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(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. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree 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 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(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 <PP|"}V "  , OHY$KNORMALIZED DETERMINANT D3_R(Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah s geometric energy det(M3) D3:= -------------------------------------- ( v12-v21)*(v13-v31)*(v23-v32) D3=1 for collinear points THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. (TH.2 => TH.1) R N LIMIT Points on  circle at N are directions in plane TH.1 and TH.2 are also true for R =N . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ det(M3) D3:= -------------------------------------- (original Atiyah s definition) ( v12-v21)*(v13-v31)*(v23-v32) Extrinsic formula: (v21  v31) (v13  v23) (v12 -v32) D3= 1 + ---------------------------------------------- (v12 - v21) (v13 - v31) (v23 - v32) INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@OINTRINSIC FORMULA for D3fxINTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) D3 = 1+exp(-p)* " sinh(p-a)/sinh(a) (=> TH2 Intrinsic proof) EUCLIDEAN CASE: If we define 3-point function by d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) then D3= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det(1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24 ) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([@%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi verified Atiyah s conjecture (Conjecture 1) for almost collinear configurations and configurations with dihedral symmetry. 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(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. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree 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 points8PPP- [    !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~y      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~              PowerPoint Document(DE' DocumentSummaryInformation8 3 !"#$%&'()*+,-./0124567B!PyB>(ʪ%IRlӶm/wԥ"`W%*oB>^vyW.MӶmn6I<Ͽ:qLM!w۶g$I; eǓ$AՃQ/AsJ???I| !(:ϗt:=gc$Ibv_<6iZE۶!rܶiD~ q=]y[UU4qH4?n[B$70⺮V < a˾JZ]םNmr]뺧1eYضG4&#@Ea:equoQ z@O9ȇ_ L.{UUGE&x:c. 7^ߦinbXދyy?~Je8RUU|k#^׷nwk0?Bo [yCmeٷip}Nݲ,'?Ȭ&am3Q Ie_ܦx[~}t@&Bb[-moy+u[ix<4Vr@#AXWv𺮿Y ;+ !d&ߴBH]!+&}eY~k>VeYts|Ј!ߠ}6E2?WW$맚u߯𧔷eY~/um 5Bok!/"2!mB k(ʲiֿ%l}u0o=2u},/L4-]  $_ڶu]7cs;h@H!^E.HDf˦i8= .E^,˺QGkPmIfsj^`}եm{^Ӻ}?ފ˲Do]rA"[m[4_>0aO\!XT@Ѷ@Qu҆"RcEQYM,G&"}ߗ1!2wG$rT<*4tE.M^=sÞ[Ř+.p{[G1/RmWd袃N( 8NI_bi|VM 7Mc۶G0G7ڶ-Ww۶e^ W}&?BP-#-s&p/Jy<`? 9؎ fY\^U՘ۦid'M"`6M @/vYZ"us`[ÒkDY5 jO}Ua[ՎO}Z\ 0&x۽[y&wb(WsZJ#TdJ[ &)&G8sy(ԯWȃWd ك c`y>z3ߵ<ϵŒU @EUUm C~<)LynW!դ*5f 4mZʪzUOmڤ*[>^PI!g]xڠr|po<\!20H{iW=vCoCu#mWMZfY ,|y.FBP9X[T0KsLMs7+h xVLa;N;ᮯop;~WML8wU?>^|O ; 9 zH߽Ps$a|Mm[m0kxUy6Ŷʻ__y6Ǯ>6iP9A2qn'{#dzaV>27=Ax^r F4KKϘS |դj?`oOa1aM&JKI#NtHZTe4@Jy/*h{_tbzGN;Że٘6'`Kߺ5W<5z|}]QC۫`saxwYeq^}bDy?=31]y0 Ѓqmh{ &0)"t:5qji#.S}%#nU]2h^?M:A:y>wXsBQ+cz4Mb&. k% j&Mv!wmZS咸,NR4vx@ uWsk_)Ld̽!. 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(leading to a connection between classical and quantum physics) ATIYAH s candidate map (2001) (via elementary construction, but not yet justified even for small n ) gave a renaissance to classical (euclidean and noneuclidean ) geometries and interconnects them with many other areas of modern mathematics. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree points 1,2,3 inside circle (|z|=R) 3 point-pairs on circle p1 (12) (13) p2 (21) (23) p3 (31) (32) point-pair u,v define quadratic with these roots (z-u)*(z-v) 3 point-pairs ---> 3 quadratics p1, p2, p3 ---> p1, p2, p3 THEOREM 1 (Atiyah 2001). For any triple 1,2,3 of distinct points inside circle the 3 quadratics are linearly independent Remark: Atiyah gave a synthetic proof which unfortunately does not generalize to more than 3 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(31)=(32)=(21) |---x----x------x--------| (12)=(13)=(23) u 1 2 3 v(`"u) p1 (z-u)^2 p2 (z-u)*(z-v) p3 (z-v)^2 clearly linearly independent || l*p2+ m*p3 always has v as root || but p1 has u,u as roots and u `" v THEOREM1 3-by-3 determinant of coefficient matrix 1  v12-v13 v12*v13 det(M3) = det( 1 -v21-v23 v21*v23 ) is nonzero 1 -v31-v32 v31*v32 <PP|"}V "  , OHY$KNORMALIZED 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 . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ 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< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@OINTRINSIC FORMULA for D3fxINTRINSIC 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= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=o-8Hilbert s Arithmetic of Ends QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det(1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24 ) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([@%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt 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.20D@'Thank you very much for your attention.((('/A&D'F(H)J*L+N,P-R.T/V0X1Z2a3b4c5d6e7f8h9j:l;n<p= = 0L0 vn  (   r   S (V * `/  V  T   C ,Abroj4orig~H   0޽h ? 3380___PPT10. - 0 (  X  C  R   V   S FV   0  V   H  0޽h&ۗ ? 3380___PPT10.r7o -k72p1ai(y / 00DArialThree0TTHG17 Slide 18 Slide 19MGeometric interpretation of the "nonplanar" part in Eastwood-Norbury formula Slide 21Remarks on Eastwood-Norbury Slide 23 Slide 24cEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS Slide 26 Slide 279POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS+RELATIONS _!' DragutinDragutin AND BASIC DISTANCES FOR 6 POINTS Slide 30*ĐOKOVIĆ’S՜.+,D՜.