Category Theory for Scientists

 

At last—a textbook on category theory for scientists! And it’s free!

• David Spivak, Category Theory for Scientists.

It’s based on a course the author taught:

This course is an attempt to extol the virtues of a new branch of mathematics, called category theory, which was invented for powerful communication of ideas between different fields and subfields within mathematics. By powerful communication of ideas I actually mean something precise. Different branches of mathematics can be formalized into categories. These categories can then be connected together by functors. And the sense in which these functors provide powerful communication of ideas is that facts and theorems proven in one category can be transferred through a connecting functor to yield proofs of an analogous theorem in another category. A functor is like a conductor of mathematical truth.

I believe that the language and toolset of category theory can be useful throughout science. We build scientific understanding by developing models, and category theory is the study of basic conceptual building blocks and how they cleanly fit together to make such models. Certain structures and conceptual frameworks show up again and again in our understanding of reality. No one would dispute that vector spaces are ubiquitous. But so are hierarchies, symmetries, actions of agents on objects, data models, global behavior emerging as the aggregate of local behavior, self-similarity, and the effect of methodological context.

Some ideas are so common that our use of them goes virtually undetected, such as set-theoretic intersections. For example, when we speak of a material that is both lightweight and ductile, we are intersecting two sets. But what is the use of even mentioning this set-theoretic fact? The answer is that when we formalize our ideas, our understanding is almost always clarified. Our ability to communicate with others is enhanced, and the possibility for developing new insights expands. And if we are ever to get to the point that we can input our ideas into computers, we will need to be able to formalize these ideas first.

It is my hope that this course will offer scientists a new vocabulary in which to think and communicate, and a new pipeline to the vast array of theorems that exist and are considered immensely powerful within mathematics. These theorems have not made their way out into the world of science, but they are directly applicable there. Hierarchies are partial orders, symmetries are group elements, data models are categories, agent actions are monoid actions, local-to-global principles are sheaves, self-similarity is modeled by operads, context can be modeled by monads.

He asks readers from different subjects for help in finding new ways to apply category theory to those subjects. And that’s the right attitude to take when reading this book. I’ve found categories immensely valuable in my work. But it took effort to learn category theory and see how it can apply to different subjects. People are just starting to figure out these things, so don’t expect instant solutions to the problems in your own favorite field.

But Spivak does the best job I’ve seen so far at explaining category theory as a general-purpose tool for thinking clearly. Since I’m busy using category theory to clarify the relationships between fields like chemistry, population biology, electrical engineering and control theory, this subject is very much on my mind.

Graph Laplacians

There’s been some new progress on graph Laplacians!

As a mathematical physicist, I’ve always been in love with the Laplacian:

   

It shows up in many of the most fundamental equations of physics: the wave equation, the heat equation, Schrödinger’s equation… and Poisson’s equation:

which says how the density of matter, affects the gravitational potential

As I’ve grown interested in network theory, I’ve gotten more and more interested in ‘graph Laplacians’. These are discretized versions of the Laplacian, where we replace 3-dimensional space by a ‘graph’, meaning something like this:

You can get a lot of interesting information about a graph from its Laplacian. You can also set up discretized versions of all the famous equations I mentioned.

The new progress is a simple algorithm for very efficiently solving Poisson’s equation for graph Laplacians:

• Jonathan A. Kelner, Lorenzo Orecchia, Aaron Sidford, Zeyuan Allen Zhu, A simple, combinatorial algorithm for solving SDD systems in nearly-linear time.

Here’s a very clear explanation of the general idea, conveying some sense of why it’s so important, without any nasty equations:

• Larry Hardesty, Short algorithm, long-range consequences, MIT News, 1 March 2013.

It begins:

In the last decade, theoretical computer science has seen remarkable progress on the problem of solving graph Laplacians — the esoteric name for a calculation with hordes of familiar applications in scheduling, image processing, online product recommendation, network analysis, and scientific computing, to name just a few. Only in 2004 did researchers first propose an algorithm that solved graph Laplacians in “nearly linear time,” meaning that the algorithm’s running time didn’t increase exponentially with the size of the problem.

At this year’s ACM Symposium on the Theory of Computing, MIT researchers will present a new algorithm for solving graph Laplacians that is not only faster than its predecessors, but also drastically simpler.

