A new proof of Atiyah’s conjecture on configurations of four points

Abstract

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 R3 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.