Speaking from a podium in Sharm el-Sheikh, where other world leaders met to hold discussions on Gaza's future, Mr Trump said: "With the historic agreement we've just signed, those prayers of millions have finally been answered."

He added: "We've achieved what everybody said was impossible, at long last, we have peace in the Middle East."

Mr Trump, Egyptian President Abdel Fattah el-Sissi, Turkish President Recep Tayyip Erdogan and Qatari Emir Tamim bin Hamad Al Thani had earlier signed a document the US president said would lay the groundwork for Gaza's future.

However, a copy was not made public.

Mr Trump flew to Sharm el-Sheikh from Jerusalem, where he got a rapturous welcome and at least 22 standing ovations as he addressed Israel's Knesset (parliament) - but was also heckled by some members.

The US president, who was hailed by Israeli Prime Minister Benjamin Netanyahu as Israel's "greatest friend", said "after so many years of unceasing war and endless danger, today the skies are calm, the guns are silent and the sirens are still".

He added: "It's the start of a grand concord and lasting harmony for Israel and all the nations of what will soon be a truly magnificent region."

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It comes after Israel and Hamas last week agreed to the first phase of a long-sought ceasefire deal aimed at bringing the devastating war in Gaza to an end after two years of bloodshed.

Israel's war in Gaza, launched following the killing of 1,200 people and capture of 251 more by Hamas during its attacks on 7 October 2023, saw more than 67,000 Gazans killed, according to Palestinian health officials.

Its figures do not differentiate between civilians and combatants but it says around half of those killed were women and children.

On Monday, Hamas freed all 20 living hostages, but only released the bodies of four dead hostages.

As part of Mr Trump's peace deal, Hamas was given 72 hours (until 10am on Monday) to return the 48 hostages left in Gaza - 20 of whom were known to be alive and 28 who have died.

Hamas previously indicated that recovering the bodies of some dead hostages may take longer, as not all burial sites are known.

The Hostages and Missing Families Forum condemned Hamas's actions, calling it "a blatant breach of the agreement".

Prime Minister Sir Keir Starmer attended the ceremony in Sharm el-Sheikh alongside other world leaders, including Italy's Giorgia Meloni and France's Emmanuel Macron.

He said the UK could play a key role in monitoring the ceasefire.

Sir Keir said: "What happens tomorrow really matters, and that's why, what I've been discussing with leaders all day is, what part can we play?

"And we, the United Kingdom in particular, I think, can play a part in monitoring the ceasefire, but also decommissioning the capability of Hamas and their weaponry drawing on our experience in Northern Ireland.

"So the question today has already moved to, how do we implement how do we make sure this moves forward? It's really important we keep that focus. We mustn't have any missteps now."

This was the Trump show. There's no question about that. It was a show called by him, pulled off for him, attended by leaders who had no other choice and all because he craves the ego boost.

Gaza deal signed - as it happened

But the day was also an unquestionable and game-changing geopolitical achievement.

Trump stopped the war, he stopped the killing, he forced Hamas to release all the hostages, he demanded Israel to free prisoners held without any judicial process, he enabled aid to be delivered to Gaza, and he committed everyone to a roadmap, of sorts, ahead.

He did all that and more.

He also made the Israel-Palestine conflict, which the world has ignored for decades, a cause that European and Middle Eastern nations are now committed to invest in. No one, it seems, can ignore Trump.

Love him or loathe him, those are remarkable achievements.

'Focus of a goldfish'

The key question now is - will he stay the course?

One person central to the negotiations which have led us to this point said to me last week that Trump has the "focus of a goldfish".

It's true that he tends to have a short attention span. If things are not going his way, and it looks likely that he won't turn out to be the winner, he quickly moves on and blames someone else.

So, is there a danger of that with this? Let's check in on it all six months from now (I am willing to be proved wrong - the Trump-show is truly hard to chart), but my judgement right now is that he will stay the course with this one for several reasons.

First, precisely because of the show he has created around this. Surely, he won't want it all to fall apart now?

He has invested so much personal reputation in all this, I'd argue that even he wouldn't want to drop it, even when the going gets tough - which it will.

Second, the Abraham Accords. They represented his signature foreign policy achievement in his first term - the normalisation of relations between Israel and the Muslim world.

