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Blog - prospects for a green mathematics (Rev #2)

This page is a blog article in progress, written by John Baez and David Tanzer. To discuss this article while it’s being written, visit the Azimuth Forum.

contribution to the MPE 2013 blog, by John Baez and David A. Tanzer

Green mathematics for the era of environmental jeopardy

It is increasingly clear that we are witnessing the beginning of a series of unfortunate environmental events. The problems include global warming, ice melting, permafrost melting, sea level rise, extreme weather events, wars, peak oil, loss of habitats, mass extinctions, deforestation, soil erosion, acid rain, air pollution, famines, pathogen mutations, antibiotic resistant diseases, epidemics, ocean acidification, ocean dead zones, radioactive waste, ozone depletion, the water crisis, and the accumulation of toxins.

Deep changes will be required to stabilize and regenerate the environment, but, unfortunately, the political will for such difficult changes may not be mustered until things get significantly worse. Whether it comes sooner, or later, the recovery of an injured biosphere will be recognized as the top priority for continued social development. This will – and does – pose a great challenge for science and its twin, mathematics. At the Azimuth project, we want to get started now to help with the mathematical foundations for this science project, which is bound to become socially urgent as things progress.

Historically, major transformations in mathematics have been linked with major transformations in society. For example, the birth of written numerals in the Middle East was closely intertwined with the agricultural revolution and the development of trade. For convenience, contracts were represented by little clay tokens, stored in sealed envelopes.

Later, these tokens were represented by marks on the outsides of the envelopes. Eventually these marks evolved into the Babylonian numeral system. Many centuries later, the Industrial Revolution was deeply connected to breakthroughs in mechanics and calculus. Now, as the 21st Century unfolds, mathematics will become increasingly driven by the great social need to understand the biosphere and our role within it.

We refer to the social enterprise of developing mathematics suitable for understanding the biosphere as green mathematics.

Since the biosphere is a massive network of interconnected elements, it is plausible that network theory will play an important role in green mathematics. Network theory is a sprawling field of investigation, just starting to become organized, which combines ideas from graph theory, systems theory, biology, ecology and sociology. Networks such as the biosphere display massive complexity. Hence network theory finds an important partner in the the science of complexity itself, which can now be seen in the particular forms like computational complexity theory, chaos theory, and organizational complexity theory. Computation itself is a partner to network theory, for it is both a network-theoretic structure – think for instance of computations being defined by networks of logic gates – and the means by which network dynamics can be simulated and therefore approached in an experimental manner.

One major application of network theory is the study of “tipping points,” through which there is an abrupt passage from one dynamic regime to another. It is critical for environmental scientists to be able to identify nearby tipping points, in order to inform policy makers and guide their decisions. in turn, scientists may call on mathematicians to make in-depth analysis of environmental time-series data.

Another key area is the study of shocks to systems. When is it possible for a system, or an organism, to recover from a major blow to one of its subsystems? What are the repair processes that take place when a tree, or a worm, is cut?

Next we assert that the network theory is not just another name for biology, etc., because in it we are seeing the outlines of new mathematical terrains. Here we describe two recent developments in mathematical network theory.

First, consider a leaf. In The Formation of a Tree Leaf, by Qinglan Xia, we see what might be the secret key to one of Nature’s algorithms – the formation of the vein patterns in a tree leaf. The system of veins, which is a transport network for nutrients and other substances, is modeled by a directed graph – a “tree” in this case – where the nodes are cells, and the edges are the “pipes” that connect the cells. Every cell generates a “revenue” of energy from photosynthesis, but it also incurs the cost of transporting substances it from the base of the leaf (along with flows in the opposite direction). The net revenue from a cell is the excess of its energy revenue over its transport cost.

The total cost of fluid transportation depends on the network structure. There is a transport cost for each pipe, and a cost for the turning of the fluid around the bends in the network. For each pipe, the cost equals Length * (CrossSectionalArea ^ Alpha) * Flow, where Alpha is a model parameter, and Flow is proportional to the total number of cells that get “fed” through this pipe. Alpha captures the cost savings from using a thicker pipe to transport materials together in parallel. Similarly, it is more efficient to pack many letters onto a single mail truck, and transport them together along the common part of their paths. The explanation of the cost for the bends, which is more involved, uses a parameter for the penalization due to bending.

Development proceeds through alternate iterations of growth and network optimization. In a growth iteration, a new layer of cells gets added, consisting of all potential cells, which, when adjoined to the network, would have a positive net revenue. In an optimization step, local adjustments are made to the transport graph, to find a local minimum of the cost function. Remarkably, by varying the two parameters, the simulations can produce realistic models of various types of natural leaves, such as oak and maple.

As compared to approaches such as L-systems which merely generate somewhat realistic images of plants, the work by Xia takes a truly biological approach, because it is based upon a plausible model of how plants actually work. Moreover, it is a network-theoretic approach to a biological subject, and it is mathematics – replete with lemmas, theorems and algorithms – from start to finish.

Here is another illustration, which shows that the dynamics of a network is an area for basic research of a mathematical character. It pertains to stochastic Petri nets, which are a simple model for reaction networks, that applies to a broad range of systems, including chemical reaction networks, predator-prey networks, and vending machines. Reactions are represented by process nodes that consume tokens at their input places, and generate tokens at their outputs Each place represents a type of entity, and each token stands for an instance of that type. The reactions proceed concurrently, and generate a data-flow of tokens through the net. In a stochastic Petri net, the reaction events are probabilistically determined by a Markov chain where the expected firing rate of a reaction is a function of the number of tokens at its inputs.

What is new is the discovery, reported in the Network Theory series of the Azimuth project, that some of the mathematics quantum mechanics is transferable to the classical realm of stochastic Petri nets. The key step here is to represent a probabilistic state of a Petri net – a distribution which assigns a probability to each potential configuration of a Petri net – by a power series. A power series representation (with complex coefficients) is also used for quantum states. The quantum particle annihilation and creation operators are represented as operators on these power series. Notably, when these formal operators are applied to the stochastic states of a Petri net, they have a meaning that rooted in the annihilation and creation of tokens by the reactions of the network.

For a given Petri net, the Hamiltonian operator for its Markov chain, which expresses the probabilistic law of motion for the network as an infinitesimal operator on the power series, is constructed as a composition of functions that include the annihilation and creation operators for the species (places) in the network. The structure of this composition is given as a function of the connectivity structure of the Petri net. Furthermore, this new and more abstract mathematics for Petri nets shows potential as a framework, because some of the fundamental theorems about network equilibrium states, can be proven in a compact and elegant way using this operator math.

We take these results as further evidence that the life of a network, and the networks of life, are brimming with mathematical content.

All of these subjects are being actively pursued in the Azimuth Project, which is an open collaboration between mathematicians, scientists, engineers and programers trying to help save the planet. Our goal is to present clear and accurate information on the relevant issues, and to help people work together on our common problems. On the Azimuth Wiki, we are explaining the main environmental and energy problems the world faces today. We are studying plans of action, network theory and climate cycles. We are investigating climate models and how to program them. Already we have developed some interactive climate models that run in a web browser. See the Azimuth Forum for active discussions on all of these topics.

If you are interested in helping out, that would be great. We need you, and your special expertise. You can write articles, contribute information, pose questions, fill in details, write software, help with research, help with writing, and more.

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