{"id":53,"date":"2021-04-18T01:14:35","date_gmt":"2021-04-18T01:14:35","guid":{"rendered":"https:\/\/josephmalkoun.com\/blog\/?p=53"},"modified":"2021-04-18T01:14:35","modified_gmt":"2021-04-18T01:14:35","slug":"the-atiyah-problem-on-configurations","status":"publish","type":"post","link":"https:\/\/josephmalkoun.com\/blog\/the-atiyah-problem-on-configurations\/","title":{"rendered":"The Atiyah Problem on Configurations"},"content":{"rendered":"\n

This post is a continuation of my previous post “the spin-statistics theorem and the Berry-Robbins problem”<\/a>. The reader is referred to that post to understand the origins of the Atiyah problem on configurations and the Atiyah-Sutcliffe conjectures.<\/p>\n\n\n\n

Let $C_n(\\mathbb{R}^3)$ denote the configuration space of $n$ distinct points in $\\mathbb{R}^3$. Given $\\mathbf{x} = (\\mathbf{x}_1,\\ldots,\\mathbf{x}_n) \\in C_n(\\mathbb{R}^3)$, so that each $\\mathbf{x}_i \\in \\mathbb{R}^3$, for $i = 1, \\ldots, n$ and the $\\mathbf{x}_i$ are distinct, we will associate to the configuration $\\mathbf{x}$ $n$ complex polynomials $p_i(\\zeta)$, for $i = 1, \\ldots, n$ of degree (at most) $n-1$, each of them only defined up to multiplication by a nonzero complex factor.<\/p>\n\n\n\n

Let $i \\in \\{1,\\ldots,n\\}$. For each $j \\neq i$, $1 \\leq j \\leq n$, we form the normalized vector<\/p>\n\n\n\n

$$v_{ij} = \\frac{\\mathbf{x}_j-\\mathbf{x}_i}{\\lVert \\mathbf{x}_j-\\mathbf{x}_i \\rVert} \\in S^2.$$<\/p>\n\n\n\n

Using stereographic projection, there is a natural complex structure on $S^2$, which turns it into the Riemann sphere $\\hat{S}$ and a natural complex coordinate $\\zeta$ defined on $S^2 \\setminus \\{N\\}$, where $N = (0,0,1)^T$ is the “North” pole, so to speak.<\/p>\n\n\n\n

So $v_{ij}$ corresponds to a complex number $\\zeta_{ij}$ if $v_{ij} \\neq N$ and to the point at infinity $\\infty$ on the Riemann sphere $\\hat{S}$ if $v_{ij} = N$.<\/p>\n\n\n\n

Let $p_i(\\zeta)$ be the complex polynomial of degree (at most) $n-1$, having the $\\zeta_{ij}$, for $1 \\leq j \\leq n$ and $j \\neq i$, as its set of roots, taking multiplicity into account. Note that each polynomial $p_i(\\zeta)$ is only defined up to a nonzero complex factor.<\/p>\n\n\n\n

Atiyah conjectured that given any $\\mathbf{x} \\in C_n(\\mathbb{R}^3)$, the corresponding polynomials $p_i(\\zeta)$, for $1 \\leq i \\leq n$, are always linearly independent over $\\mathbb{C}$. This is also known as the Atiyah-Sutcliffe conjecture $1$. If it is true, then it would imply that the map<\/p>\n\n\n\n

$$\\mathbf{x} \\mapsto (p_1(\\zeta), \\ldots, p_n(\\zeta))$$<\/p>\n\n\n\n

gives, after orthogonalization, a smooth solution of the Berry-Robbins problem.<\/p>\n\n\n\n

This is a problem I have been working on for a very long time. It is still open for general $n$, though it was proved to be true by Atiyah for $n = 3$ and by Eastwood and Norbury for $n = 4$.<\/p>\n\n\n\n

There is more to the story. Indeed, Atiyah and Sutcliffe have defined a very interesting normalized determinant function $D$<\/p>\n\n\n\n

$$D: C_n(\\mathbb{R}^3) \\to \\mathbb{C}$$<\/p>\n\n\n\n

whose non-vanishing is equivalent to the linear independence conjecture above (the Atiyah-Sutcliffe conjecture 1). However, Atiyah and Sutcliffe made a stronger conjecture, known as the Atiyah-Sutcliffe conjecture 2, which states that<\/p>\n\n\n\n

$$|D(\\mathbf{x})| \\geq 1 \\text{ for any $\\mathbf{x} \\in C_n(\\mathbb{R}^3)$.}$$ <\/p>\n\n\n\n

However, I should tell the reader first how to define $D$. This will be the subject of a future post.<\/p>\n","protected":false},"excerpt":{"rendered":"

This post is a continuation of my previous post “the spin-statistics theorem and the Berry-Robbins problem”. The reader is referred to that post to understand the origins of the Atiyah problem on configurations and the Atiyah-Sutcliffe conjectures. Let $C_n(\\mathbb{R}^3)$ denote the configuration space of $n$ distinct points in $\\mathbb{R}^3$. Given $\\mathbf{x} = (\\mathbf{x}_1,\\ldots,\\mathbf{x}_n) \\in C_n(\\mathbb{R}^3)$, so that each $\\mathbf{x}_i \\in \\mathbb{R}^3$, for $i = 1, \\ldots, n$ and the $\\mathbf{x}_i$ are distinct, we will associate to the configuration $\\mathbf{x}$ $n$…<\/p>\n

Read More Read More<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3,4,2],"tags":[16],"class_list":["post-53","post","type-post","status-publish","format-standard","hentry","category-geometry","category-mathematical-physics","category-mathematics","tag-atiyah-sutcliffe-conjectures"],"_links":{"self":[{"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/posts\/53","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/comments?post=53"}],"version-history":[{"count":9,"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/posts\/53\/revisions"}],"predecessor-version":[{"id":63,"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/posts\/53\/revisions\/63"}],"wp:attachment":[{"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/media?parent=53"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/categories?post=53"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/tags?post=53"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}