+,@    On-screen Show E' "" 'Arial WingdingsSymbol ESSTIXThreeDefault DesignMATH/CHEM/COMP 2010rEuclidean and Hyperbolic Geometry of point particles: A progress on the tantalizing Atiyah-Sutcliffe conjectures 3 POINTS INSIDE CIRCLE#SPECIAL CASE OF 3 COLLINEAR POINTSNORMALIZED DETERMINANT D3_REXPLICIT FORMULAS FOR D3Hilbert’s Arithmetic of EndsINTRINSIC FORMULA for D3,SEVEN NEW ATIYAH-TYPE TRIANGLE’S ENERGIES"Equations for Atiyah 3pt energies Slide 11 Slide 124 POINTS INSIDE CIRCLENORMALIZED DETERMINANT D4=D4_R+Eastwood-Norbury formulas for euclidean D4*New proof of the Eastwood-Norbury formula Slide $ [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(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 <PP|"}V "  , OHY$KNORMALIZED DETERMINANT D3_R(Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah s geometric energy det(M3) D3:= -------------------------------------- ( v12-v21)*(v13-v31)*(v23-v32) D3=1 for collinear points THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. (TH.2 => TH.1) R N LIMIT Points on  circle at N are directions in plane TH.1 and TH.2 are also true for R =N . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ det(M3) D3:= -------------------------------------- (original Atiyah s definition) ( v12-v21)*(v13-v31)*(v23-v32) Extrinsic formula: (v21  v31) (v13  v23) (v12 -v32) D3= 1 + ---------------------------------------------- (v12 - v21) (v13 - v31) (v23 - v32) INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) D3 = 1+exp(-p)* " sinh(p-a)/sinh(a) (=> TH2 Intrinsic proof) EUCLIDEAN CASE: If we define 3-point function by d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) then D3= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det(1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24 ) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([@%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt 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.20D@'Thank you very much for your attention.((('/A&D'F(H)J*L+N,P-R.T/V0X1Z2a3b4c5d6e7f8h9j:l;n<p=rr-Np1ai(y / 00DArialThree0TTHGܖ 0ܖ"DWingdingse0TTHGܖ 0ܖ DSymbolgse0TTHGܖ 0ܖ0DESSTIXThree0TTHGܖ 0ܖ@ .  @n?" dd@  @@`` 2*02HH HH,,(C  b$ߛ&Liw9U%b$48&N8oU%b$o'h֫jIM/r 5b$;Kd_67|Ȱ֙$$$$$$$$$$$$$"$G H f`Oz8P2$}٥Q R2$N9vVոX2$qk;b4P\&Z2$iRJX@]$2$8P>Rwͪ\ 0e2$гd17v2$̛rꑧ3gKH{2$#s6 ;j*WE | 0e0e     A@ A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab0 f@Bwאʚ;ij8ʚ;g4{d{dS 0<ppp@ <4dddd w 0T$G<4!d!d w 0T$G 80___PPT10 33?  %Og)qEuclidean and Hyperbolic Geometry of point particles: A progress on the tantalizing Atiyah-Sutcliffe conjectures rr TMotivation: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. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree 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 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(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       !"%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ <PP|"}V "  , OHY$KNORMALIZED DETERMINANT D3_R(Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah s geometric energy det(M3) D3:= -------------------------------------- ( v12-v21)*(v13-v31)*(v23-v32) D3=1 for collinear points THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. (TH.2 => TH.1) R N LIMIT Points on  circle at N are directions in plane TH.1 and TH.2 are also true for R =N . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ det(M3) D3:= -------------------------------------- (original Atiyah s definition) ( v12-v21)*(v13-v31)*(v23-v32) Extrinsic formula: (v21  v31) (v13  v23) (v12 -v32) D3= 1 + ---------------------------------------------- (v12 - v21) (v13 - v31) (v23 - v32) INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) D3 = 1+exp(-p)* " sinh(p-a)/sinh(a) (=> TH2 Intrinsic proof) EUCLIDEAN CASE: If we define 3-point function by d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) then D3= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt turned out later that some of our generalizations are related to hyperbolic version for almost collinear configurations (with only 1 point aside a line). 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(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. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree 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 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(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 <PP|"}V "  , OHY$KNORMALIZED DETERMINANT D3_R(Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah s geometric energy det(M3) D3:= -------------------------------------- ( v12-v21)*(v13-v31)*(v23-v32) D3=1 for collinear points THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. (TH.2 => TH.1) R N LIMIT Points on  circle at N are directions in plane TH.1 and TH.2 are also true for R =N . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ det(M3) D3:= -------------------------------------- (original Atiyah s definition) ( v12-v21)*(v13-v31)*(v23-v32) Extrinsic formula: (v21  v31) (v13  v23) (v12 -v32) D3= 1 + ---------------------------------------------- (v12 - v21) (v13 - v31) (v23 - v32) INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) D3 = 1+exp(-p)* " sinh(p-a)/sinh(a) (=> TH2 Intrinsic proof) EUCLIDEAN CASE: If we define 3-point function by d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) then D3= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {q.)New proof of the Eastwood-Norbury formula**((  Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt turned out later that some of our generalizations are related to hyperbolic version for almost collinear configurations (with only 1 point aside a line). 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(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. