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~~~~ that~~*network 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~~~~ of~~tipping 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 of*new 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 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~~ 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 of*tokens*~~ ~~ .~~ 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.

We are pursuing these subjects in the Azimuth Project, an open collaboration between mathematicians, scientists, engineers and programmers trying to help save the planet. On the Azimuth Wiki and Azimuth Blog we are trying to explain the main environmental and energy problems the world faces today. We are~~ also~~ studying plans of action, network theory, climate cycles, the programming of climate models, and more.

If you would like to help, 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. Just drop us a line, either here or on the Azimuth Blog.

category: blog