The Prime Minister will meet Vladimir Putin for the first time at the G20 summit in China. Faced with the prospect of a Hillary Clinton presidency in the US, Putin will be keener than might be expected to establish a positive relationship.
He will, as ever, be formidably well briefed and direct. He will appreciate, and reciprocate, Mrs May's forensic, no-nonsense style. But he will react sharply to any challenge on Russia's key concerns.
His brittleness comes from his conviction that the vastly stronger West is out to do Russia down. Even so, he probably feels that, currently, things are moving his way.
China, where they meet, is warming relations with Russia even as they deteriorate with the West. Russia is increasingly setting the agenda on Syria. Putin can live with what he has got in Ukraine, while awaiting the erosion of Western sanctions. Russia's economy seems to be recovering from the oil shock. He faces awkward parliamentary elections, but knows there is no real threat to his survival as president.
Putin will not expect dramatic results from this meeting. He knows that the UK has been amongst the toughest in the EU on relations with Russia, and views us as something of a US clone on international security issues (which is why he lost interest in his relationship with Tony Blair). He will be interested to hear about Brexit, on which his feelings are mixed (the EU is Russia's largest trading partner). A characteristic joke would be an offer of advice on becoming a proud, free-standing, nation state.
But his main aim will be to warm up the strikingly cold UK-Russia relationship left by Mrs May's predecessor. This would include more high-level political contact and some strengthening of trade and investment links (he would no doubt like to sell us more gas if we pull out of the Hinkley Point nuclear power project), as well as the security links cut off after the Litvinenko murder.
Mrs May too will not have exaggerated expectations. As Home Secretary during the Litvinenko inquiry, she will have no illusions about Russian ruthlessness. And she will have been fully briefed on Russia's current military posturing. She might want to break the ice with a reference to last week's anniversary of the Arctic convoys, which the Russians still deeply appreciate. But she will need to be clear on our support for Nato's increased preparedness in Europe.
On Ukraine, her message that Russia must observe the Minsk agreement would have more effect if she could undertake to say the same to the Ukrainians. And on Syria she should acknowledge that Russian concerns about militant Islamism closely match our own, and we should strengthen our co-operation accordingly.
But the key point for her to make is that the current escalation of Russia-Nato tensions is in no one's long-term interests. We should be looking together for ways to reverse it.