论文标题
Atiyah对四个配置的猜想的新证明
A new proof of Atiyah's conjecture on configurations of four points
论文作者
论文摘要
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 $始终是线性独立的。第一个``硬''案例是$ n = 4 $点，伊斯特伍德和诺伯里在几何和拓扑中证明了线性独立的猜想（2），2001年。 我们提出了Atiyah在四个点（即伊斯特伍德和诺伯里定理）配置的线性独立性猜想的新证明。我们的证明包括表明，$ 4 $多项式的革兰氏矩阵与Euclidean $ 4 $点的配置相关的$ 4 $ -3 $ -space始终是正定确定的。它利用$ 2 $ spinor的微积分和Hermitian阳性半足质矩阵的理论。
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.