Back in his first presidency, he tried to push the accords through without solving the Palestinian question. It didn't work.

This time, he's grasped the nettle. Now he wants to bring it all together in a grand bargain. He's doing it for peace but also, of course, for the business opportunities - to help "make America great again".

Peace - and prosperity - in the Middle East is good for America. It's also good for Trump Inc. He and his family are going to get even richer from a prosperous Middle East.

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Then there is the Nobel Peace Prize. He didn't win it this year. He was never going to - nominations had to be in by January.

But next year he really could win - especially if he solves the Ukraine challenge too.

If he could bring his coexistence and unity vibe to his own country - rather than stoking the division - he may stand an even greater chance of winning.

She's waiting for two men to walk her home from the gurdwara - a Sikh place of worship - at a time of rising fears over physical and verbal racial abuse.

Resham reaches for her walking stick and puts on her shoes.

"When I came to this country, I didn't fear for my safety. But now I do - every day. It's a lot worse now," she says.

It's a disturbing admission at a time when some fear Britain's communities are more fragmented than ever.

It comes as new figures last week revealed religious hate crime recorded by police in England and Wales has reached a record high.

Two volunteers arrive, opening the exit door for Resham at the gurdwara in Smethwick in the West Midlands.

Sarbjit Singh - who works in a bakery - and Mangat Singh - who works at a salad farm - are part of the congregation and walk people to and from the gurdwara on their days off to keep them safe.

With attendance numbers at the gurdwara falling because of safety fears, Sarbjit says it's something they need to do "until we get a bit of peace and tranquillity back in society and the community".

"We have to do something. We can't just sit in silence. And we can't let the congregation just stay at home," Sarbjit says.

Other groups of Sikh men are also taking action - joining patrols along a road in nearby Oldbury where a young Sikh woman was allegedly raped a few weeks ago.

That alleged attack - and an assault on two taxi drivers in Wolverhampton in August - have horrified the Sikh community. Police are treating both incidents as racially aggravated.

Monty Singh, who is taking part in the patrols, says they are "not vigilantes".

"We need to make it crystal clear that we're good people, we're just trying to do the right thing and support our community," Monty adds.

Pervinder Kaur is the vice president of the gurdwara.

She believes racial abuse is being emboldened after a summer of demonstrations outside hotels housing asylum seekers and renewed debate about immigration.

"People are more verbal about it now. They are not scared of the consequences," she said.

A sense of vulnerability is common among minority communities and security measures have also been stepped up at mosques around the country.

Imam Adam Kantar from the Rumi Mosque in Edmonton, north London, said: "Many people are now telling their children not to be outside after certain hours.

"Their [husbands], they prefer to go shopping instead of their hijab [wearing] wives and spouses.

"We have to engage with the wider community and prove that we love this country and its people. The Union Jack is our flag."

What Britain's fragmented society has in common is fear among minority communities - even if the causes are different.

Rabbi Josh Levy, the co-lead of Progressive Judaism, has spent years trying to advance interfaith dialogue.

He said: "Single individual(s) or small groups can cause a huge amount of pain and distress.

"There are lots of really great examples of community cohesion around the country. But generally, there is definitely a sense of fragmentation. And whether it's driven by political concerns or what's happening internationally.

"We've got huge work to do in taking the work that happens on a national level and bringing it into local communities."

An obscure suite of elements that sit in their own corner of the periodic table, they were mostly renowned among chemists and geologists for being tricky and fiddly - incredibly hard to refine, but with chemical facets that made them, well… interesting.

Not so much for a single thing they did by themselves, but for what they did in conjunction with other elements.

Added to alloys, rare earths can make them stronger, more ductile, more heat-resistant, and so on. Think of them as a sort of metallic condiment: a seasoning you add to other substances to make them stronger, harder, better.

The best example is probably neodymium. On its own, there's nothing especially spectacular about this rare earth element. But add it to iron and boron, and you end up with the strongest magnets in the world. Neodymium iron boron magnets are everywhere.

If you have a pair of headphones or earbuds, the speakers inside them ("drivers" is the technical term) are driven by these rare earth magnets.

If you have a pair of Apple AirPods, those magnets aren't just in the speakers; they're what's responsible for the satisfying "click" when the case snaps shut.

