In the article entitled Atiyah, “The geometry of classical particles,” Surv. Differ. Geom. 7, 1–15 (2000) and in the article entitled Atiyah, “Configurations of points,” Philos. Trans. R. Soc., A 359, 1375–1387 (2001), Sir Michael Atiyah introduced what is known as the Atiyah problem on configurations of points, which can be briefly described as the conjecture that the n polynomials (each defined up to a phase factor) associated geometrically to a configuration of n distinct points in are always linearly independent. The first “hard” case is for n = 4 points, for which the linear independence conjecture was proved by Eastwood and Norbury- in the article entitled “A proof of Atiyah’s conjecture on configurations of four points in Euclidean three-space,” Geom. Topol. 5(2), 885–893 (2001). We present a new proof of Atiyah’s linear independence conjecture on configurations of four points, i.e., of Eastwood and Norbury’s theorem. Our proof consists in showing that the Gram matrix of the four polynomials associated with a configuration of four points in Euclidean 3-space is always positive definite. It makes use of 2-spinor calculus and the theory of Hermitian positive semidefinite matrices.
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May 2023
Research Article|
May 25 2023
A new proof of Atiyah’s conjecture on configurations of four points
Joseph Malkoun
Joseph Malkoun
a)
(Conceptualization, Writing – original draft, Writing – review & editing)
Independent Researcher
, Charlotte, North Carolina 28213, USA
a)Author to whom correspondence should be addressed: [email protected]
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a)Author to whom correspondence should be addressed: [email protected]
J. Math. Phys. 64, 053507 (2023)
Article history
Received:
March 26 2023
Accepted:
May 06 2023
Citation
Joseph Malkoun; A new proof of Atiyah’s conjecture on configurations of four points. J. Math. Phys. 1 May 2023; 64 (5): 053507. https://doi.org/10.1063/5.0151938
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