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Rick the Explainer

Hi my name is Rick, and although some people think that I am a fiction, I don’t agree with them. You can actually find me in my home town, which is close to the border between the North and South poles. But never mind that, because I’m here to talk about something else.

Some friends told me about the Azimuth project, which is a group of scientists, engineers and programmers working together to understand environmental problems. I found the Azimuth Blog, which has topics ranging from the cycles of the ice ages, to the geometry of information, to some kind of mathematical “Pachinko machine” they refer to as either a Petri net or a reaction network. On their forum, they discuss things such as complex networks, and a new, more ecologically friendly brand of mathematics.

This is great, I said, it could be the start of some entirely new highways of thinking! So let’s roll up our sleeves, sharpen some pencils, brew the coffee and start digging into it! The music played: Roll on, roll on.

Yet, when I took a closer look towards Mount Azimuth, I saw some steep hills to climb! Even the trail signs had math symbols. A professor named John Baez was giving a vibrant talk about math categories, networks of connections, and troubles in the environment. I wanted to get it, but the words were foggy. As far as I could tell, his main point was that ideas from quantum micro-bits can help us to understand ecology problems such as how frogs and rabbits get along in a community forum. That sounded like a far fetch, yet he did have good credentials as a Professor of gravity and other subjects.

Despite the haze on the mountain, I could see some outlines of the group, which has professors, students, programmers, researchers, enthusiasts and other interested folks. They want to work together on topics in science that really matter to people today. They are putting together a research wiki, intended to provide a bird’s-eye view of our main environmental problems; writing software for interactive climate models; and publishing a blog, which aims to create an “ecosystem” of people teaching science to each other; conducting discussions on forum; and running a Google+ channel. The Azimuth website invites us to: write articles, contribute information, ask questions, fill in details, write software, help with research, help with writing, and more. It’s an open place.

But the scene is not only roses, because the sciences such as ecology and evolution, with all the intellectual adventures they offer to us, are the same sciences that warn of major dangers in the environment. The top priority for research should therefore go to sustainable development. In effect, science itself is now saying that we need to change our relationship with nature. This recognition, which has been part of an ages-old wisdom, in its modern form calls for an advance in our understanding of the biosphere – meaning everything living, along with its environment – as a connected whole.

In the Azimuth article Prospects for a Green Mathematics, John Baez and David Tanzer argue that, because the biosphere is a massive network of relationships, our need to understand it for sustainable development will urge on the theory of networks. Then they suggest that this “green network theory” has already started, by citing Qinglan Xia’s network model of the growth of a plant leaf. There, the veins of a leaf are depicted as a network of “pipes” for transporting fluids to the cells. The model contains a growth mechanism, which explains how at each point in time the shape of the vein network determines the pattern of growth at its edges. When this model is run on a computer, pictures of leaves are produced. With a suitable choice of parameters, the resulting images can strikingly resemble the shape of leaves such as Oak and Elm. Moral: network theory has the potential to illuminate the workings of nature.

A major application area for network theory is known as reaction networks. Examples include networks of interconnected chemical reactions, and networks of predator-prey relationships in a food chain. Here, a “reaction” means any process that converts some entities into some other entities. For instance, consider a magical lizard that eats 100 crickets, forms two clones of itself, and then disappears. This reaction can be written as: 1 Lizard + 100 Cricket –> 2 Lizard. Or a chemical reaction, in which two isolated atoms of oxygen combine to form an oxygen molecule: O + O –> O2. Each activation of a reaction changes the species counts by a definite amount. For instance, each activation of our Lizard-Cricket reaction causes the Cricket count to decrease by 100, and the Lizard count to increase by 1. The activity rates of the various reactions will determine the ways in which the population counts are changing. For instance, if reaction O + O –> O2 is happening faster than the opposite reaction O2 –> O + O, that means molecular oxygen O2 is building up, and atomic oxygen O is going down. Equilibrium is reached when the ongoing reactions balance each other out, so that the population counts remain stable.

Here are some illustrations of the relevance of network theory to our understanding of the biosphere. The atmosphere itself is a massive chemical reaction network, involving many species of molecules and reactions between them. This “soup” includes water molecules H20, hydrogen gas H2, oxygen gas O2, nitrogen gas N2, methane CH4, carbon dioxide CO2, ozone O3, …, and reactions such as the formation of water from hydrogen gas and oxygen gas. In biochemical reaction networks, the reactions involve large, biological molecules, such as proteins and DNA. Lastly, note that the global circulations involved in weather and climate consist of complex networks of interconnected processes, and so network theory may eventually make a contribution to climatology.

The subject of computer modelling of natural processes, including those based on network models, is a great point of intersection between mathematics, science, and programming, and it has led to fruitful collaborations between Azimuth members from different fields. In an informal “seminar,” they went through a text on climate models, learned the simpler ones, and programmed them for educational use in a web browser. In order to start a new round of development, the Azimuth project extends an open invitation to any programmers who are in interested in collaborating to learn about the models and flesh them out in working code. In the Azimuth tutorial series Petri Net Programming, David Tanzer is providing some of the background concepts on reaction networks, for an audience of programmers. Any realistic modelling must take into consideration the random, or stochastic character of natural processes. This is a rich subject, which I hope to talk about more in future blog articles.

There is also the study of the math of what a network is, which includes the investigation of categories of networks. Here, a category means a collection of things of the same type along with a system of relationships between them. The relationships in a category are constituted into a system through an operation by which relationships can be composed together to form new relationships. When this category theory is applied to networks, each network is treated as a relationship between its connection points, and the composition operator is defined by concatenating networks together to form larger networks. Here are some things that we may hope will result from this very new area of research in network theory: a mathematical unification of the many, disparate existing network languages; theorems to reveal the deeper structures of networks; and abstractions for interpreting, organizing and computing with the accumulating masses of data on real-world networks. This category theory of networks is being actively pursued by John Baez and a research group that he is leading at U.C. Riverside.

Now I am heading back to the rain forests of Azimuth to acquaint myself with the regional dialects, and, upon my return, I will invite you on a tour of some of the more colorful trails. Although I can’t promise that the journey will be completely effortless, we shall prudently steer clear of the most jagged peaks, and we will take frequent breaks to drink water and pat ourselves on the back. If nothing else, we may learn a bit about the anthropology of the place.

Finally, in case you have any concerns about my qualifications, I have just obtained my permit as an Azimuth tour guide. See my green and white badge, which says: Rick the Explainer.

category: blog