NorburyREMARK1: 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.

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”)

POSITIVE PARAMETRIZATION OF DISTANCESBETWEEN 4 POINTS :

b12: = (r13+r24 -r12-r34 )/2a12: = (r13+r24-r14-r23)/2

If r12+r34 < r13+r24 > r14+r23 then :

t1: = (r12+r14-r24)/2, t2: = (r12+r23-r13)/2, t3:=(r23+r24-r34)/2, t4:= (r14+r34-r13)/2

r34 = t3 + a12 + t4

r24 = t2 + a12 + b12 + t4

r14 = t1 + b12 + t4

r23 = t2 + b12 + t3r13 = t1 + b12 + a12 + t3

r12 = t1 + a12 + t2

t4

t4

t1

b12

t1

c12

a12

t3 t3

c12

a12t2

b12

t2

BC = 16,30 cmAD = 12,71 cm

BD = 11,23 cmAC = 21,61 cm

CD = 10,63 cmAB = 21,54 cm

A1''B2" = 1,58 cmC1C2 = 1,58 cm

A1'C2" = 0,34 cmB1B2 = 0,34 cm

A1A2 = 1,92 cm B2'C2' = 1,92 cmParameterization ofdistances between 4points

+b12t4

+a12t3

+b12t1

+a12t2

A2

C1

B1

B2'

B2"

B2

A1''A1'

A1

C2"

C2'

C2

A

C

D

B

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 C3for 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.

Atiyah – Sutcliffe 4 point determinant

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

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)

POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS

Generating distances:t1,t2,t3,t4,t5,t6,a12,a23,a34, b12,b23,b34,c12,c23,c34, d12,d23,d34,e12,e23,e34, f12,f23,f34

r34=t3+c12+c23+c34 +t4 etc.

r24=BD=BB4+C1D =t2+b12+b23+b34+ c12+c23+c34 +t4

r23=t2+b12+b23+b34+t3

r13=AC=AA4+B1C =t1+a12+a23+a34+ b12+b23+b34+t3

r12=AB=t1+a12+a23+a34 +t2

r14=AD=AA3'+A3'C2'+C2'D=AA3+B1B4+C2D =t1+a12+a23+b12+b23+b34+c23+c34+t4

A4B = 0,72 cm

A3A4 = 2,34 cm

RELATIONS AND BASIC DISTANCES FOR 6 POINTS

Basic distances: t1,t2,t3,t4,t5,t6,a12,a23,a34,b12,b23,b34,c23,c34,d34

Relations:

c12=a34,d12=b34,d23=a23,e12=c34,e23=b23,e34=a12,f12=d34,f23=c23,f34=b12.

Parameterizationof distancesbetween 6 points(convex case)

B1B4 = 12,87 cm

A3'C2' = 12,87 cm

F4A = 3,76 cm

F3F4 = 2,76 cm

F2F3 = 3,13 cm

F1F2 = 3,79 cm

FF1 = 3,10 cm

E4F = 3,10 cm

E3E4 = 5,88 cm

E2E3 = 4,25 cm

E1E2 = 2,60 cm

EE1 = 1,33 cm D4E = 1,33 cm

D3D4 = 3,79 cm

D2D3 = 5,59 cm

D1D2 = 5,86 cm

DD1 = 2,83 cm

C4D = 2,83 cm

C3C4 = 2,60 cm

C2C3 = 3,13 cm

C1C2 = 2,28 cm

CC1 = 2,39 cm

B4C = 2,39 cm

B3B4 = 5,86 cm

B2B3 = 4,25 cm

B1B2 = 2,76 cm

BB1 = 0,64 cmAA1 = 3,76 cm

A3A4 = 2,28 cm

A2A3 = 5,59 cmA1A2 = 5,88 cm

E3

F2A3'

A3

F3

A2

D2E2

B4

C1

B3

C4

D1

D3

C2'

C2

A4B1

F1

E4

E1 D4

C3

B2

F4

A1

F

A B

C

D

E

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... )

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.

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.

Thank you very much for your attention.