Transcript A Progress on Atiyah-Sutcliffe geometric Conjectures

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
• 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
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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
(32)
(12)
(31)
2
1
(21)
3
(13)
(23)
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
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
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det(M3)
• D3:= -------------------------------------•
( v12-v21)*(v13-v31)*(v23-v32)
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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
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det(M3)
D3:= -------------------------------------- (original Atiyah’s definition)
( v12-v21)*(v13-v31)*(v23-v32)
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Extrinsic formula:
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(v21 – v31) (v13 – v23) (v12 -v32)
• D3= 1 + ---------------------------------------------•
(v12 - v21) (v13 - v31) (v23 - v32)
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INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< π):
--------------------------------------------------------------------------------------------------------------
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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 !)
Hilbert’s Arithmetic of Ends
INTRINSIC FORMULA for D3
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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
SEVEN NEW ATIYAH-TYPE
TRIANGLE’S ENERGIES
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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!
Equations for Atiyah 3pt energies
4 POINTS INSIDE CIRCLE
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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
( 1 -v31-v32-v34
( 1 -v41-v42-v43
– v21*v23*v24)
– v31*v32*v34)
– v41*v42*v43)
v21*v23 +v21*v24+v23*v24
v31*v32 +v31*v34+v32*v34
v41*v42 +v41*v43+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
4
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)+...
c'
a'
b'
1
c
D4 / 64abca’b’c’ = D4
(=>eucl. Conjecture 1, and
“almost”(=60/64 ) of euclidean
Conjecture2
3
b
a
2
a'((b'+c')^2-a^2)*d3(a,b,c)
New proof of the EastwoodNorbury formula
Geometric interpretation of the "nonplanar"
part in Eastwood-Norbury formula
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.
POSITIVE PARAMETRIZATION OF DISTANCES
BETWEEN 4 POINTS :
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
a12: = (r13+r24-r14-r23)/2
b12: = (r13+r24 -r12-r34 )/2
r12 = t1 + a12 + t2
r13 = t1 + b12 + a12 + t3
r23 = t2 + b12 + t3
r14 = t1 + b12 + t4
r24 = t2 + a12 + b12 + t4
r34 = t3 + a12 + t4
C
Parameterization of
distances between 4
points
A1A2 = 1,92 cm B2'C2' = 1,92 cm
t3 +a12
B1B2 = 0,34 cm A1'C2" = 0,34 cm
t3
t3
C1C2 = 1,58 cm A1''B2" = 1,58 cm
C1
c12
C2
B2'
C2"
AC = 21,61 cm
BD = 11,23 cm
t4
D
t4 +b12
c12
A1'b12 A1''
B2"
t4
t1 +b12
B2
b12
B1
a12
C2'
AB = 21,54 cm
CD = 10,63 cm
AD = 12,71 cm
BC = 16,30 cm
t2
t1
t2 +a12
A
t1
A1 a12
A2
t2
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 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.
Atiyah – Sutcliffe 4 point determinant
Verification of four point conjecture of Svrtan – Urbiha (implying Atiyah – Sutcliffe C3)
POSITIVE PARAMETRIZATIONS FOR
DISTANCES BETWEEN 6 POINTS
A3A4 = 2,34 cm
r12=AB=t1+a12+a23+a34 +t2
A4B = 0,72 cm
r13=AC=AA4+B1C
=t1+a12+a23+a34+ b12+b23+b34+t3
r23=t2+b12+b23+b34+t3
r14=AD=AA3'+A3'C2'+C2'D=AA3+B1B4+C2D
=t1+a12+a23+b12+b23+b34+c23+c34+t4
r24=BD=BB4+C1D
=t2+b12+b23+b34+c12+c23+c34+t4
r34=t3+c12+c23+c34+t4 etc.
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
RELATIONS AND BASIC
DISTANCES FOR 6 POINTS
Relations:
c12=a34,
d12=b34,
d23=a23,
e12=c34,
e23=b23,
e34=a12,
f12=d34,
f23=c23,
f34=b12.
Basic distances: t1,t2,t3,t4,t5,t6,a12,a23,a34, b12,b23,b34,c23,c34,d34
Parameterization
of distances
between 6 points
(convex case)
EE1 = 1,33 cm
E1
E
D4E = 1,33 cm
D4
D3D4 = 3,79 cm
D3
E1E2 = 2,60 cm
D2D3 = 5,59 cm
E2
D2
E2E3 = 4,25 cm
D1D2 = 5,86 cm
D1
E3
DD1 = 2,83 cm
D
E3E4 = 5,88 cm
C4D = 2,83 cm
E4
C4
E4F = 3,10 cm
F
C3C4 = 2,60 cm
C2'
C3
FF1 = 3,10 cm
C2C3 = 3,13 cm
F1
C2
F1F2 = 3,79 cm
C1C2 = 2,28 cm
A3'C2' = 12,87 cm
C1
F2
A3'
CC1 = 2,39 cm
F2F3 = 3,13 cm
F3
B4
F3F4 = 2,76 cm
B1B4 = 12,87 cm
B3B4 = 5,86 cm
F4
B3
B2B3 = 4,25 cm
F4A = 3,76 cm
B2
B1B2 = 2,76 cm
A
AA1 = 3,76 cm
A1
A1A2 = 5,88 cm
A2
A2A3 = 5,59 cm
A3
A4
A3A4 = 2,28 cm
B1
B
BB1 = 0,64 cm
C
B4C = 2,39 cm
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
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[1] Atiyah M, Sutcliffe P, The Geometry of Point Particles. arXiv: hepth/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.