This animation shows two different “spanning trees” for a simple graph, a grid like those used in much scientific computing. The speedups promised by a new MIT algorithm require “low-stretch” spanning trees (green), in which the paths between neighboring nodes don’t become excessively long (red).

I can’t beat this article at its own game… except to clarify that ‘solving graph Laplacians’ means solving Poisson’s equation with a graph Laplacian replacing the usual Laplacian.

So, let me just supplement this article with the nasty equations saying what a graph Laplacian actually is. Start with a graph. More precisely, start with a simple graph. Such a graph has a set of vertices and a set of edges such that

which says the edges are undirected, and

which says there are no loops.

The graph Laplacian is an operator that takes a function on the vertices of our graph,

and gives a new such function as follows:

The version of Poisson’s equation for this graph Laplacian is thus

But I should warn you: this operator has eigenvalues that are less than equal to zero, like the usual Laplacian People often insert a minus sign to make the eigenvalues ≥ 0.

There is a huge amount to say about graph Laplacians! If you want, you can learn more here:

• Michael William Newman, The Laplacian Spectrum of Graphs, Masters Thesis, Department of Mathematics, University of Manitoba, 2000.

I’ve been learning about some of their applications here:

• Ernesto Estrada, The Structure of Complex Networks: Theory and Applications, Oxford University Press, Oxford, 2011.

I hope sometime to summarize a bit of this material and push the math forward a bit. So, it was nice to see new progress on efficient algorithms for doing computations with graph Laplacians!

The Search For Budget-Conscious Life

 

Lisa and I had dinner with Gregory Benford and his wife when I visited U.C. Irvine a couple of weekends ago, and he raised an interesting point. So far, radio searches for extraterrestrial life have only seen puzzling brief signals – not long transmissions. But what if this is precisely what we should expect?

A provocative example is Sullivan, et al. (1997). This survey lasted about 2.5 hours, with 190 1.2 minute integrations. With many repeat observations, they saw nothing that did not seem manmade. However, they “recorded intriguing, non-repeatable, narrowband signals, apparently not of manmade origin and with some degree of concentration toward the galactic plane…” Similar searches also saw one-time signals, not repeated (Shostak & Tarter, 1985; Gray & Marvel, 2001 Gray, 2001). These searches had slow times to revisit or reconfirm, often days (Tarter, 2001). Overall, few searches lasted more than hour, with lagging confirmation checks (Horowitz & Sagan, 1993). Another striking example is the “WOW” signal seen at the Ohio SETI site…

That’s a quote from a paper Benford wrote with his brother and nephew:

• Gregory Benford, James Benford, and Dominic Benford, Searching for cost optimized interstellar beacons.

They claim the cheapest way a civilization could communicate to lots of planets is a pulsed, broadband, narrowly focused microwave beam that scans the sky. So, for anyone receiving this signal, there would be a lot of time between pulses. That might explain some of the above mysteries, or this one:

As an example of using cost optimized beacon analysis for SETI purposes, consider in detail the puzzling transient bursting radio source, GCRT J17445-3009, which has extremely unusual properties. It was discovered in 2002 in the direction of the Galactic Center (1.25° south of GC) at 330 MHz in a VLA observation and subsequently re-observed in 2003 and 2004 in GMRT observations (Hyman, 2005, 2006, 2007). It is a pulsed coherent source, with the ‘burst’ lasting as much as 10 minutes, with 77-minute period. Averaged over all observations, Hyman et al. give a duty cycle of 7% (1/14), although since some observations may have missed part of bursts, the duty cycle might be as high as 13%.

Even if these are red herrings, it seems very smart to figure out the cheapest ways to transmit signals and use that to guess what signals we should look for. We can easily make the mistake of assuming all extraterrestrial civilizations who bother to send signals through space will be willing to beam signals of enormous power toward us all the time. That could be true of some, but not necessarily all.

The cost analysis is here:

• James Benford, Gregory Benford, Dominic Benford, Messaging with cost optimized interstellar beacons.

and you can see a summary in this talk by Gregory’s brother James, who works on high-power microwave technologies:

Quantum Techniques for Studying Equilibrium in Reaction Networks

 

The summer before last, I invited Brendan Fong to Singapore to work with me on my new ‘network theory’ project. He quickly came up with a nice new proof of a result about mathematical chemistry. We blogged about it, and I added it to my book, but then he became a grad student at Oxford and got distracted by other kinds of networks—namely, Bayesian networks.