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree 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 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(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 <PP|"}V "  , OHY$KNORMALIZED DETERMINANT D3_R(Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah s geometric energy det(M3) D3:= -------------------------------------- ( v12-v21)*(v13-v31)*(v23-v32) D3=1 for collinear points THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. (TH.2 => TH.1) R N LIMIT Points on  circle at N are directions in plane TH.1 and TH.2 are also true for R =N . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ det(M3) D3:= -------------------------------------- (original Atiyah s definition) ( v12-v21)*(v13-v31)*(v23-v32) Extrinsic formula: (v21  v31) (v13  v23) (v12 -v32) D3= 1 + ---------------------------------------------- (v12 - v21) (v13 - v31) (v23 - v32) INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) D3 = 1+exp(-p)* " sinh(p-a)/sinh(a) (=> TH2 Intrinsic proof) EUCLIDEAN CASE: If we define 3-point function by d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) then D3= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {q.)New proof of the Eastwood-Norbury formula**((  s/  Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt turned out later that some of our generalizations are related to hyperbolic version for almost collinear configurations (with only 1 point aside a line). 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"&h_OjŌ+g0D,'&j_\mYs 筺ݱtkbes{K'E$p:9sʐSMU5i|nJګF5ÝϽM)g;GrR+Ô-{vdx*ni\}}74>\oLL}փ=`F,n0k"5&NVh\4SՏ2_J5:s7wV?J R[=J*Yͬ%hl=#N|^Y˞>NF#u4βYoesK bV^gȗ2`w=n"Rwͪ\ 0e2$гd17v2$̛rꑧ3gKH{2$#s6 ;j*WE | 0e0e     A@ A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab0 f@Bwאʚ;ij8ʚ;g4{d{dS 0pppp@ <4dddd w 0T$G<4!d!d w 0T$G 80___PPT10 33?  %Pg)qEuclidean and Hyperbolic Geometry of point particles: A progress on the tantalizing Atiyah-Sutcliffe conjectures rr TMotivation: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. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree 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 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(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 <PP|"}V "  , OHY$KNORMALIZED DETERMINANT D3_R(Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah s geometric energy det(M3) D3:= -------------------------------------- ( v12-v21)*(v13-v31)*(v23-v32) D3=1 for collinear points THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. (TH.2 => TH.1) R N LIMIT Points on  circle at N are directions in plane TH.1 and TH.2 are also true for R =N . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ det(M3) D3:= -------------------------------------- (original Atiyah s definition) ( v12-v21)*(v13-v31)*(v23-v32) Extrinsic formula: (v21  v31) (v13  v23) (v12 -v32) D3= 1 + ---------------------------------------------- (v12 - v21) (v13 - v31) (v23 - v32) INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) D3 = 1+exp(-p)* " sinh(p-a)/sinh(a) (=> TH2 Intrinsic proof) EUCLIDEAN CASE: If we define 3-point function by d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) then D3= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {q.)New proof of the Eastwood-Norbury formula**((  s/  u0  Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt 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.20D@'Thank you very much for your attention.((('/A&D'F(H)J*L+N,P-R.T/V0X1Z2a3b4c5d6e7f8h9j:l;n<p=r>t?v@ @ 0L0 P$(  $T $ C ,Abroj7origSH $ 0޽h ? 3380___PPT10.i 0 0 `((  (X ( C  R   V  ( S V   0  V   H ( 0޽h&ۗ ? 3380___PPT10.@rXu _ 0Xv1|l(y / 00DArialThree0TTHGܖ 0ܖ"DWingdingse0TTHGܖ 0ܖ DSymbolgse0TTHGܖ 0ܖ0DESSTIXThree0TTHGܖ 0ܖ@ RESULTS AND GENERALIZATIONSRemark References(Thank you very much for your attention.  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 %Qg)qEuclidean and Hyperbolic Geometry of point particles: A progress on the tantalizing Atiyah-Sutcliffe conjectures rr TMotivation: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. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree 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 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(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 <PP|"}V "  , OHY$KNORMALIZED DETERMINANT D3_R(Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah s geometric energy det(M3) D3:= -------------------------------------- ( v12-v21)*(v13-v31)*(v23-v32) D3=1 for collinear points THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. (TH.2 => TH.1) R N LIMIT Points on  circle at N are directions in plane TH.1 and TH.2 are also true for R =N . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ det(M3) D3:= -------------------------------------- (original Atiyah s definition) ( v12-v21)*(v13-v31)*(v23-v32) Extrinsic formula: (v21  v31) (v13  v23) (v12 -v32) D3= 1 + ---------------------------------------------- (v12 - v21) (v13 - v31) (v23 - v32) INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) D3 = 1+exp(-p)* " sinh(p-a)/sinh(a) (=> TH2 Intrinsic proof) EUCLIDEAN CASE: If we define 3-point function by d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) then D3= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {q.)New proof of the Eastwood-Norbury formula**((  s/  u0  w1LGeometric interpretation of the "nonplanar" part in Eastwood-Norbury formulaMM :!   Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt turned out later that some of our generalizations are related to hyperbolic version for almost collinear configurations (with only 1 point aside a line). 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(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. