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Blog - prospects for a green mathematics (Rev #6)

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 Tanzer

The Mathematics of Planet Earth

It is increasingly clear that we are witnessing the start of a series of unfortunate environmental events. The problems include habitat loss, an increased rate of extinction, global warming, the melting of ice caps and permafrost, an increase in extreme weather events, gradually rising sea levels, ocean acidification, the spread of oceanic “dead zones”, a depletion of natural resources, and ensuing social strife.

These are not separate problems. They all come from a way of life that views the Earth as essentially infinite, human civilization as a negligible perturbation, and exponential economic growth as the norm. Deep changes will occur as these simplified views bring us crashing into the brick wall of reality. If we do not muster the will to take action before things get significantly worse, we will need to so later. While we may plead that it is “too difficult” or “too late”, this won’t matter: a transformation is inevitable. All we can do is start where we find ourselves, and begin adapting to life on a finite-sized planet.

Where does mathematics fit into all this? While the biggest issues facing us are cultural, major transformations in society have always caused and been helped along by major transformations in mathematics. Starting near the end of the last ice age, the Agricultural Revolution eventually led to the birth of written numerals and geometry. Centuries later, the Industrial Revolution brought us calculus, and eventually a flowering of mathematics unlike any before. Now, as the 21st century unfolds, mathematics will become increasingly driven by our need to understand the biosphere and our role within it.

We refer to mathematics suitable for understanding the biosphere as green mathematics. It is just being born, but we can already speculate on what it will be like.

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 beginning to become organized, which combines ideas from graph theory, probability theory, biology, ecology, sociology and more. Computation plays a specially important role, 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 studied.

One application of network theory is the study of tipping points, through which a system abruptly passes from one regime to another. It is critical for scientists to identify nearby tipping points in the biosphere, to inform policy makers and guide their decisions. Scientists need mathematicians and statisticians to develop ways of analyzing data to detect incipient tipping points, and find the best ways to head off catastrophic changes. Another key area is the study of shocks and resilience. When can a system recover from a major blow to one of its subsystems?

We claim that network theory is not just another name for biology, ecology, or any other existing science, 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 network for transporting nutrients and other substances, is modeled by a directed graph—a “tree” in this case—where nodes are cells, and edges are “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 is proportional to its length, times its cross-sectional area to some power α, times the number of cells that get “fed” through this pipe. The exponent α captures the savings from using a thicker pipe to transport materials together in parallel. There is also another parameter β that measures the cost of each bend in a pipe.

Development proceeds through cycles of growth and network optimization. In each stage of growth, a new layer of cells gets added, consisting of all potential cells that would give a revenue exceeding the cost of bringing fluid to it. 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 that merely create pretty images of plants, Xia’s approach is based on a simple but illuminating 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. A stochastic Petri net is a model for networks of reactions. A stochastic Petri net has “tokens”, which represent entities, and “places” which hold the tokens, and represent types of entities. “Reactions” remove tokens from their input places, and deposit tokens at their output places. The reactions proceed concurrently, and generate a flow of tokens. The reaction events occur probabilistically, by a Markov chain in which the expected firing rate of a reaction depends on the number of tokens at its inputs.

None of this is new. The surprising part is that many techniques from quantum field theory can be transferred to the realm of stochastic Petri nets. The key idea is to represent a stochastic state by a power series. Here the monomials represent states in which there is a definite number of tokens in each place. There is one variable for each place in the network, and its exponent in the monomial indicates the number of tokens stored there. In a linear combination of these monomials, the coefficients represent probabilities.

In quantum field theory, states are often represented by power series with complex coefficients. The annihilation and creation of particles are described by operators on the space of power series. The first interesting result is that when these operators are applied to the stochastic states of a Petri net, they describe the annihilation and creation of tokens in the network. Remarkably, the commutation relations between annihilation and creation operators, often viewed as a hallmark of quantum theory, make perfect sense in this purely classical, probabilistic context.

Next, each stochastic Petri net gives a “Hamiltonian” describing the probabilistic law of motion for that networks. The Hamiltonian is an operator on power series built from annihilation and creation operators. The precise formula for the this operator depends on the reactions in the Petri net. Moreover, this approach lets us prove many theorems about stochastic Petri nets, already known to chemists, in a compact and elegant way.

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

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