For me, the answer to this year's Edge question is clear: The Continuity Equations.
These are already familiar to you, at least in anecdotal form. Most everyone has heard of the law of "Conservation of Mass" (sometimes using the word "matter" instead of mass) and probably its partner "Conservation of Energy" too. These laws tell us that for practical, real-world (i.e. non-quantum, non-general relativity) phenomena, matter and energy can never be created or destroyed, only shuffled around. That concept has origins tracing at least as far back as the ancient Greeks, was formally articulated in the 18th century (a major advance for modern chemistry), and today underpins virtually every aspect of the physical, life, and natural sciences. Conservation of Mass (matter) is what finally quashed the alchemists' quest to transform lead to gold; Conservation of Energy is what consigns the awesome power of a wizard's staff to the imaginations of legions of Lord of the Rings fans.
The Continuity Equations take these laws an important step further, by providing explicit mathematical formulations that track the storage and/or transfers of mass (Mass Continuity) and energy (Energy Continuity) from one compartment or state to another. As such, they are not really a single pair of equations but instead written into a variety of forms, ranging from the very simple to the very complex, in order to best represent the physical/life science/natural world phenomenon they are supposed to describe. The most elegant forms, adored by mathematicians and physicists, have exquisite detail and are therefore the most complex. A classic example is the set of Navier-Stokes equations (sometimes called the Saint-Venant equations) used to understand the movements and accelerations of fluids. The beauty of Navier-Stokes lies in their explicit partitioning and tracking of mass, energy and momentum through space and time. However, in practice such detail also makes them difficult to solve, requiring either hefty computing power or simplifying assumptions to be made to the equations themselves.
But the power of the Continuity Equations is not limited to complex forms comprehensible solely to mathematicians and physicists. A forest manager, for example, might use a simple, so-called "mass balance" form of a mass continuity equation to study her forest, by adding up the number, size, and density of trees, the rate at which seedlings establish, and subtracting the trees' mortality rate and number of truckloads of timber removed to learn if its total wood content (biomass) is increasing, decreasing, or stable. Automotive engineers routinely apply simple "energy balance" equations when, for example, designing a hybrid electric car to recapture kinetic energy from its braking system. None of the energy is truly created or destroyed just recaptured (e.g. from a combustion engine, which got it from breaking apart ancient chemical bonds, which got it from photosynthetic reactions, which got it from the Sun). Any remaining energy not recaptured from the brakes is not really "lost", of course, but instead transferred to the atmosphere as low-grade heat.
The cardinal assumption behind these laws and equations is that mass and energy are conserved (constant) within a closed system. In principle, the hybrid electric car only satisfies energy continuity if its consumption is tracked from start (the Sun) to finish (dissipation of heat into the atmosphere). This a bit cumbersome so it is usually treated as an open system. The metals used in the car's manufacture satisfy mass continuity only if tracked from their source (ores) to landfill. This is more feasible, and such "cradle-to-grave" resource accounting—a high priority for many environmentalists—is thus more compatible with natural laws than our current economic model, which tends to externalize (i.e., assume an open system) such resource flows.
Like the car, our planet, from a practical standpoint, is an open system with respect to energy and a closed system with respect to mass (while it's true that Earth is still being bombarded by meteorites, that input is now small enough to be neglected). The former is what makes life possible: Without the Sun's steady infusion of fresh, external energy, life as we know it would quickly end. An external source is required because although energy cannot be destroyed, it is constantly degraded into weaker, less useful forms in accordance with the 2nd law of thermodynamics (consider again the hybrid-electric car's brake pads—their dissipated heat is of not much use to anyone). The openness of this system is two-way, because Earth also streams thermal infrared energy back out to space. Its radiation is invisible to us, but to satellites with "vision" in this range of the electromagnetic spectrum the Earth is a brightly glowing orb, much like the Sun.
Interestingly, this closed/open dichotomy is yet another reason why the physics of climate change are unassailable. By burning fossil fuels, we shuffle carbon (mass) out of the subsurface—where it has virtually no interaction with the planet's energy balance—to the atmosphere, where it does. It is well understood that carbon in the atmosphere alters the planet's energy balance (the physics of this have been known since 1893) and without carbon-based and other greenhouse gases our planet would be a moribund, ice-covered rock. Greenhouse gases prevent this by selectively altering the Earth's energy balance in the troposphere (the lowest few miles of the atmosphere, where the vast majority of its gases reside), thus raising the amount of thermal infrared radiation that it emits. Because some of this energy streams back down to Earth (as well as out to space) the lower troposphere warms, to achieve energy balance. Continuity of Energy commands this.
Our planet's carbon atoms, however, are stuck here with us forever—Continuity of Mass commands that too. The real question is what choices will make about how much, and how fast, to shuffle them out the ground. The physics of natural resource stocks, climate change and other problems can often be reduced to simple, elegant, equations—if only we had such masterful tools to dictate their solution.