A New Perspective of Entropy
Hello world! Last summer I wrote a short paper entitled "Entropy as a Topological Operad Derivation," which describes a small but interesting connection between information theory, abstract algebra, and topology. I blogged about it here in June 2021, and the paper was later published in an open-access journal called Entropy in September 2021. In short, it describes a correspondence between Shannon entropy and functions on topological simplices that obey a version of the Leibniz rule from calculus, which I call "derivations of the operad of topological simplices," hence the title.
By what do those words mean? And why is such a theorem interesting?
To help make the ideas more accessible, I've recently written a new article aimed at a wide audience to explain it all from the ground up. I'm very excited to share it with you! It's entitled "A New Perspective of Entropy," and a trailer video is below:
As mentioned in the video, the reader is not assumed to have prior familiarity with the words "information theory" or "abstract algebra" or "topology" or even "Shannon entropy." All these ideas are gently introduced from the ground up.
The "only" prerequisite for this new article is an interest in learning about higher mathematics. The reader is also assumed to be a bit ambitious, as the ideas do get a bit advanced at times. In particular, the paper draws on concepts that will be familiar to some math undergrads, grad students, and research mathematicians. So there's hopefully something in it for everyone. But since the paper is targeted at such a wide audience, some portions might be too easy for some, while other parts might be too challenging. Even so, I hope each reader can find an enjoyable sweet spot.
Here is the abstract:
This article describes a new connection between two seemingly disparate topics in science, namely entropy and higher mathematics. It does not assume prior knowledge of either subject and begins with a brief introduction to information theory and a concept known as Shannon entropy, which we simply refer to as entropy.
We then survey the vast landscape of higher mathematics, giving special attention to advanced analogues of high-school algebra and geometry known as abstract algebra and topology, respectively. Our goal is then to show that entropy, abstract algebra, and topology are inextricably linked through a version of a well-known formula from calculus known as the Leibniz rule.
This result is given in the author’s recent work in [Bra21], and this present article is intended to give an overview of the ideas by gently introducing them from the ground up.
Without further ado, below is the PDF! You can also click here to open the PDF in a new browser window.
Hope you enjoy!