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Blog - prospects for a green mathematics (Rev #4)

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 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. 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 a partner in the science of complexity itself, which can now be seen in particular forms such as 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 – e.g. computations that are 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 identify nearby tipping points, in order to inform policy makers and guide their decisions. This is a mathematical topic.

Another key area is the study of shocks to systems. When can a system, or an organism, 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 see the outlines of new mathematical terrains. We illustrate with two recent developments.

First, consider a leaf. In The Formation of a Tree Leaf, by Qinglan Xia, we see what could be the secret key to one of Nature’s algorithms – the growth of the veins in a leaf. The vein system, which is a transport network for nutrients and other substances, is modeled by a directed graph – a “tree” in this case – where nodes are cells, and edges are the “pipes” that connect them. Each cell generates a “revenue” of energy, and it incurs the cost of transporting substances between it and the base of the leaf.

The total transport cost depends on the network structure. There is a 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 parameter, and Flow is proportional to the number of cells that get “fed” through this pipe. Alpha captures the 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. There is another parameter, Beta, for the cost of turning the fluid.

Development proceeds through cycles of growth and network optimization. In a growth iteration, a new layer of cells gets added, consisting of all potential cells that would give a positive net revenue. During optimization, 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 give realistic models of various types of natural leaves.

Unlike approaches such as L-systems which use an imaginative model to generate images of plants, Xia takes a biological approach, which is based on 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 that network dynamics is an area for mathematical investigation. It pertains to stochastic Petri nets, which are a model for networks of reactions. A network contains “tokens”, that represent entities, and places, which hold the tokens, and represent entity types. Reactions are process nodes that remove tokens from their input places, and deposit tokens at their outputs. The reactions proceed concurrently, and generate a data-flow of tokens. The reaction events are probabilistically determined, by a Markov chain, in which the expected firing rate of a reaction depends on the number of tokens at its inputs.

What is new is the discovery, reported in the Azimuth Network Theory series, that part of the quantum mathematics is transferable to the realm of stochastic Petri nets. The key idea, inspired from quantum mechanics, is to represent a probabilistic state by a power series. Here the monomials represent possible states of the network. There is one variable for each place in the network, and its exponent in the monomial indicates the number of tokens stored there. The coefficient of the monomial gives the probability of being in that state.

Now in quantum mechanics, the states are represented by power series that use complex coefficients. The annihilation and creation of particles are represented by operators over these power series. The first interesting result is that when these formal operators are applied to the stochastic states of a Petri net, they take on a meaning that is rooted in the annihilation and creation of tokens in the network. There is an annihilation operator, and a creation operator, for each place in the network.

Next, the Hamiltonian operator for the Markov chain of a Petri net gives the probabilistic law of motion for the network. It is an operator on power series, which is composed using the annihilation and creation operators for the places. The structure of this composition reflects the connections in the Petri net. Moreover, this mathematics shows potential as a framework for network dynamics, because in it some of the basic theorems about network equilibrium states, are proven in a compact and elegant way.

Conclusion: The life of a network, and the networks of life, are brimming with mathematical content.

These subjects are being pursued in the Azimuth Project, which is an open collaboration between mathematicians, scientists, engineers and programers trying to help save the planet. We aim to present clear and accurate information on relevant issues, and to help people work together on our common problems. On the wiki, we are explaining the main environmental and energy problems the world faces today. We are studying plans of action, network theory, climate cycles, and the programming of climate models.

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