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree 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 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(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 <PP|"}V "  , OHY$KNORMALIZED DETERMINANT D3_R(Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah s geometric energy det(M3) D3:= -------------------------------------- ( v12-v21)*(v13-v31)*(v23-v32) D3=1 for collinear points THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. (TH.2 => TH.1) R N LIMIT Points on  circle at N are directions in plane TH.1 and TH.2 are also true for R =N . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ det(M3) D3:= -------------------------------------- (original Atiyah s definition) ( v12-v21)*(v13-v31)*(v23-v32) Extrinsic formula: (v21  v31) (v13  v23) (v12 -v32) D3= 1 + ---------------------------------------------- (v12 - v21) (v13 - v31) (v23 - v32) INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) D3 = 1+exp(-p)* " sinh(p-a)/sinh(a) (=> TH2 Intrinsic proof) EUCLIDEAN CASE: If we define 3-point function by d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) then D3= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {q.)New proof of the Eastwood-Norbury formula**((  s/  u0  w1LGeometric interpretation of the "nonplanar" part in Eastwood-Norbury formulaMM :!   y2  Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt 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.20D@'Thank you very much for your attention.((('/,A&D'F(H)J*L+N,P-R.T/V0X1Z2a3b4c5d6e7f8h9j:l;n<p=r>t?v@xAzB B 0L0 4(  4T 4 C ,A broj9origb< H 4 0޽h ? 3380___PPT10.Ū 2 0 8(  8X 8 C  R   V  8 S T   0  V   H 8 0޽h&ۗ ? 3380___PPT10.p6Ȫr]9y e299%z1l(y / 00DArialThree0TTHGܖ 0ܖ"DWingdingse0TTHGܖ 0ܖ DSymbolgse0TTHGܖ 0ܖ0DESSTIXThree0TTHGܖ 0ܖ@ .  @n?" dd@  @@`` z8<HH HH,,(C  b$ߛ&Liw9U%b$48&N8oU%b$o'h֫jIM/r 5b$;Kd_67|Ȱ֙b$]Zb<#vqn'$b$7nTʺpbΰb$ z/fwRb$4"(r52Tu o`b$pThYK Di($$$$$$$"$G H f`Oz8P2$}٥Q R2$N9vVոX2$qk;b4P\&Z2$iRJX@]$2$8P>Rwͪ\ 0e2$гd17v2$̛rꑧ3gKH{2$#s6 ;j*WE | 0e0e     A@ A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab0 f@Bwאʚ;ij8ʚ;g4{d{dS 0pppp@ <4dddd w 0T$G<4!d!d w 0T$G 80___PPT10 33?  %ARg)qEuclidean and Hyperbolic Geometry of point particles: A progress on the tantalizing Atiyah-Sutcliffe conjectures rr TMotivation: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. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree 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 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(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 <PP|"}V "  , OHY$KNORMALIZED DETERMINANT D3_R(Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah s geometric energy det(M3) D3:= -------------------------------------- ( v12-v21)*(v13-v31)*(v23-v32) D3=1 for collinear points THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. (TH.2 => TH.1) R N LIMIT Points on  circle at N are directions in plane TH.1 and TH.2 are also true for R =N . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ det(M3) D3:= -------------------------------------- (original Atiyah s definition) ( v12-v21)*(v13-v31)*(v23-v32) Extrinsic formula: (v21  v31) (v13  v23) (v12 -v32) D3= 1 + ---------------------------------------------- (v12 - v21) (v13 - v31) (v23 - v32) INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) D3 = 1+exp(-p)* " sinh(p-a)/sinh(a) (=> TH2 Intrinsic proof) EUCLIDEAN CASE: If we define 3-point function by d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) then D3= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {q.)New proof of the Eastwood-Norbury formula**((  s/  u0  w1LGeometric interpretation of the "nonplanar" part in Eastwood-Norbury formulaMM :!   y2  Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt turned out later that some of our generalizations are related to hyperbolic version for almost collinear configurations (with only 1 point aside a line). 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(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. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree 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 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(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 <PP|"}V "  , OHY$KNORMALIZED DETERMINANT D3_R(Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah s geometric energy det(M3) D3:= -------------------------------------- ( v12-v21)*(v13-v31)*(v23-v32) D3=1 for collinear points THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. (TH.2 => TH.1) R N LIMIT Points on  circle at N are directions in plane TH.1 and TH.2 are also true for R =N . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ det(M3) D3:= -------------------------------------- (original Atiyah s definition) ( v12-v21)*(v13-v31)*(v23-v32) Extrinsic formula: (v21  v31) (v13  v23) (v12 -v32) D3= 1 + ---------------------------------------------- (v12 - v21) (v13 - v31) (v23 - v32) INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) D3 = 1+exp(-p)* " sinh(p-a)/sinh(a) (=> TH2 Intrinsic proof) EUCLIDEAN CASE: If we define 3-point function by d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) then D3= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {q.)New proof of the Eastwood-Norbury formula**((  s/  u0  w1LGeometric interpretation of the "nonplanar" part in Eastwood-Norbury formulaMM :!   y2  Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff{3  ^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt turned out later that some of our generalizations are related to hyperbolic version for almost collinear configurations (with only 1 point aside a line). 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 %Rg)qEuclidean and Hyperbolic Geometry of point particles: A progress on the tantalizing Atiyah-Sutcliffe conjectures rr TMotivation: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. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree 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 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(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 <PP|"}V "  , OHY$KNORMALIZED DETERMINANT D3_R(Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah s geometric energy det(M3) D3:= -------------------------------------- ( v12-v21)*(v13-v31)*(v23-v32) D3=1 for collinear points THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. (TH.2 => TH.1) R N LIMIT Points on  circle at N are directions in plane TH.1 and TH.2 are also true for R =N . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ det(M3) D3:= -------------------------------------- (original Atiyah s definition) ( v12-v21)*(v13-v31)*(v23-v32) Extrinsic formula: (v21  v31) (v13  v23) (v12 -v32) D3= 1 + ---------------------------------------------- (v12 - v21) (v13 - v31) (v23 - v32) INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) D3 = 1+exp(-p)* " sinh(p-a)/sinh(a) (=> TH2 Intrinsic proof) EUCLIDEAN CASE: If we define 3-point function by d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) then D3= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {q.)New proof of the Eastwood-Norbury formula**((  s/  u0  w1LGeometric interpretation of the "nonplanar" part in Eastwood-Norbury formulaMM :!   y2  Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff{3  }4  ^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt turned out later that some of our generalizations are related to hyperbolic version for almost collinear configurations (with only 1 point aside a line). 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(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. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree 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 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(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 <PP|"}V "  , OHY$KNORMALIZED DETERMINANT D3_R(Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah s geometric energy det(M3) D3:= -------------------------------------- ( v12-v21)*(v13-v31)*(v23-v32) D3=1 for collinear points THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. (TH.2 => TH.1) R N LIMIT Points on  circle at N are directions in plane TH.1 and TH.2 are also true for R =N . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ det(M3) D3:= -------------------------------------- (original Atiyah s definition) ( v12-v21)*(v13-v31)*(v23-v32) Extrinsic formula: (v21  v31) (v13  v23) (v12 -v32) D3= 1 + ---------------------------------------------- (v12 - v21) (v13 - v31) (v23 - v32) INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) D3 = 1+exp(-p)* " sinh(p-a)/sinh(a) (=> TH2 Intrinsic proof) EUCLIDEAN CASE: If we define 3-point function by d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) then D3= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {q.)New proof of the Eastwood-Norbury formula**((  s/  u0  w1LGeometric interpretation of the "nonplanar" part in Eastwood-Norbury formulaMM :!   y2  Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff{3  }4  ^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt 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.20D@'Thank you very much for your attention.(((' /dA&D'F(H)J*L+N,P-R.T/V0X1Z2a3b4c5d6e7f8h9j:l;n<p=r>t?v@xAzB|C~D C 0L0 ZR<(  <T < C ,A brojAorig"V < HPXT  ?"6@ NNN?Nk LAtiyah  Sutcliffe 4 point determinant'(2'&H < 0޽h ? 3380___PPT10.#r{m3&p~1n(y / 00DArialThree0TTHGܖ 0ܖ"DWingdingse0TTHGܖ 0ܖ DSymbolgse0TTHGܖ 0ܖ0DESSTIXThree0TTHGܖ 0ܖ@ .  @n?" dd@  @@`` H@HH HH,,(C  b$ߛ&Liw9U%b$48&N8oU%b$o'h֫jIM/r 5b$;Kd_67|Ȱ֙b$]Zb<#vqn'$b$7nTʺpbΰb$ z/fwRb$4"(r52Tu o`b$pThYK Di($b$~˸{qXB?zb$G -ARwͪ\ 0e2$гd17v2$̛rꑧ3gKH{2$#s6 ;j*WE | 0e0e     A@ A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab@f@Bwאʚ;ij8ʚ;g4{d{dS 0<ppp@ <4dddd w 0T$G<4!d!d w 0T$G 80___PPT10 33?  %Rg)qEuclidean and Hyperbolic Geometry of point particles: A progress on the tantalizing Atiyah-Sutcliffe conjectures rr TMotivation: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. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree 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 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(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 <PP|"}V "  , OHY$KNORMALIZED DETERMINANT D3_R(Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah s geometric energy det(M3) D3:= -------------------------------------- ( v12-v21)*(v13-v31)*(v23-v32) D3=1 for collinear points THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. (TH.2 => TH.1) R N LIMIT Points on  circle at N are directions in plane TH.1 and TH.2 are also true for R =N . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ det(M3) D3:= -------------------------------------- (original Atiyah s definition) ( v12-v21)*(v13-v31)*(v23-v32) Extrinsic formula: (v21  v31) (v13  v23) (v12 -v32) D3= 1 + ---------------------------------------------- (v12 - v21) (v13 - v31) (v23 - v32) INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) D3 = 1+exp(-p)* " sinh(p-a)/sinh(a) (=> TH2 Intrinsic proof) EUCLIDEAN CASE: If we define 3-point function by d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) then D3= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {q.)New proof of the Eastwood-Norbury formula**((  s/  u0  w1LGeometric interpretation of the "nonplanar" part in Eastwood-Norbury formulaMM :!   y2  Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff{3  }4  ^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt turned out later that some of our generalizations are related to hyperbolic version for almost collinear configurations (with only 1 point aside a line). 