Joseph Malkoun's Blog

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Joseph Malkoun's blog, discusses Mathematics and Programming. Joseph obtained his PhD in Mathematics from the State University of New York at Stony Brook in 2012. His area of expertise is Differential Geometry. He also holds a Master of Science degree in Mathematics from McGill University (2006).

Joseph Malkoun's Blog

2w ago

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 $i = 1, 2$. We could define the cross ratio of $L_1$ and $L_2$ to be the cross-ratio of the $4$ corresponding ideal points, namely
$$ \operatorname{C ..read more

Joseph Malkoun's Blog

1y ago

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 still open for a general $n > 4$. I just wanted to clear some confusion which may have arisen. It is a long story which is painful to rec ..read more

Joseph Malkoun's Blog

1y ago

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 bundle of $P$, satisfying the so-called Bogomolny equation
$$ F = *d_{\omega} \Phi $$
where $F$ is the curvature of ..read more

Joseph Malkoun's Blog

1y ago

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 own humble point of view, I liked the fact that if you wanted to do some scientific programming, you had many official Python libraries to he ..read more

Joseph Malkoun's Blog

1y ago

A picture with Sir Michael on the right and me on the left, in Edinburgh. I think it was taken on July 16, 2018.
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 2000. The talk was about an “elementa ..read more

Joseph Malkoun's Blog

1y ago

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 $\mathbb{R}^3$, I have explained in my prev ..read more

Joseph Malkoun's Blog

1y ago

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 $\mathbf{x}$ $n$ complex polyn ..read more

Joseph Malkoun's Blog

1y ago

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 configuration. Moreover, let $U(n)$ denote the ..read more