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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, blog byJohn Baez and David A. TanzerJohn Baez and David Tanzer

Green The mathematics Mathematics for the era of environmental Planet jeopardy Earth

It is increasingly clear that we are witnessing the beginning start of a series of unfortunate environmental events. The problems include habitat loss, an increased rate of extinction, global warming, the melting of ice melting, caps permafrost and melting, permafrost, sea an level increase rise, in extreme weather events, wars, gradually peak rising oil, sea loss levels, 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 spread crisis, and the accumulation of toxins. 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 be occur required as to these stabilize simplified and views regenerate bring us crashing into the environment, brick but, wall unfortunately, of reality. If we do not muster the political will for to such take difficult action changes before may not be mustered until things get significantly worse. worse, Whether we will need to so later. While we may plead that it comes is sooner, “too difficult” or later, “too the late”, recovery this of won’t an matter: injured biosphere will be recognized as the top priority for continued social development. This will and does pose a great transformation challenge is for inevitable. science All and its twin, mathematics. At the Azimuth project, we want can do is start where we find ourselves, and begin adapting to get life started on a finite-sized planet.now to help with the mathematical foundations for this science project, which is bound to become socially urgent as things progress.

Historically, Where does mathematics fit into all this? While the biggest issues facing us are cultural, major transformations in mathematics society have always caused and been linked helped with along by major transformations in society. mathematics. For Starting example, near the end of the last ice age, the Agricultural Revolution eventually led to the birth of written numerals in the Middle East was intertwined with the agricultural revolution and geometry. Centuries later, the development Industrial Revolution brought us calculus, and eventually a flowering of trade. mathematics For unlike convenience, any contracts before. were Now, represented as the 21st century unfolds, mathematics will become increasingly driven by little our clay need tokens, to stored understand in the sealed biosphere envelopes. 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.

Later, Since these tokens were represented by marks on the outsides biosphere is a massive network of the interconnected envelopes. elements, Eventually it these is marks plausible evolved that 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 network the theory biosphere and our role within it . 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 besimulated and studied.

We One refer application to the social enterprise of developing network mathematics theory suitable is for understanding the biosphere study as ofgreen mathematicstipping 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?

Since We the claim biosphere that is a massive network of theory interconnected elements, it is plausible not that just another name for biology, ecology, or any other existing science, because in it we see the outlines of network new theory mathematical terrains . will We play illustrate an with important two role recent in developments. 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 displaymassive complexity. The study of network complexity, and the related study of the computational complexity of network simulations, are therefore important aspects of network theory. Computation itself is also 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 First, major consider 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. In turn, scientists need mathematicians and statisticians to analyze environmental data. Another key area is the study of shocks to systems. When can a system, leaf. or In an organism, recover from a major blow to one of its subsystems?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.

We The claim total now transport that cost depends on the network theory structure. There is not a just cost another name for biology, each etc., pipe, because and a cost for the turning of the fluid around the bends in it we see the outlines network. For each pipe, the cost is proportional to its length, times its cross-sectional area to some power α, times the number of new 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.mathematical terrains. We illustrate with two recent developments.

First, Development consider proceeds a through leaf. 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.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 “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 that as merely L-systems create which pretty use an imaginative model to generate images of plants, Xia Xia’s takes approach a biological approach, which is based on a plausible simple but illuminating model of how plants actually work. Moreover, it is anetwork-theoretic approach to a biological subject, and it is mathematics —replete replete with lemmas, theorems and algorithms algorithms—from 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.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.

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 None in of this is new. The surprising part is that many techniques from quantum mechanics, field theory can be transferred to the states realm are of represented stochastic Petri nets. The key idea is to represent a stochastic state by a power series series. that Here use the complex monomials coefficients. represent The states annihilation in which there is a definite number of tokens in each place. There is one variable for each place in the network, and creation its exponent in the monomial indicates the number of particles tokens are stored represented there. by In a linear combination of these monomials, the coefficients represent probabilities.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, In the quantum Hamiltonian field operator theory, for states the are Markov often chain represented by power series with complex coefficients. The annihilation and creation of a particles Petri are net described gives by operators on the probabilistic space law of motion for the network. It is an operator on power series, series. which The first interesting result is composed that using when these operators are applied to the stochastic states of a Petri net, they describe 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 tokens for network dynamics, because in it some of the basic network. theorems Remarkably, about the network commutation equilibrium relations states, between are annihilation proven in a compact and elegant creation way. 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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