“Systems of systems” is a fashionable buzzword for complicated systems that are themselves made of complicated systems, often of disparate sorts. They’re important in modern engineering, and it takes some thought to keep them from being unmanageable. Biology and ecology are full of systems of systems.

David Spivak has been working a lot on operads as a tool for describing systems of systems. Here’s a nice programmatic talk advocating this approach:

But if you know a bit about operads, it may help give you an idea of their flexibility as a formalism for describing ways of sticking together components to form bigger systems!

I’ll probably talk about this kind of thing more pretty soon. So far I’ve been using category theory to study networked systems like electrical circuits, Markov processes and chemical reaction networks. The same ideas handle all these different kind of systems in a unified way. But I want to push toward biology. Here we need more sophisticated ideas. My philosophy is that while biology seems “messy” to physicists, living systems actually operate at higher levels of abstraction, which call for new mathematics.

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I have been following David’s work for a while, but I hadn’t seen that paper yet, thanks for the share. It’s very nice to see how his work has developing, I’m really enjoying this Operad direction.

I hope to see more writing from you on the subject too!

“My philosophy is that while biology seems “messy” to physicists, living systems actually operate at higher levels of abstraction, which call for new mathematics.”
This is one of the coolest insights into modelling living systems that I’ve seen :)

I’ve been following you since I discovered your wonderful three lecture series about the numbers 5,8 and 24 – thank you :)

For a higher level overview of biological systems you might find the work of Geoffrey West of interest (if you don’t know his work already). He seems to have been studying biological systems for a while now. Here is a lecture of his that I found interesting:

P.S.
In one of your 5,8,24 lectures you mention ‘spinors’ and the fact that they have the special property that you must rotate them twice to get them back to where they started from (something that at first seems rather ludicrous). You then go on to say that there is no such “real world” example of this (or words to that effect).

Would I be the first to notice that if one takes a Mobius strip, holds the twisted section in one hand (steady) and rotates the loop through the hand gripping the twist – such that the twist never moves, that you have to turn the darn loop twice to get the strip back to the starting orientation?

Thanks, I’ll try that lecture by Geoffrey West sometime. Actually I don’t enjoy videos very much, because they tend to be rather slow unless the material being presented is very hard, so I’ll start by seeing if Geoffrey West wrote anything on this topic.

You then go on to say that there is no such “real world” example of this (or words to that effect).

I doubt I said exactly that, because electrons, protons and neutrons are examples of spinors. All matter is largely made of spinors!

Would I be the first to notice…

No, but it’s a nice observation. Another way to say what you’re saying is that the edge of the Möbius strip is a ‘connected double cover’ of the circle at the center of the Möbius strip: two points at the edge correspond to each point at the center, and yet the edge is connected. The group of symmetries of a spinor, called Spin(n), is similarly a connected double cover of the group of rotations in n dimensions, called SO(n). So there’s a relationship, which is strongest in 2-dimensional space: the group SO(2) is a circle and the group Spin(2) can be visualized as the edge of a Möbius strip.

So no, the Möbius strip is not a spinor, but the math of Möbius strips is related to the math of spinors. This relation is used here to help explain spinors:

Metron already designs systems of systems used in Coast Guard search and rescue missions. Their grant proposal was to use category theory and operads to do this better. They needed an academic mathematician as part of their team: that was one of the program’s requirements. So they asked if I was interested.

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You need the word 'latex' right after the first dollar sign, and it needs a space after it. Double dollar signs don't work, and other limitations apply, some described here. You can't preview comments here, but I'm happy to fix errors.

Very interesting topic, I’d love to read more about that! You got a typo in the link “So far”, namely “networs” instead of “networks” in the address.

Thanks. Fixed!

Impressive! Thought provoking!

I have been following David’s work for a while, but I hadn’t seen that paper yet, thanks for the share. It’s very nice to see how his work has developing, I’m really enjoying this Operad direction.

I hope to see more writing from you on the subject too!

“My philosophy is that while biology seems “messy” to physicists, living systems actually operate at higher levels of abstraction, which call for new mathematics.”

This is one of the coolest insights into modelling living systems that I’ve seen :)

P.S. for anyone interested, here’s a direct link to David’s video https://mbi.osu.edu/video/player/?id=3902&title=Operads+as+a+potential+foundation+for+systems+of+systems

Hello John,

I’ve been following you since I discovered your wonderful three lecture series about the numbers 5,8 and 24 – thank you :)

For a higher level overview of biological systems you might find the work of Geoffrey West of interest (if you don’t know his work already). He seems to have been studying biological systems for a while now. Here is a lecture of his that I found interesting:

P.S.

In one of your 5,8,24 lectures you mention ‘spinors’ and the fact that they have the special property that you must rotate them twice to get them back to where they started from (something that at first seems rather ludicrous). You then go on to say that there is no such “real world” example of this (or words to that effect).

Would I be the first to notice that if one takes a Mobius strip, holds the twisted section in one hand (steady) and rotates the loop through the hand gripping the twist – such that the twist never moves, that you have to turn the darn loop twice to get the strip back to the starting orientation?

Does this count as a ‘spinor’ ?

Thanks, I’ll try that lecture by Geoffrey West sometime. Actually I don’t enjoy videos very much, because they tend to be rather slow unless the material being presented is very hard, so I’ll start by seeing if Geoffrey West wrote anything on this topic.

I doubt I said exactly that, because electrons, protons and neutrons are examples of spinors. All matter is largely made of spinors!

No, but it’s a nice observation. Another way to say what you’re saying is that the edge of the Möbius strip is a ‘connected double cover’ of the circle at the center of the Möbius strip: two points at the edge correspond to each point at the center, and yet the edge is connected. The group of symmetries of a spinor, called Spin(n), is similarly a connected double cover of the group of rotations in n dimensions, called SO(n). So there’s a relationship, which is strongest in 2-dimensional space: the group SO(2) is a circle and the group Spin(2) can be visualized as the edge of a Möbius strip.

So no, the Möbius strip is not a spinor, but the math of Möbius strips is related to the math of spinors. This relation is used here to help explain spinors:

• Wikipedia, spinor.

Metron already designs systems of systems used in Coast Guard search and rescue missions. Their grant proposal was to use category theory and operads to do this better. They needed an academic mathematician as part of their team: that was one of the program’s requirements. So they asked if I was interested.