On Sunday, an estimated 300,000 people gathered in New York City to demand action on climate change. It was an impressive crowd — possibly the largest US rally on any issue since the Iraq war protests in 2003.

Global climate talks are still bogged down by a deep dispute between rich and poor countries

But now comes the hard part.

On Tuesday, representatives from around the world will meet in New York to kick off a year-long negotiation over a new climate treaty. By the end of 2015, they hope to hammer out an agreement with "legal force" that commits all nations to reductions in heat-trapping greenhouse gases.

The main obstacle here is that there are still deep, deep divisions among different countries about how best to tackle global warming. Poorer countries argue that the US, Europe, and other rich nations are responsible for most of the extra carbon-dioxide in the atmosphere today, so they should bear most of the burden for addressing global warming. Richer countries, by contrast, say that you also have to look at future emissions when allocating blame — so fast-growing nations like China and India need to do more.

This core dispute pops up everywhere, from arguments over how much, exactly, each country should reduce emissions to disputes over climate aid. (Since 2010, wealthy nations have offered $35 billion in aid to help poorer countries adapt to the effects of climate change, but additional promised money has been slow to arrive.)

We can also break this dispute down in graph form, with the help of the Global Carbon Project's excellent new report on global emissions (which is worth exploring in full). Here are the 7 key facts:

1) T he world is totally failing to meet its own climate goals

Let's start with the big picture on climate change. The black line above shows the current growth of annual carbon-dioxide emissions worldwide. The colored lines show various future pathways and what they'd mean for climate change.

We're currently on pace for somewhere between 5.8°F and 9.7°F of warming

The world's nations have all agreed that it would be "dangerous" to allow global average temperatures to rise more than 2°C (or 3.6°F) above pre-industrial levels. (See here for why.) But to have a decent chance of staying below 2°C, global emissions would likely have to follow the blue or yellow trajectories — peaking in the very near future and declining sharply.

Instead, emissions keep rising. And if emissions continue on their present course, scientists estimate, we can expect somewhere between 3.2°C and 5.4°C of warming by the end of the century (that's between 5.8°F and 9.7°F). Various reports have warned that this would entail drastic and irreversible changes, like destabilizing Greenland's ice sheet or large-scale extinctions. The World Bank, for one, thinks this much warming could be impossible for many countries to adapt to.

Now, granted, countries like the United States, Europe, and China have all made various pledges to reduce their emissions in the future. But even if you take all those pledges seriously, analysts at the Climate Action Tracker have found that we're still on pace for between 3°C and 4.6°C of warming by century's end. That's not much better. (The range is largely due to uncertainty over exactly how the climate would respond to so much carbon — but scientists are quite confident that it would get a lot hotter.)

2) We're on pace to blow through our "carbon budget" in 20 to 30 years

Here's another way to look at the climate issue. If we want to avoid more than 2°C of global warming, scientists estimate, then humans can only put about 3,200 gigatons of carbon dioxide into the atmosphere. (This gives us a two-thirds chance of staying below the limit.)

We're about two-thirds of the way through our carbon budget

Since the Industrial Revolution, we've loaded roughly 2,000 gigatons of carbon dioxide into the atmosphere. So we only have about 1,200 gigatons left. And at current growth rates, we'll blow through that in about two or three decades.

So that's our "carbon budget." Meeting the 2°C goal would, in theory, entail divvying up the remaining 1,200 gigatons among various countries. Country A gets to emit this much carbon. Country B gets to emit this much carbon. And so on. But that's incredibly hard to do. For one, setting these sorts of top-down goals hasn't had much success to date. And, as we'll see below, allocating responsibility isn't easy.

By the way, the Global Carbon Project estimates that we've already "committed" to using up 50 percent of the carbon budget with our existing infrastructure. All the coal plants we've already built and cars we've already bought are expected to last for years or decades. If they're used as intended, that will use up about half of the budget.

3) The US and Europe are responsible for much of the CO2 already in the atmosphere...

Now comes the blame game. The chart above shows cumulative emissions from fossil fuels and cement over time. For any given year, it more or less shows who is responsible for what fraction of the carbon-dioxide in the atmosphere.