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(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. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree 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 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(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 <PP|"}V "  , OHY$KNORMALIZED DETERMINANT D3_R(Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah s geometric energy det(M3) D3:= -------------------------------------- ( v12-v21)*(v13-v31)*(v23-v32) D3=1 for collinear points THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. (TH.2 => TH.1) R N LIMIT Points on  circle at N are directions in plane TH.1 and TH.2 are also true for R =N . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ det(M3) D3:= -------------------------------------- (original Atiyah s definition) ( v12-v21)*(v13-v31)*(v23-v32) Extrinsic formula: (v21  v31) (v13  v23) (v12 -v32) D3= 1 + ---------------------------------------------- (v12 - v21) (v13 - v31) (v23 - v32) INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) D3 = 1+exp(-p)* " sinh(p-a)/sinh(a) (=> TH2 Intrinsic proof) EUCLIDEAN CASE: If we define 3-point function by d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) then D3= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {q.)New proof of the Eastwood-Norbury formula**((  s/  u0  5  w1LGeometric interpretation of the "nonplanar" part in Eastwood-Norbury formulaMM :!   y2  Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff{3  }4  ^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi s conjectural strengthening of Atiyah-Sutcliffe-Conjecture 2 for dihedral configurations and Atiyah-Sutcliffe Conjec      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuwxz{|}~ture 3 for 9 points on a line and one outside by extensive computer help.^Pi_ERemarkfIt 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.20D@'Thank you very much for your attention.(((' /A&D'F(H)J*L+N,P-R.T/V0X1Z2a3b4c5d6e7f8h9j:l;n<p=r>t?v@xAzB|C~DE E 0L0 XPL(  LT L C ,A brojCorigT L C ,AbrojDorig&0H L 0޽h ? 3380___PPT10.P 5 0 P(  PX P C  R   V  P S ̊T   0  V   H P 0޽h&ۗ ? 3380___PPT10. r .M6O5|JQ1n(y / 00DArialThree0TTHGܖ 0ܖ"DWingdingse0TTHGܖ 0ܖ DSymbolgse0TTHGܖ 0ܖ0DESSTIXThree0TTHGܖ 0ܖ@ .  @n?" dd@  @@`` PBHH HH,,(C  b$ߛ&Liw9U%b$48&N8oU%b$o'h֫jIM/r 5b$;Kd_67|Ȱ֙b$]Zb<#vqn'$b$7nTʺpbΰb$ z/fwRb$4"(r52Tu o`b$pThYK Di(b$+ՙ>k u{.b$~˸{qXB?zb$G -AU$$$"$G H f`Oz8P2$}٥Q R2$N9vVոX2$qk;b4P\&Z2$iRJX@]$2$8P>Rwͪ\ 0e2$гd17v2$̛rꑧ3gKH{2$#s6 ;j*WE | 0e0e     A@ A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab@f@Bwאʚ;ij8ʚ;g4{d{dS 0<ppp@ <4dddd w 0T$G<4!d!d w 0T$G 80___PPT10 33?  %ISg)qEuclidean and Hyperbolic Geometry of point particles: A progress on the tantalizing Atiyah-Sutcliffe conjectures rr TMotivation: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. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree 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 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(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 <PP|"}V "  , OHY$KNORMALIZED DETERMINANT D3_R(Atiyah defined the normalized determinant D3=D3_R (continuous on unordered triples of distinct points in open disk of radius R) ...Atiyah s geometric energy det(M3) D3:= -------------------------------------- ( v12-v21)*(v13-v31)*(v23-v32) D3=1 for collinear points THEOREM2 (ATIYAH): D3R1 AND D3=1 ONLY FOR COLLINEAR TRIPLES. (TH.2 => TH.1) R N LIMIT Points on  circle at N are directions in plane TH.1 and TH.2 are also true for R =N . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ det(M3) D3:= -------------------------------------- (original Atiyah s definition) ( v12-v21)*(v13-v31)*(v23-v32) Extrinsic formula: (v21  v31) (v13  v23) (v12 -v32) D3= 1 + ---------------------------------------------- (v12 - v21) (v13 - v31) (v23 - v32) INTRINSIC FORMULA in terms of hyperbolic angles A,B,C (0< A+B+C< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC FORMULA in terms of side lengths a,b,c (p=(a+b+c)/2 semiperimeter) D3 = 1+exp(-p)* " sinh(p-a)/sinh(a) (=> TH2 Intrinsic proof) EUCLIDEAN CASE: If we define 3-point function by d3(a,b,c):=(-a+b+c)*(a-b+c)*(a+b-c) then D3= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {q.)New proof of the Eastwood-Norbury formula**((  s/  u0  5  w1LGeometric interpretation of the "nonplanar" part in Eastwood-Norbury formulaMM :!   y2  Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff{3  }4  ^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt 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.20D@'Thank you very much for your attention.(((' /A&D'F(H)J*L+N,P-R.T/V0X1Z2a3b4c5d6e7f8h9j:l;n<p=r>t?v@xAzB|C~DErQ5fQ1p(y / 00DArialThree0TTHGܖ 0ܖ"DWingdingse0TTHGܖ 0ܖ DSymbolgse0TTHGܖ 0ܖ0DESSTIXThree0TTHGܖ 0ܖ@ .  @n?" dd@  @@`` XDHH HH,,(C  b$ߛ&Liw9U%b$48&N8oU%b$o'h֫jIM/r 5b$;Kd_67|Ȱ֙b$]Zb<#vqn'$b$7nTʺpbΰb$ z/fwRb$4"(r52Tu o`b$pThYK Di(b$+ՙ>k u{.b$~˸{qXB?zb$G -AU$$$"$G H f`Oz8P2$}٥Q R2$N9vVոX2$qk;b4P\&Z2$iRJX@]$2$8P>Rwͪ\ 0e2$гd17v2$̛rꑧ3gKH{2$#s6 ;j*WE | 0e0e     A@ A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab@f@Bwאʚ;ij8ʚ;g4{d{dS 0pppp@ <4dddd w 0T$G<4!d!d w 0T$G 80___PPT10 33?  %T6MATH/CHEM/COMP 2010H INTRINSIC FORMULA FOR FIVE POINTS ATIYAH DETERMINANT Dragutin SvrtanX85$,6g)qEuclidean and Hyperbolic Geometry of point particles: A progress on the tantalizing Atiyah-Sutcliffe conjectures rr TMotivation:BERRY-ROBBINS PROBLEM(1997) coming from spin-statistics in particle physics Cn(R^3):=configuration space of n ordered distinct points/particles in R^3 PROBLEM: Does there exists a continuous equivariant map f_n:C_n(R^3)U(n)/T^n(=space of n orthogonal complex lines) ? (leading to a connection between classical and quantum physics) ATIYAH s candidate map (2001) (via elementary construction, but not yet justified even for small n ) gave a renaissance to classical (euclidean and noneuclidean ) geometries and interconnects them with many other areas of modern mathematics. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree points 1,2,3 inside circle (|z|=R) 3 point-pairs on circle p1 (12) (13) p2 (21) (23) p3 (31) (32) point-pair u,v define quadratic with these roots (z-u)*(z-v) 3 point-pairs ---> 3 quadratics p1, p2, p3 ---> p1, p2, p3 THEOREM 1 (Atiyah 2001). For any triple 1,2,3 of distinct points inside circle the 3 quadratics are linearly independent Remark: Atiyah gave a synthetic proof which unfortunately does not generalize to more than 3 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(31)=(32)=(21) |---x----x------x--------| (12)=(13)=(23) u 1 2 3 v(`"u) p1 (z-u)^2 p2 (z-u)*(z-v) p3 (z-v)^2 clearly linearly independent || l*p2+ m*p3 always has v as root || but p1 has u,u as roots and u `" v THEOREM1 3-by-3 determinant of coefficient matrix 1  v12-v13 v12*v13 det(M3) = det( 1 -v21-v23 v21*v23 ) is nonzero 1 -v31-v32 v31*v32 <PP|"}V "  , OHY$KNORMALIZED 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 . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ 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< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC 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= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {q.)New proof of the Eastwood-Norbury formula**((  s/  u0  5  w1LGeometric interpretation of the "nonplanar" part in Eastwood-Norbury formulaMM :!   y2  Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff{3  }4  ^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt 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.20D@'Thank you very much for your attention.(((' /A&D'F(H)J*L+N,P-R.T/V0X1Z2a3b4c5d6e7f8h9j:l;n<p=r>t?v@xAzB|C~DEF< F 0L0 T$(  Tr T S sT  `}  V  r T S DT  ` V  H T 0޽h ? 3380___PPT10.`< 6 0  X(  XX X C  R   V  X S T   0  V   H X 0޽h&ۗ ? 3380___PPT10.PJCrL W026(41p(y / 00DArialThree0TTHGܖ 0ܖ"DWingdingse0TTHGܖ 0ܖ DSymbolgse@ o@vfVG|g  5(  y--$xx--'@"Arial-. $2 MATH/CHEM/COMP 2010O  ."System8-@"Arial-. 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Gl6^2tKȺ^*j> 1ugG|n`<6yOYw\z bxdh%γFIpr~U+2iZϳgvNv^KzH\ӆP2k}vqĬ-{Y̔dem带&SY'B=mt9ڛEөo[qs73{W<φ1ZD z!60#/b)d< UY}eduQV=k~vH/?gxddN顈\xS#jfIJ=MqYR`q~ zCޙeOm)m=:u_rG|]s~}fxU슀0w#p_BG GwXa' ?>瓽xW2#VWYrݬUfܻ&kAWM/o#Ho;<>_c6.ϑUMl3y=dz-~,vWWm9(z) Lm1/?zibS꥕md]jm_}RHbxDCRgn*aMW <#jOs+qݩU>ﺢ  g^/x&?{x̵ ӻp>oʕ IENDB`0TTHGܖ 0ܖ0DESSTIXThree0TTHGܖ 0ܖ@ .  @n?" dd@  @@`` XDHH HH,,(C  b$ߛ&Liw9U%b$48&N8oU%b$o'h֫jIM/r 5b$;Kd_67|Ȱ֙b$]Zb<#vqn'$$b$ z/fwRb$4"(r52Tu o`b$pThYK Di(b$+ՙ>k u{.b$~˸{qXB?zb$G -AUb$y'.=-wZi$$"$G H f`Oz8P2$}٥Q R2$N9vVոX2$qk;b4P\&Z2$iRJX@]$2$8P>Rwͪ\ 0e2$гd17v2$̛rꑧ3gKH{2$#s6 ;j*WE | 0e0e     A@ A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab@f@Bwאʚ;ij8ʚ;g4{d{dS 0pppp@ <4dddd w 0T$G<4!d!d w 0T$G 80___PPT10 33?  %T6MATH/CHEM/COMP 2010H INTRINSIC FORMULA FOR FIVE POINTS ATIYAH DETERMINANT Dragutin SvrtanX85$,6g)qEuclidean and Hyperbolic Geometry of point particles: A progress on the tantalizing Atiyah-Sutcliffe conjectures rr TMotivation:BERRY-ROBBINS PROBLEM(1997) coming from spin-statistics in particle physics Cn(R^3):=configuration space of n ordered distinct points/particles in R^3 PROBLEM: Does there exists a continuous equivariant map f_n:C_n(R^3)U(n)/T^n(=space of n orthogonal complex lines) ? (leading to a connection between classical and quantum physics) ATIYAH s candidate map (2001) (via elementary construction, but not yet justified even for small n ) gave a renaissance to classical (euclidean and noneuclidean ) geometries and interconnects them with many other areas of modern mathematics. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree points 1,2,3 inside circle (|z|=R) 3 point-pairs on circle p1 (12) (13) p2 (21) (23) p3 (31) (32) point-pair u,v define quadratic with these roots (z-u)*(z-v) 3 point-pairs ---> 3 quadratics p1, p2, p3 ---> p1, p2, p3 THEOREM 1 (Atiyah 2001). For any triple 1,2,3 of distinct points inside circle the 3 quadratics are linearly independent Remark: Atiyah gave a synthetic proof which unfortunately does not generalize to more than 3 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(31)=(32)=(21) |---x----x------x--------| (12)=(13)=(23) u 1 2 3 v(`"u) p1 (z-u)^2 p2 (z-u)*(z-v) p3 (z-v)^2 clearly linearly independent || l*p2+ m*p3 always has v as root || but p1 has u,u as roots and u `" v THEOREM1 3-by-3 determinant of coefficient matrix 1  v12-v13 v12*v13 det(M3) = det( 1 -v21-v23 v21*v23 ) is nonzero 1 -v31-v32 v31*v32 <PP|"}V "  , OHY$KNORMALIZED 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 . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ 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< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC 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= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c = D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {q.)New proof of the Eastwood-Norbury formula**((  s/  u0  5  w1LGeometric interpretation of the "nonplanar" part in Eastwood-Norbury formulaMM :!   y2  Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff{3  }4  ^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt 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.20D@'Thank you very much for your attention.(((' /A&D'F(H)J*L+N,P-R.T/V0X1Z2a3b4c5d6e7f8h9j:l;n<p=r>t?v@xAzB|C~DEF ? 0L0 0(  T  C ,Abroj6origqDAyH  0޽h ? 3380___PPT10.P5rr4s/41{(y  L 8http://www.math.missouri.edu/archive/Miller-Lectures/atiyah/atiyah.htmlhttp://www.math.missouri.edu/archive/Miller-Lectures/atiyah/atiyah.html/ 00DArialThree0TTHGܖ 0ܖ"DWingdingse0TTHGܖ 0ܖ DSymbolgse0TTHGܖ 0ܖ0DESSTIXThree0TTHGܖ 0ܖ@ .  @n?" dd@  @@`` `FHH HH,,(C  b$ߛ&Liw9U%b$48&N8oU%b$o'h֫jIM/r 5b$;Kd_67|Ȱ֙b$]Zb<#vqn'$$b$ z/fwRb$4"(r52Tu o`b$pThYK Di(b$+ՙ>k u{.b$~˸{qXB?zb$G -AUb$y'.=-wZi$$"$G H f`Oz8P2$}٥Q R2$N9vVոX2$qk;b4P\&Z2$iRJX@]$2$8P>Rwͪ\ 0e2$гd17v2$̛rꑧ3gKH{2$#s6 ;j*WE | 0e0e     A@ A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab@f@Bwאʚ;ij8ʚ;g4{d{dS 0pppp@ <4dddd w 0T$G<4!d!d w 0T$G 80___PPT10 33?  %^6MATH/CHEM/COMP 2010H INTRINSIC FORMULA FOR FIVE POINTS ATIYAH DETERMINANT Dragutin SvrtanX85$,6g)qEuclidean and Hyperbolic Geometry of point particles: A progress on the tantalizing Atiyah-Sutcliffe conjectures rr TMotivation:BERRY-ROBBINS PROBLEM(1997) coming from spin-statistics in particle physics Cn(R^3):=configuration space of n ordered distinct points/particles in R^3 PROBLEM: Does there exists a continuous equivariant map f_n:C_n(R^3)U(n)/T^n(=space of n orthogonal complex lines) ? (leading to a connection between classical and quantum physics) ATIYAH s candidate map (2001) (via elementary construction, but not yet justified even for small n ) gave a renaissance to classical (euclidean and noneuclidean ) geometries and interconnects them with many other areas of modern mathematics. n_Z MfORf2f2^G3 POINTS INSIDE CIRCLEfThree points 1,2,3 inside circle (|z|=R) 3 point-pairs on circle p1 (12) (13) p2 (21) (23) p3 (31) (32) point-pair u,v define quadratic with these roots (z-u)*(z-v) 3 point-pairs ---> 3 quadratics p1, p2, p3 ---> p1, p2, p3 THEOREM 1 (Atiyah 2001). For any triple 1,2,3 of distinct points inside circle the 3 quadratics are linearly independent Remark: Atiyah gave a synthetic proof which unfortunately does not generalize to more than 3 points8PPP- [   [fI"SPECIAL CASE OF 3 COLLINEAR POINTS##(f"H(31)=(32)=(21) |---x----x------x--------| (12)=(13)=(23) u 1 2 3 v(`"u) p1 (z-u)^2 p2 (z-u)*(z-v) p3 (z-v)^2 clearly linearly independent || l*p2+ m*p3 always has v as root || but p1 has u,u as roots and u `" v THEOREM1 3-by-3 determinant of coefficient matrix 1  v12-v13 v12*v13 det(M3) = det( 1 -v21-v23 v21*v23 ) is nonzero 1 -v31-v32 v31*v32 <PP|"}V "  , OHY$KNORMALIZED 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 . P[fvf'f'2vMEXPLICIT FORMULAS FOR D3fZ 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< ): -------------------------------------------------------------------------------------------------------------- D3 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2))  "(cos(A+B+C)/2*cos(-A+B+C)/2*cos(A-B+C)/2*cos(A+B-C)/2) (algebraic trigonometric !) RP[PP6}   &o+ " 6ff@o-8Hilbert s Arithmetic of Ends OINTRINSIC FORMULA for D3fxINTRINSIC 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= *(cos^2(A/2)+ cos^2(B/2)+ cos^2(C/2)) D3=1+ d3(a,b,c)/8*a*b*c 8=Z>$ f$f&,=QRSEVEN NEW ATIYAH-TYPE TRIANGLE S ENERGIES**(f)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_ , =100,...,111 (with D3_ =D3 for  =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 = *(cos^2(A/2)+ cos^2(B/2)+ cos^2(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_ R 1, for  = 000 , 111. (ii) 0<D3_ # 1, for  `" 000 , 111. (iii) D3_000+ D3_100+...+D3_110+...+D3_111=6=3! fyPPPdf1"&( &  &%&   $&$(( ((,&,001000000f4&4f688f88f88f 88f<&<f&f & f "  & f "  f&f &f  $&$f$"$ $&$f$"$1((f((f,&,,&,00448"821  2i*!Equations for Atiyah 3pt energies""( k+  m,  S4 POINTS INSIDE CIRCLEfFour points 1,2,3,4 inside circle (|z|=R) 4 point-triples on circle p1 (12) (13) (14) p2 (21) (23) (24) p3 (31) (32) (34) p4 (41) (42) (43) point-triple u,v,w define cubic (polynomial) with these roots (z-u)*(z-v)*(z-w) 4 point-triples ---> 4 cubics p1, p2, p3 ,p4 ---> p1, p2, p3,p4 pP  U NORMALIZED DETERMINANT D4=D4_R(T4-by-4 determinant of coefficient matrix ( 1 -v12-v13-v13 v12*v13+ v12*v14+ v13*v14  v12*v13*v14) |M4| =det( 1 -v21-v23-v23 v21*v23 +v21*v24+v23*v24  v21*v23*v24) ( 1 -v31-v32-v34 v31*v32 +v31*v34+v32*v34  v31*v32*v34) ( 1 -v41-v42-v43 v41*v42 +v41*v43+v42*v43  v41*v42*v43) Det(M4) D4:= -------------------------------------------------------------------------------- (v12-v21)*(v13-v31)*(v14-v41)*(v23-v32)*(v24-42)*(v34-v43) CONJECTURES : C1(Atiyah): D4 `"0 (<--> p1, p2, p3, p4 linearly independent) C2(Atiyah-Sutcliffe): D4 R 1 C3(Atiyah-Sutcliffe): |D4|^2 R D3(1,2,3)*D3(1,2,4)*D3(1,3,4)*D3(2,3,4) ,+P([?%f&f",ff & f3ff*&f*W!*Eastwood-Norbury formulas for euclidean D4++(f*In 2001 they proved , by tricky use of MAPLE that for n=4 points in Eucl. 3-space Re(D4) = 64abca b c - 4*d3(a.a ,b.b ,c.c ) +SUM +288*VOLUME^2, where SUM: = a [(b +c )^2-a^2)]*d3(a,b,c)+... D4 / 64abca b c =               D4 (=>eucl. Conjecture 1, and  almost (=60/64 ) of euclidean Conjecture2 t{PPf3#f ff % J {q.)New proof of the Eastwood-Norbury formula**((  s/  u0  5  w1LGeometric interpretation of the "nonplanar" part in Eastwood-Norbury formulaMM :!   y2  Y"[$  \#](bEUCLIDEAN ATIYAH-SUTCLIFFE CONJECTURES VIA POSITIVE PARAMETRIZATION OF DISTANCES FOR ANY 4 POINTS (caffbLBy 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. 3ZtZ37? D f*&fMff{3  }4  ^%8POSITIVE PARAMETRIZATIONS FOR DISTANCES BETWEEN 6 POINTS99f8 _&*RELATIONS AND BASIC DISTANCES FOR 6 POINTS++(f* `'CJOKOVI S RESULTS AND GENERALIZATIONS&&(f%zIn 2002 okovi 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 okovi 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.^Pi_ERemarkfIt 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.20D7 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-uminhoP2 %&$tm < 1@s 0:z@'Thank you very much for your attention.(((' /A&D'F(H)J*L+N,P-R.T/V0X1Z2a3b4c5d6e7f8h9j:l;n<p=r>t?v@xAzB|C~DEFG< G 0L0 0\$(  \r \ S 8c `}  V  r \ S $T ` V  H \ 0޽h ? 3380___PPT10.05 7 0 @`(  `X ` C  R   V  ` S  Z  0  V   H ` 0޽h&ۗ ? 3380___PPT10.0r " $ '' 1Root EntrydO) cPicturessCurrent User8SummaryInformation(