So, we’ve just now finally written up this result as a self-contained paper:

• John Baez and Brendan Fong, Quantum techniques for studying equilibrium in reaction networks.

Check it out and let us know if you spot mistakes or stuff that’s not clear!

The idea, in brief, is to use math from quantum field theory to give a somewhat new proof of the Anderson–Craciun–Kurtz theorem.

This remarkable result says that in many cases, we can start with an equilibrium solution of the ‘rate equation’ which describes the behavior of chemical reactions in a deterministic way in the limit of a large numbers of molecules, and get an equilibrium solution of the ‘master equation’ which describes chemical reactions probabilistically for any number of molecules.

The trick, in our approach, is to start with a chemical reaction network, which is something like this:

and use it to write down a Hamiltonian describing the time evolution of the probability that you have various numbers of each kind of molecule: A, B, C, D, E, … Using ideas from quantum mechanics, we can write this Hamiltonian in terms of annihilation and creation operators—even though our problem involves probability theory, not quantum mechanics! Then we can write down the equilibrium solution as a ‘coherent state’. In quantum mechanics, that’s a quantum state that approximates a classical one as well as possible.

All this is part of a larger plan to take tricks from quantum mechanics and apply them to ‘stochastic mechanics’, simply by working with real numbers representing probabilities instead of complex numbers representing amplitudes!

I should add that Brendan’s work on Bayesian networks is also very cool, and I plan to talk about it here and even work it into the grand network theory project I have in mind. But this may take quite a long time, so for now you should read his paper:

• Brendan Fong, Causal theories: a categorical perspective on Bayesian networks.

Localizing and Networking Basic Technology

guest post by Iuval Clejan

Natural philosophy (aka science) is distinguished from pure philosophy or mathematics by coupling theory to experiment. Engineering is distinguished from science in its focus on solving practical problems rather than merely coming up with more accurate models of the universe. Climate change will not be fixed by pure philosophy or argumentation. We need to use the methods of science and engineering to make progress towards a solution. The problem is complicated and involves not just climate dynamics and ecology, but psychology, economics and technology. Besides theory and experiment, we now have the tool of simulation. I propose a think-tank (or more properly, a think/do/simulate-tank) analogous to the Manhattan Project, which developed the first atomic bomb. However, this project would involve social and physical scientists, computer programmers, engineers, farmers and craftspeople who are trying to collaboratively solve the problem of how to provide food, shelter, water, clothes, medicine and recreation for a self contained village in a sustainable way. Sustainability has psychological dimensions, not just ecological. For example, it implies that people would want to keep living in this village, or similar villages. If we are interested in sustainability beyond the initial village, then sustainability implies replicability—that the village would inspire many other people to live similarly.

Initial outputs of this project would be well-founded suggestions regarding what kinds of production skills are needed and how to effectively network them, how many people, how much land, how much time spent on production in order to achieve village-scale independence and sustainability. An eventual outcome would be an actual demonstration of a functional village.

Why village? The word village is used here to mean a group of people who are economically networked in isolation from the rest of the global economy. It also implies choosing a particular geographic location, so not all outputs would be transferable to other locations, though with the initial simulation stage many locations could be tried.

Why economic isolation? Without putting a boundary on the experiment, the problem is too complex, even for simulation. Entropy reduction is the same reason cells have membranes and scientists have labs. The membrane could be permeable to sunlight, wind (and emissions) and water, but at first it might be simpler to keep it impermeable to economic exchange. In addition, it is easy to externalize all unsustainable practices without a membrane. But the size of the membrane is not predetermined. One possible conclusion might be that the village has to be the size of the whole earth. Another reason for starting with a village is that changes in biological (and probably other complex) systems always proceed from small populations that can spread out by replication. It is more practical to achieve a global change in lifestyle and technology starting with a small group of willing people who can then inspire others by example, rather than try to impose a change on a large population, the way fascist and communist experiments have proceeded. Another reason for keeping things smaller and more local is that a stronger feedback between production and consumption may arise, which would regulate unsustainable consumption, because the environmental, social and psychological costs of production are visible in the village, as opposed to hidden or abstracted from the consumers. There are other reasons for localization (e.g. resilience, freedom, more meaningful employment for more people, better relations among people or between people and nature), less directly related to climate change, and more speculative.