The US and Europe have had their fossil fuel party, now India and China want theirs

As you can see, the United States and Europe are responsible for 49 percent of all carbon emissions from fossil fuels and cement that have been emitted since 1870. They've long enjoyed the growth benefits from using fossil fuels — and now other countries want theirs. This chart also explains why many people put the blame for current global warming squarely on the United States and Europe and are asking for aid to adapt.

But the US and Europe, by contrast, argue that the situation is now shifting rapidly. If you only look since 1990, China has been responsible for 20 percent of the cumulative emissions in the atmosphere — more than Europe (14 percent) and on par with the United States (20 percent). That's why wealthier nations often point instead to charts like the one below:

4) ...But nowadays it's developing countries who emit the most carbon

The previous climate treaty — the Kyoto Protocol of 1997 — divided the world into two sections: wealthy nations ("Annex B") and developing countries ("non-Annex B"). The former were supposed to cut their emissions. The latter were allowed a free pass so that they could keep growing.

Emissions from wealthy nations are declining

As the chart above shows, that more or less happened. Annual emissions from Europe, the United States, and other wealthy "Annex B" countries are declining, albeit slowly. But emissions from developing countries have exploded.

Today, developing countries produce 58 percent of the world's annual carbon-dioxide emissions. China alone is responsible for 27 percent. This is why negotiators from wealthy countries often insist that China and other poorer countries need to commit to reductions in any new treaty.

5) China's per capita emissions are now higher than the EU's

Here's another chart on the same theme. China's overall emissions are currently higher than those of the US or Europe. But Chinese negotiators have long argued that that's only fair — after all, China has 1.3 billion people, so of course it's emissions will be higher.

But now China is crossing a different threshold. It's per capita emissions are actually higher than those in Europe. (They are still far below America's, although America's have been dropping sharply.)

By the way, that chart above also shows why India is so reluctant to heed calls to reduce its emissions — especially when its per-capita emissions are still so low and 300 million people still have no access to electricity.

6) That's partly because rich nations are "outsourcing" their carbon

Here's another tricky issue. Emissions can also be "outsourced" abroad. Say, for instance, a US factory moves to China and produces goods that are then shipped back to the United States. America's emissions decrease. China's emissions increase. But who's responsible for that carbon, really?

This isn't a trivial issue. The Global Carbon Budget 2014 report notes that virtually all of the reductions in emissions made by wealthy countries like the US and Europe since 1990 have been offset by "outsourced" emissions to places like China. These emissions transfers are now growing at a rate of 11 percent per year.

7) Divvying up the remaining carbon budget is... complicated

So now, if we want to stay below the 2°C limit, negotiators have to take all of the factors above and come up with some way to divvy up the remaining carbon budget.

As this chart from the Global Carbon Project shows, there are lots of ways to do this. Negotiators could agree to let everyone maintain their current share of emissions (an "inertia"). The United States emits 18 percent of the world's emissions today and gets 18 percent of the remaining carbon budget. But that's not terribly fair, since India would get penalized for being poor — it would miss out on the growth benefits of burning fossil fuels.

Alternatively, you could take an "equity" approach. India is a big and poor country in dire need of more growth. So it should be allowed to take up a bigger share of the carbon budget. The United States and Europe get a much smaller share, by contrast — they need to enact draconian emissions cuts. The problem is that it's not clear whether this is even technically feasible.

A third approach, described here, is a "blended" approach that tries to compromise on all these different issues. A recent paper in Nature Climate Change argued that this was the best way to stay below the carbon budget. Whether national governments agree with that, of course, is another story.

More realistically, the world's nations will set their own individual goals based on what they each think makes the most sense for themselves. Most analysts think this is the most likely outcome of the current UN talks — each nation sets voluntary goals for themselves and the new treaty sets up some sort of monitoring and verification mechanism.

Now, whether all those voluntary pledges add up to staying within the carbon budget and below 2°C is much less clear. Which brings us all the way back to chart #1 up top…

Further reading

Past UN climate talks have failed. Will this one be any different?

An in-depth look at the 2°C climate goal — and what happens if it's no longer feasible

Here's what the world would look like if we took global warming seriously