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

TL;DR

This paper presents a novel proof confirming that four-point configurations in three-dimensional space always produce linearly independent polynomials, resolving a longstanding conjecture by Atiyah using advanced mathematical techniques.

Contribution

The paper introduces a new proof of Atiyah's conjecture for four points, employing 2-spinor calculus and hermitian matrix theory, providing a different approach from previous proofs.

Findings

The Gram matrix of the four associated polynomials is always positive definite.

The proof confirms the linear independence of the polynomials for four-point configurations.

The approach utilizes 2-spinor calculus and positive semidefinite matrix theory.

Abstract

In Surveys in Differential Geometry, Volume 7, published in 2002 and Philosophical Transactions of the Royal Society A, Volume 359, published in 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 $\mathbb{R}^3$ 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 Geometry & Topology (2), in 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 $4$ polynomials associated to a configuration of $4$ points in Euclidean $3$-space is always positive definite. It makes use of $2$-spinor calculus and the theory of hermitian positive semidefinite matrices.