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Blog - Azimuth explained (part 1)

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Copyright (c) 2014, David Tanzer, All Rights Reserved

guest post by David Tanzer</i>

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 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 call either a Petri net or a reaction network. On their forum, they talk about things like 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.

But 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 on topics in science that really matter to people today. They are building a research wiki to give 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; chatting on their 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. So the top priority for research should now go to sustainable development. In effect, science itself is telling us that we need to change our relationship with nature. This recognition has indeed been part of an ages-old wisdom, but in its modern form it calls for an advance our understanding of the biosphere — 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, since the biosphere is a massive network of relationships, our need to understand it for sustainable development will urge on the theory of networks. Then, by citing the example of Qinglan Xia’s network model of a growing tree leaf, they suggest that the “green network theory” is already underway. There, the veins of a leaf are modeled as a network of pipes for transporting fluids to the cells, and a growth mechanism is presented, which explains how the shape of the network determines its pattern of growth at each point in time. When this process is simulated on a computer, nice pictures of leaves are generated, and, with a suitable choice of parameters, they can take on a form similar to leaf types such as Maple and Mulberry. Moral: network theory is capable of illuminating the actual workings of nature.

A major application area of network theory is to reaction networks. These include networks of 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: a mythical lizard eats twelve crickets, forms two clones of itself, and then disappears. As a formula, this reaction is written: 1 Lizard + 12 Cricket → 2 Lizard. In the chemical reaction O + O → O2, two separate atoms of oxygen combine to form one molecule of oxygen. Each activation of a reaction changes the population counts by a definite amount. One activation of our first reaction decreases the Cricket count by 12, and increases the Lizard count by 1. The movement of the population counts is called the dynamics of the network, and it is determined by the combined effects of the separate rates of each of the reactions. For a simple example, consider a network with two reactions: the formation of oxygen molecules, and its opposite, the splitting of oxygen molecules. If formation is happening faster than splitting, then the net effect, of course, is that molecular oxygen is increasing and atomic oxygen is decreasing. Equilibrium is reached when the effects of all the ongoing reactions balance out, so the populations stabilize.

Here are a couple of illustrations of the relevance of network theory to the biosphere. First, the atmosphere is a massive chemical reaction network, containing many types of molecules and reactions: water molecules H2O, 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. Second, in biochemical reaction networks, the reactions involve large, biological molecules, such as proteins and DNA.

There has been activity at Azimuth on the computer simulation of models, which is an inter-disciplinary field that brings together scientists, programmers, and mathematicians. In an informal seminar, the group went through a text on climate models, learned some of the simpler ones, and programmed them for the web browser — see the results here. Towards a new seminar, the Azimuth project extends an open invitation to any programmers interested in collaborating to study the models and develop them into applications. Also, to give background to programmers who are new to these applications, David Tanzer has started a tutorial blog series on reaction networks and their underlying mathematics. Note that a crucial part of this background will need to address the subject of random processes, since this is a fundamental part of our description of reality. In recent months, Tanzer seems to have slacked off a bit, and so for the time being I’ll be picking up the beat.

There is also the pure mathematics of what a network is, which includes the study 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 system is defined by an operation of composing relationships together to form new relationships. In this view, every network gets treated as the relationship between its connection points, and networks are composed by joining them together to form bigger networks. This is a new area of research for network theory, and one may hope for the following results from it down the line: a mathematical unification of the many, disparate existing network languages; theorems to reveal the deeper structures of networks; abstractions for interpreting and organizing the accumulating masses of data on real-world networks; abstractions for building software tools to process this data. This approach to network theory is being actively pursued by John Baez, who is leading a research group on it at U.C. Riverside.

Now I am headed back to the rain forests of Azimuth to acquaint myself with the regional dialects. Upon my return, I will invite you to tour some of the more colorful trails. Although I can’t promise you 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