This is probably the place to admit my main bias. I am a Gandhist Luddite (who has a PhD in Physics, worked as a semiconductor engineer and a molecular biologist) , not the angry, machine-smashing kind, and I like not only to tinker with technology, but to think how it affects people and nature. I don’t think all technology can be equated with progress. I call this project the Luddite Manhattan Project (or or Localizing and Networking Basic Technology project) for that reason and because it parallels the project that produced the nuclear bomb. I think that the craftspeople and farmers would contribute more to this project than the scientists and engineers. I think that in the multidimensional optimization of technology, we have focused too much on efficiency (disregarding other human values) and that the industrial revolution was largely a mistake (though some good things came out of it, like global communication). If we focus on other human values, we can optimize technology better. I think that localism of basic-needs production (when coupled to non-technological things like democracy) is a constraint from which many other good things such as sustainability, full, meaningful employment, freedom, and good social relations would follow, though it too can be taken to extremes. Given my bias, I suspect that the kind of technology network that would be most sustainable would be pre-industrial, with a few modern innovations. If we really did the book-keeping accurately we would probably find that industrial production is unsustainable. Or rather we would find that pre-industrial production can be sustainable, while current industrial production is not (I leave open the possibility that industrial production might be sustainable in the future, with new innovations, but even then it tramples too many human values). But these conclusions would be outputs of the project, not pre-assumptions or inputs of the project. I welcome some discussion of these ideas, followed by computation, testing and implementation.

The technical part of the project is basically a networking problem. It would allow initial imports (in a way that would allow replicability—that is don’t hog a disproportionate fraction of resources into the village) into a specific location and then network existing technologies so that the system is self-sustaining. What one craftsperson produces, others in the village must use so that the village can continue in perpetuity. A blacksmith needs some fuel, but also customers who need his products and can exchange stuff that he needs. A cooper is mostly useless in the current industrial economy, but would probably find some use in a local village economy, where people need ways to store water and other liquids.

Here are some typical challenges and questions the project would face: How can antibiotics be made on a village scale with no external inputs? What can’t be made and can we find substitutes? Are there missing technology links and can we invent them, or do we need to start with another scenario? What food needs to be produced to provide basic caloric needs to all inhabitants of the village? How much area is required? How can water be captured and transported without plastic or rubber? How much carbon is emitted in production of everything? Where does garbage go? How can metals be recycled? Can plastic be produced? Can electronics be produced? Is there enough time for art, science, scholarship and other forms of edifying human activity? What kind of economic systems work? Is there an optimal one as far as sustainability, or is it a matter of personal preference? These are all questions that can be tackled, if we face them with curiosity and realism, instead of with fear and the kind of magical thinking that most people have towards technology and other things they don’t understand. I’ve heard that Leonardo Da Vinci was the last man to understand the technology of his age, but we have computers to help us.

It might be appropriate at this stage to mention that I do not advocate giving up entirely the industrial mode of production, or the global trade it requires. The Localizing and Networking Basic Technology project would address only food, shelter, water, medicine, all the subsidiary crafts necessary to sustain these, and a few edifying human activities like art, music and scholarship. Computers and internet hardware are almost certainly best left to industrial production, and so are cars, airplanes (but the need for these will drastically decrease if this project is successful), some of the parts for particle accelerators and fancy biotech equipment, etc.

The initial computational stage of the project could model itself on online multiplayer games like Warcraft and planning games like Sim City (I have tried to contact Will Wright, to no avail). I do not play these games (I prefer simple low tech games personally), but I see the usefulness of online collaboration and computation for this project, as a sort of in-silico evolution. Programmers and mathematicians could set up the software to allow both online collaboration and some central planning. I think the simplest solutions should be tried first, i.e. the most primitive technologies, like hunting and gathering. My educated guess is that they will be shown incapable of providing basic needs given the current world population. The same conclusion would probably follow for current industrial production, except the incapacity would be with regards to sustainability. I predict the sweet spot where both sustainability and capacity to “feed the world” (meaning provide a decent life) would be achieved by pre-industrial, agrarian and craft-based production.

