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Category: Geometry

Sir Michael Atiyah

Sir Michael Atiyah

When CAMS (Center for Advanced Mathematical Sciences) was first founded circa 1999, I was an Electrical Engineering student at the American University of Beirut (AUB) where CAMS is physically located. I was curious and wanted to attend a talk by Sir Michael Atiyah, whom I will refer to as Sir Michael for the rest of the post. It took place in Hotel Al-Bustan actually. I think it may have been at some point in December 1999, or perhaps in January…

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The Atiyah-Sutcliffe determinant

The Atiyah-Sutcliffe determinant

This post is a continuation of my series of posts, which includes previous posts such as “the spin-statistics theorem and the Berry-Robbins problem” and “the Atiyah problem on configurations”. I will make use of notation introduced there, particularly in the latter post. I will explain in this post, how to define the Atiyah-Sutcliffe normalized determinant function, which is a smooth complex-valued function on $C_n(\mathbb{R}^3)$. Given a configuration $\mathbf{x} = (\mathbf{x}_1, \ldots, \mathbf{x}_n) \in C_n(\mathbb{R}^3)$ of $n$ distinct points in $\mathbb{R}^3$,…

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The Atiyah Problem on Configurations

The Atiyah Problem on Configurations

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 $\mathbf{x}$ $n$…

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The spin-statistics theorem and the Berry-Robbins problem

The spin-statistics theorem and the Berry-Robbins problem

In ​1​ , Berry and Robbins propose an interesting way to obtain the spin-statistics theorem, which is close to the famous belt trick, though expressed more mathematically. They completely explain their construction for $2$ particles, but while attempting to extend their construction to $n$ particles, they faced a technical obstruction. This led to the Berry-Robbins problem. Let $C_n(\mathbb{R}^3)$ denote the configuration space of $n$ distinct particles. Then the symmetric group $S_n$ acts on $C_n(\mathbb{R}^3)$ by permuting the components of any…

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