MATH/CHEM/COMP 2010

INTRINSIC FORMULA FOR FIVE POINTS ATIYAH

DETERMINANT

Dragutin Svrtan

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 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

3

2

1

(31)

(13)

(12)

(21)

(23)

(32)

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 || p2+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

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 .

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))D3 = ½*(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))

– – ½½√√(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) (algebraic trigonometric !)

Hilbert’s Arithmetic of Ends

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

semiperimeter)

• D3 = 1+exp(-p)* D3 = 1+exp(-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 by by • d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c)d3(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)) • D3=1+ d3(a,b,c)/8*a*b*cD3=1+ d3(a,b,c)/8*a*b*c

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)/ = 1-exp(-p+c)*sinh(p)*sh(p-a)* sh(p-b)/ ∏∏ sinh(a) sinh(a)• D3_110= = 1-exp(p-c)*sinh(p)*sh(p-a)* sh(p-b)/ 1-exp(p-c)*sinh(p)*sh(p-a)* sh(p-b)/ ∏∏ sinh(a) sinh(a)• D3_111= = 1+exp(p)*1+exp(p)*∏∏ sinh(p-a)/sinh(a) sinh(p-a)/sinh(a)• D3_111 = ½*(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3_111 = ½*(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))

+ ½*+ ½*√√(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)

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 (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

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)

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

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).

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.

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

POSITIVE PARAMETRIZATION OF DISTANCESBETWEEN 4 POINTS :

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

• By using our positive parametrization we obtain a proof of the strongest Atiyah- Sutcliffe conjecture C3for arbitrary 4 points in 3-dimensional Euclidean space.It is remarkable that the “huge” 4500-term polynomial (inr12,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.

Atiyah – Sutcliffe 4 point determinant

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

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

POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS

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 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.

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

Thank you very much for your attention.