I am totally willing to be proven wrong by this experiment about my anti-industrialization bias. With regards to scientific experimentation, there needs to be well posed hypotheses that can be proven wrong, and good controls. The engineering approach is an alternative. Who is willing to work on this project? Let’s make amends for unleashing the horror of the Bomb on the earth, tackle climate change realistically and have some technical fun. For further information please see:

• Iuval Clejan, Luddite Manhattan Project, first stage, 16 April 2012.

• Iuval Clejan, A proposal for funding a blueprint of a village-based technology ecosystem, 5 February 2012.

The Foundations of Applied Mathematics

Suppose we take “applied mathematics” in an extremely broad sense that includes math developed for use in electrical engineering, population biology, epidemiology, chemistry, and many other fields. Suppose we look for mathematical structures that repeatedly appear in these diverse contexts — especially structures that aren’t familiar to pure mathematicians. What do we find? The answers may give us some clues about how to improve the foundations of mathematics!

This is what I’m talking about at the Category-Theoretic Foundations of Mathematics Workshop at U.C. Irvine this weekend.

You can see my talk slides here. You can click on any picture or anything written in blue in these slides to get more information — for example, references.

Bridging the Greenhouse-Gas Emissions Gap

I could use some help here, finding organizations that can help cut greenhouse gas emissions. I’ll explain what I mean in a minute. But the big question is:

How can we bridge the gap between what we are doing about global warming and what we should be doing?

That’s what this paper is about:

• Kornelis Blok, Niklas Höhne, Kees van der Leun and Nicholas Harrison, Bridging the greenhouse-gas emissions gap, Nature Climate Change 2 (2012), 471-474.

According to the United Nations Environment Programme, we need to cut CO₂ emissions by about 12 gigatonnes/year by 2020 to hold global warming to 2 °C.

After the UN climate conference in Copenhagen, many countries made pledges to reduce CO₂ emissions. But by 2020 these pledges will cut emissions by at most 6 gigatonnes/year. Even worse, a lot of these pledges are contingent on other people meeting other pledges, and so on… so the confirmed value of all these pledges is only 3 gigatonnes/year.

The authors list 21 things that cities, large companies and individual citizens can do, which they claim will cut greenhouse gas emissions by the equivalent of 10 gigatonnes/year of CO₂ by 2020. For each initiative on their list, they claim:

(1) there is a concrete starting position from which a significant up-scaling until the year 2020 is possible;

(2) there are significant additional benefits besides a reduction of greenhouse-gas emissions, so people can be driven by self-interest or internal motivation, not external pressure;

(3) there is an organization or combination of organizations that can lead the initiative;

(4) the initiative has the potential to reach an emission reduction by about 0.5 Gt CO₂e by 2020.

21 Initiatives

Now I want to quote the paper and list the 21 initiatives. And here’s where I could use your help! For each of these, can you point me to one or more organizations that are in a good position to lead the initiative?

Some are already listed, but even for these I bet there are other good answers. I want to compile a list, and then start exploring what’s being done, and what needs to be done.

By the way, even if the UN estimate of the greenhouse-emissions gap is wrong, and even if all the numbers I’m about to quote are wrong, most of them are probably the right order of magnitude—and that’s all we need to get a sense of what needs to be done, and how we can do it.

Companies

1. Top 1,000 companies’ emission reductions. Many of the 1,000 largest greenhouse-gas-emitting companies already have greenhouse-gas emission-reduction goals to decrease their energy use and increase their long-term competitiveness, as well as to demonstrate their corporate social responsibility. An association such as the World Business Council for Sustainable Development could lead 30% of the top 1,000 companies to reduce energy-related emissions 10% below business as usual by 2020 and all companies to reduce their non-carbon dioxide greenhouse-gas emissions by 50%. Impact in 2020: up to 0.7 Gt CO₂e.

2. Supply-chain emission reductions. Several companies already have social and environmental requirements for their suppliers, which are driven by increased competitiveness, corporate social responsibility and the ability to be a front-runner. An organization such as the Consumer Goods Forum could stimulate 30% of companies to require their supply chains to reduce emissions 10% below business as usual by 2020. Impact in 2020: up to 0.2 Gt CO₂e.

