When Presidents Xi Jinping and Barack Obama meet in the United States in a few days time, they will not be short of things to talk about. The state of the global economy, trade barriers, the much touted but hard to finalize bilateral investment treaty, climate change, allegations of computer hacking and the South China Sea issue all need to be addressed. And they should really try to find time to discuss events in Iraq, Afghanistan, Syria and other hotspots as well. The perennial subject of Taiwan is likely to get a mention, as is the proposed legislation on how to treat foreign NGOs. Such an agenda does not leave much room for even a word on Hong Kong - which may be no bad thing.

Apart from the obvious differences between a Chinese communist and the African American head of the world's most important capitalist country, the two are at different stages of their careers. Obama is approaching the last act, and already has notched up quite an impressive list of achievements. He has pulled military forces out of Iraq to correct the biggest blunder of his predecessor, retreated from the folly of Afghanistan, secured a historic deal over Iran's nuclear program and restored ties with Cuba. Domestically, he has presided over a long period of steady economic growth, given millions of Americans health coverage for the first time and lifted the threat of deportation hanging over the heads of millions of long-stay illegal immigrants. Not a bad record, but it would be nice to top it off with a respectable accommodation with China.

Xi on the other hand is closer to the beginning of what is expected to be a full 10 years holding all the most senior positions in the Party, the government and the military. Most of his major challenges - and potential achievements - lie ahead. The rebalancing of the economy, getting to grips with environmental issues, deciding at what stage he can consider the corruption epidemic contained - these are all problems of the highest order and none can be ducked. Xi needs to be publicly accepted as leader of an important country in the world, with the largest population and second-largest economy. Appearances matter, so the 21-gun salute on the South Lawn of the White House should help.

The usual procedure is that separate teams of officials from both sides work hard before the summit itself trying to agree on statements on each of the major issues. Ideally some of these represent actual substantive agreements, or agreements in principle with a commitment to follow up on the details later. No doubt we will see some sort of pledge to keep the world economy growing and a renewed effort to lower trade barriers. Probably the best hope for a high-profile specific agreement will be on the environment, given the importance both leaders attach to this. For China pollution has now reached critical levels, and in the US the White House is reviewing what tools it has in its regulatory armoury to reduce CO2 levels. Meanwhile, the two might congratulate each other on their respective roles in the Iran nuclear deal.

In other areas where reaching agreement is harder, there may just be some sentences in the final communiqu to say that the subjects were raised and each side reflected its concerns to the other. That might be where Hong Kong gets a mention, if at all.

Two matters which might not make the final statement are the creation by China of the Asian Infrastructure Investment Bank, and the US attempt to negotiate a major trade deal that goes under the name of Trans-Pacific Partnership. Much of the impetus for the former came about because the US Congress declined to approve changes to the International Monetary Fund that would have enhanced the position of China and other developing economies. Congress thought it had the whip hand, Beijing proved otherwise.

The summit is best seen in its full context. The US is still by far the world's biggest economy and its preeminent military power. But it has learned from experience that these two things do not necessarily enable the country to achieve its objectives acting unilaterally. As Obama realises (though his Republican rivals continue to deny it), this will often mean the only way to move things forward will be to work together with other countries. That will involve diplomacy and require compromise, not qualities for which the US has been famous in recent times. China is a rising power, and it is natural for the country to flex its muscles to show that the century of humiliation is well and truly buried in history. In doing so, it would be wise to note the lessons the US has learned the hard way.

If the US and China are at loggerheads, there will not be much progress. If they join together, they can lead the world in tackling its many problems.