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Blog - prospects for a green mathematics (Rev #10, changes)

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This page is a blog article in progress, written by John Baez and David Tanzer . To seee see discussions of this article while it was being written, visit theAzimuth Forum.

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

It is increasingly clear that we are initiating a sequence of dramatic events across our planet. These They 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 events are all connected. 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 a permanent condition. Deep changes will occur as these idealizations bring us crashing into the brick wall of reality. If we do not muster the will to take act action before things get significantly worse, we will need to do so later. While we may plead that it is “too difficult” or “too late”, this doesn’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 math fit into all this? While the problems we face have deep roots, major transformations in society have always caused and been helped along by revolutions 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 Although it is just being born, but we can already speculate see on some what of it its will outlines. be like.

Since the biosphere is a massive network of interconnected elements, it we is expect plausible thatnetwork theory will to play an important role in the green mathematics. Network theory is a sprawling field field, of investigation, just beginning to become organized, which combines ideas from graph theory, probability theory, biology, ecology, sociology and more. Computation plays a an specially important role, role for here, it is both because it has a network-theoretic network structure structure—think of e.g. computations that are defined by networks of logic gates gates—and because and it provides the means by for which network dynamics can be simulated simulating and network studied. processes.

One application of network theory is the study oftipping points through where which a system abruptly passes from one regime to another. It Scientists is need critical for scientists to identify nearby tipping points in the biosphere, in order to inform help 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. Mathematicians, in turn, are challenged to develop new data analysis techniques for detecting incipient tipping points. Another key application area of network theory is the study of shocks and resilience. When can a system network 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 shows see the outlines ofnew mathematical terrains . We Here illustrate we with portray two recent developments.

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

The total transport cost depends on the network structure. There is are a costs cost for each pipe, of the pipes, and a cost for the turning of the fluid around the bends bends. in the network. For each pipe, the cost is proportional to the product of its length, times its cross-sectional area raised to some a power α, times and the number of leaf cells that get it “fed” feeds. through this pipe. The exponent α captures the savings from using a thicker pipe to transport materials together together. in Another parallel. There is also another parameter β that expresses measures the cost turning of cost. each bend in a pipe.

Development proceeds through cycles of growth and network optimization. In During each stage of growth, a new layer of cells gets added, consisting containing of each all potential cells cell that with would give a revenue exceeding that the would cost exceed of its bringing cost. fluid to it. During optimization, local adjustments are made to the transport graph graph, is adjusted to find a local minimum of the cost function. minimum. Remarkably, by varying the α two and parameters, β, the simulations give realistic schematic models images of various types of natural leaves. leaves like maple and mulberry.

A growing network.

Unlike approaches that merely just create pretty images of which plants, resemble Xia’s leaves, approach Xia presents an algorithmic model, which is based simplified on yet a illuminating, simple but illuminating model of how plants leaves actually work. develop. Moreover, It it is anetwork-theoretic approach to a biological subject, and it is mathematics—replete with lemmas, theorems and algorithms—from start to finish.

Here is another illustration illustration, that in network the dynamics field is of an area for mathematical investigation. A stochastic Petri net nets , is which are a model for networks of reactions. A Entities stochastic are Petri designated net by has “tokens” “tokens”, which represent entities, and entity types by “places” which hold the tokens, tokens. and represent types of entities. “Reactions” remove tokens from their input places, and deposit tokens at their output outputs. places. Concurrently, The they reactions proceed concurrently, and generate a flow of tokens. The reaction reactions events fire occur probabilistically, by in a Markov chain in where which each the expected firing rate of a reaction rate depends on the number of tokens at its inputs. input tokens.

A stochastic Petri net.

Perhaps Notably, surprisingly, many techniques from quantum field theory can are be transferable transferred to stochastic Petri nets. The key idea is to represent a stochastic state states by a power series. Monomials represent states pure in states, which there have is a definite number of tokens in at each place. There Each is one variable for each place in the network, monomial stands for a place, and its exponent in the monomial indicates the number token of count. tokens stored there. In a linear real-valued combination of these monomials, each coefficient represents the coefficients probability represent of probabilities. being in the associated state.

In Analogously, in quantum field theory, states are often representable represented by power series, series but with complex coefficients. The annihilation and creation of particles are described cast by as operators on power series. But These same operators, when these operators are applied to the stochastic states of a Petri net, they describe the annihilation and creation oftokens . in the network. Remarkably, the commutation relations between annihilation and creation operators, which are often viewed as a hallmark of quantum theory, make perfect sense in this classical probabilistic context.

Each stochastic Petri net gives has a “Hamiltonian” describing which the gives its probabilistic law of motion motion. for It that networks. The Hamiltonian is an operator on power series built from the annihilation and creation operators. operators, The precise formula for this operator depends on the reactions in a way that reflects the Petri network net. connection Moreover, structure. this With approach these lets representations, us one can prove many theorems about stochastic reaction Petri networks, nets, which are already known to chemists, in a compact and elegant way. See the Azimuthnetwork theory series for details.

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

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