3. Green financial institutions. More than 200 financial organizations are already members of the finance initiative of the United Nations Environment Programme (UNEP-FI). They are committed to environmental goals owing to corporate social responsibility, to gain investor certainty and to be placed well in emerging markets. UNEP-FI could lead the 20 largest banks to reduce the carbon footprint of 10% of their assets by 80%. Impact in 2020: up to 0.4 Gt of their assets by 80%. Impact in 2020: up to 0.4 Gt CO₂e.

4. Voluntary-offset companies. Many companies are already offsetting their greenhouse-gas emissions, mostly without explicit external pressure. A coalition between an organization with convening power, for example UNEP, and offset providers could motivate 20% of the companies in the light industry and commercial sector to calculate their greenhouse-gas emissions, apply emission-reduction measures and offset the remaining emissions (retiring the purchased credits). It is ensured that offset projects really reduce emissions by using the ‘gold standard’ for offset projects or another comparable mechanism. Governments could provide incentives by giving tax credits for offsetting, similar to those commonly given for charitable donations. Impact by 2020: up to 2.0 Gt CO₂e.

Other actors

5. Voluntary-offset consumers. A growing number of individuals (especially with high income) already offset their greenhouse-gas emissions, mostly for flights, but also through carbon-neutral products. Environmental NGOs could motivate 10% of the 20% of richest individuals to offset their personal emissions from electricity use, heating and transport at cost to them of around US$200 per year. Impact in 2020: up to 1.6 Gt CO₂e.

6. Major cities initiative. Major cities are large emitters of greenhouse gases and many have greenhouse-gas reduction targets. Cities are intrinsically highly motivated to act so as to improve local air quality, attractiveness and local job creation. Groups like the C40 Cities Climate Leadership Group and ICLEI — Local Governments for Sustainability could lead the 40 cities in C40 or an equivalent sample to reduce emissions 20% below business as usual by 2020, building on the thousands of emission-reduction activities already implemented by the C40 cities. Impact in 2020: up to 0.7 Gt CO₂e.

7. Subnational governments. Several states in the United States and provinces in Canada have already introduced support mechanisms for renewable energy, emission-trading schemes, carbon taxes and industry regulation. As a result, they expect an increase in local competitiveness, jobs and energy security. Following the example set by states such as California, these ambitious US states and Canadian provinces could accept an emission-reduction target of 15–20% below business as usual by 2020, as some states already have. Impact in 2020: up to 0.6 Gt CO₂e.

Energy efficiency

8. Building heating and cooling. New buildings, and increasingly existing buildings, are designed to be extremely energy efficient to realize net savings and increase comfort. The UN Secretary General’s Sustainable Energy for All Initiative could bring together the relevant players to realize 30% of the full reduction potential for 2020. Impact in 2020: up to 0.6 Gt CO₂e.

9. Ban of incandescent lamps. Many countries already have phase-out schedules for incandescent lamps as it provides net savings in the long term. The en.lighten initiative of UNEP and the Global Environment Facility already has a target to globally ban incandescent lamps by 2016. Impact in 2020: up to 0.2 Gt CO₂e.

10. Electric appliances. Many international labelling schemes and standards already exist for energy efficiency of appliances, as efficient appliances usually give net savings in the long term. The Collaborative Labeling and Appliance Standards Program or the Super-efficient Equipment and Appliance Deployment Initiative could drive use of the most energy-efficient appliances on the market. Impact in 2020: up to 0.6 Gt CO₂e.

11. Cars and trucks. All car and truck manufacturers put emphasis on developing vehicles that are more efficient. This fosters innovation and increases their long-term competitive position. The emissions of new cars in Europe fell by almost 20% in the past decade. A coalition of manufacturers and NGOs joined by the UNEP Partnership for Clean Fuels and Vehicles could agree to save one additional liter per 100 km globally by 2020 for cars, and equivalent reductions for trucks. Impact in 2020: up to 0.7 Gt CO₂e.

