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A Progress on Atiyah-Sutcliffe Geometric Conjectures

A PROGRESS ON ATIYAH-SUTCLIFFE GEOMETRIC CONJECTURES Dragutin SVRTAN 1 Department of Mathematics, University of Zagreb, 10000 Zagreb, Croatia dsvrtan@math.hr In 2001 Sir M. F. Atiyah formulated a conjecture C1 and later with P. Sutcliffe two stronger conjectures C2 and C3. These conjectures, inspired by physics (spin-statistics theorem of quantum mechanics), are geometrically defined for any configuration of points in the Euclidean three space. The conjecture C1 is proved for n = 3, 4 and for general n only for some special configurations (M. F. Atiyah, M. Eastwood and P. Norbury, D.Đoković). Interestingly the conjecture C2 (and also stronger C3) is not yet proven even for arbitrary four points in a plane. So far we have verified the conjectures C2 and C3 for parallelograms, cyclic quadrilaterals and some infinite families of tetrahedra. We have also proposed a strengthening of the conjecture C3 for configurations of four points (Four Points Conjectures). For almost collinear configurations (with all but one point on a line) we propose several new conjectures (some for symmetric functions) which imply C2 and C3. By using computations with multi-Schur functions we can do verifications up to n=9 of our conjectures. We can also verify stronger conjecture of Đoković which imply C2 for his nonplanar configurations with dihedral symmetry. This was done jointly with I.Urbiha. Finally we mention that by minimizing a geometrically defined energy, figuring in these conjectures, one gets a connection to some complicated physical theories, such as Skyrmions and Fullerenes. The speaker found recently two different methods, by which a general case of the Four Points Conjectures in three space and a planar hyperbolic version (of C2) for five points is verified. References: 1. M.Atiyah, P.Sutcliffe: The Geometry of Point Particles. arXiv: hep-th/0105179. (32 pages) 2. M.Atiyah, P.Sutcliffe: Polyhedra in physics, chemistry and geometry., arXiv: math- ph/03030701 (22 pages). 3. D.Svrtan, I.Urbiha: Atiyah-Sutcliffe Conjectures for almost Collinear Configurations and Some new Conjectures for Symmetric Functions, arXiv: math/0406386 (23 pages). 4. D.Svrtan, I.Urbiha: Verification and Strengthening of the Atiyah-Sutcliffe Conjectures for several types of Configurations, arXiv: math/0609174 (49 pages).

configurations of points

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