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Author: Malkoun

A Mathematician who is also interested in Mathematical Physics and Programming.
Complex distance between two hyperbolic lines

Complex distance between two hyperbolic lines

An oriented hyperbolic line in hyperbolic $3$-space $H^3$ is uniquely determined by its “initial” ideal point (by ideal point, I mean a point on the sphere at infinity $S^2_\infty$) and its “final” ideal point. We can thus write, by abuse of language, that an oriented hyperbolic line $L$ “is”: $L = (a, b)$, where $a$, resp. $b$, is its initial, resp. final, point. Now suppose we are given two oriented hyperbolic lines, say $L_i = (a_i, b_i)$, for $i =…

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The Atiyah problem on configurations is still open

The Atiyah problem on configurations is still open

I feel like I should mention this somewhere, though part of me is a bit hesitant. Sir Michael and I had written an article entitled “The Geometry and Dynamics of Electrons” where we had claimed to prove the Atiyah problem on configurations of points. However, I should mention it somewhere, that this article contains some gaps. So if someone wishes to work on the Atiyah problem on configurations, please note that, as of the time of writing, this problem is…

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Flat Space as a Gravitational Instanton

Flat Space as a Gravitational Instanton

This post is about $M = \mathbb{H}$ with metric $g = dq d\bar{q}$. The Gibbons-Hawking ansatz is an ansatz for $4$-dimensional gravitational instantons with a $1$ parameter group of symmetries (think of $U(1)$ or $\mathbb{R}$). One may construct such an instanton from a connection $1$-form $\omega$ for a circle bundle $P$ over an open subset $B$ of $\mathbb{R}^3$ (assuming the group is $U(1)$) and from a smooth function $\Phi$ on $B$, thought of as a smooth section of the adjoint…

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Python: from hobby to current profession

Python: from hobby to current profession

My path has been complex: I have studied Electrical Engineering during my undergraduate studies. I have then studied Mathematics at the graduate level. I have worked as a Mathematics Professor for about 7 years. I am now a professional Python programmer. Q: How did that happen? A: Python started out as being a hobby of mine, maybe since 2014 or 2015. I first started using it because I was curious about it. My first programming language was C++. From my…

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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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