Energy supply

12. Boost solar photovoltaic energy. Prices of solar photovoltaic systems have come down rapidly in recent years, and installed capacity has increased much faster than expected. It created a new industry, an export market and local value added through, for example, roof installations. A coalition of progressive governments and producers could remove barriers by introducing good grid access and net metering rules, paving the way to add another 1,600 GW by 2020 (growth consistent with recent years). Impact in 2020: up to 1.4 Gt CO₂e.

13. Wind energy. Cost levels for wind energy have come down dramatically, making wind economically competitive with fossil-fuel-based power generation in many cases. The Global Wind Energy Council could foster the global introduction of arrangements that lead to risk reduction for investments in wind energy, with, for example, grid access and guarantees. This could lead to an installation of 1,070 GW by 2020, which is 650 GW over a reference scenario. Impact in 2020: up to 1.2 Gt CO₂e.

14. Access to energy through low-emission options. Strong calls and actions are already underway to provide electricity access to 1.4 billion people who are at present without and fulfill development goals. The UN Secretary General’s Sustainable Energy for All Initiative could ensure that all people without access to electricity get access through low-emission options. Impact in 2020: up to 0.4 Gt CO₂e.

15. Phasing out subsidies for fossil fuels. This highly recognized option to reduce emissions would improve investment in clean energy, provide other environmental, health and security benefits, and generate income. The International Energy Agency could work with countries to phase out half of all fossil-fuel subsidies. Impact in 2020: up to 0.9 Gt CO₂e.

Special sectors

16. International aviation and maritime transport. The aviation and shipping industries are seriously considering efficiency measures and biofuels to increase their competitive advantage. Leading aircraft and ship manufacturers could agree to design their vehicles to capture half of the technical mitigation potential. Impact in 2020: up to 0.2 Gt CO₂e.

17. Fluorinated gases (hydrofluorocarbons, perflourocarbons, SF6). Recent industry-led initiatives are already underway to reduce emissions of these gases originating from refrigeration, air-conditioning and industrial processes. Industry associations, such as Refrigerants, Naturally!, could work towards meeting half of the technical mitigation potential. Impact in 2020: up to 0.3 Gt CO2e.

18. Reduce deforestation. Some countries have already shown that it is strongly possible to reduce deforestation with an integrated approach that eliminates the drivers of deforestation. This has benefits for local air pollution and biodiversity, and can support the local population. Led by an individual with convening power, for example, the United Kingdom’s Prince of Wales or the UN Secretary General, such approaches could be rolled out to all the major countries with high deforestation emissions, halving global deforestation by 2020. Impact in 2020: up to 1.8 Gt CO₂e.

19. Agriculture. Options to reduce emissions from agriculture often increase efficiency. The International Federation of Agricultural Producers could help to realize 30% of the technical mitigation potential. (Well, at least it could before it collapsed, after this paper was written.) Impact in 2020: up to 0.8 Gt CO2e.

Air pollutants

20. Enhanced reduction of air pollutants. Reduction of classic air pollutants including black carbon has been pursued for years owing to positive impacts on health and local air quality. UNEP’s Climate and Clean Air Coalition To Reduce Short-Lived Climate Pollutants already has significant political momentum and could realize half of the technical mitigation potential. Impact in 2020: a reduction in radiative forcing impact equivalent to an emission reduction of greenhouse gases in the order of 1 Gt CO₂e, but outside of the definition of the gap.

21. Efficient cook-stoves. Cooking in rural areas is a source of carbon dioxide emissions. Furthermore, there are emissions of black carbon, which also leads to global warming. Replacing these cook-stoves would also significantly increase local air quality and reduce pressure on forests from fuel-wood demand. A global development organization such as the UN Development Programme could take the lead in scaling-up the many already existing programs to eventually replace half of the existing cook-stoves. Impact in 2020: a reduction in radiative forcing impact equivalent to an emission reduction of greenhouse gases of up to 0.6 Gt CO₂e, included in the effect of the above initiative and outside of the definition of the gap.

For more

For more, see the supplementary materials to this paper, and also:

• Niklas Höhne, Wedging the gap: 21 initiatives to bridge the greenhouse gas emissions gap.

The size of the emissions gap was calculated here:

• The Emissions Gap Report 2012, United Nations Environment Programme (UNEP).

If you’re in a rush, just read the executive summary.