Rare earth magnets are in your car: in the little motors that raise and lower the windows, inside the functioning of the airbag and the seat adjustment mechanism.

And not just the little things. Most electric vehicles use rare-earth magnets in their motors, enabling them to accelerate more efficiently than the old all-copper ones.

More sensitively, from the perspective of Western governments, in the military, there are tonnes of rare earths to be found in submarines, in fighter jets, in tanks and frigates. Much of this is in the form of magnets, but some is in the form of specialised alloys.

So, for instance, there is no making a modern jet engine without yttrium and zirconium, which, together, help those metallic fan blades withstand the extraordinary temperatures inside the engine. Without rare earths, the blades would simply melt.

Yet the amount of this stuff we mine from the ground each year is surprisingly small.

According to Rob West of Thunder Said Energy, the total size of the rare earth market is roughly the same as the North American avocado market. But, says West, those numbers underplay its profound importance.

"Buyers would likely pay over 10-100x more for small but essential quantities of rare earths, if supplies were ever disrupted," he says.

"You cannot make long-distance fibre cables without erbium. You cannot make a gas turbine or jet engine without yttrium."

China's dominance

In short, these things matter. And that brings us to the politics.

Right now, about 70% of the world's rare earth elements are mined in China.

Roughly 90% of the finished products (in other words, those magnets) are made in China. China is dominant in this field in an extraordinary way.

This is not, it's worth saying, for geological reasons.

Contrary to what the name suggests, rare earth elements aren't all that rare. Pull a chunk of soil out of the ground and there will be trace amounts of most of them in there.

True: finding concentrated ores is a bit harder, but even here, it's not as if they are all in China.

There are plenty of rich rare earth ores in Brazil, India, Australia, and even the US (indeed, the Mountain Pass mine in California is where rare earth mining really began in earnest).

Low cost of Chinese rare earths

The main explanation for Chinese dominance is that China has simply become very good at extracting lots of rare earths at relatively low cost.

According to figures from Benchmark Mineral Intelligence, the prevailing cost of Chinese rare earths is at least three times lower than the cost of similar minerals refined in Europe (to the extent that such things are available).

At this point, perhaps you're wondering how China has managed to do it - to dominate global production at such low prices.

Part of the explanation, says West, probably comes down to "transfer pricing" - in other words, China being China, refiners and producers are probably able to buy raw materials at below market prices.

Another part of the explanation is that refining rare earth ores is phenomenally energy and carbon-intensive.

Most European and American firms have pulled out of the sector because it is hideously dirty.

Such qualms are less of an issue in China, especially since most of their mines, including the biggest of all, Bayan Obo in Inner Mongolia, are hundreds if not thousands of miles from the nearest city.

Energy costs are less of a constraint in a country whose grid is still built mostly on a foundation of cheap thermal coal.

Add it all up, and you end up with the situation we have today: where the vast majority of the world's rare earths, that go into all our devices, come from dirty mines in China, produced at such a low cost that device manufacturers are happy to put them anywhere.

Anyway, that brings us to the politics.

Global trade war flaring up again

In recent months and years, China has periodically introduced controls on rare earth exports.

Last week, it announced the most serious rule change yet, essentially insisting that anyone using Chinese rare earths would have to apply for a licence from them.

It has been seen, in Washington at least, as a declaration of economic war, and, in response, Donald Trump has announced a fresh set of tariffs on China.

In short, the global trade war seems to be flaring up all over again.

Where this ends up is anyone's guess. Tim Worstall, a former scandium expert who has been in and out of the rare earths sector for decades, suspects China might have overplayed its hand.

"The end result here is that there can be two outcomes," he says.

"A: The entire world's usage of rare earths is mapped out in detail, end uses, end users, quantities, and times for the Chinese state and depends upon their bureaucracy to administer.

"B: The plentiful rare earths of elsewhere are dug up, and the supply chain is rebuilt outside China.

"My insistence is that B is going to be the outcome, and it'll be done, intervention or no."

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In practice, the new rules may simply represent an element in China's trade negotiations with the US.

So it's hard to know whether they, or for that matter America's 100% extra tariffs, will ever really bite.

Either way, it's yet more evidence of the rocky road the global economy remains on.

A damning report by the National Audit Office (NAO) said "clear failures" in the Energy Company Obligation (ECO) scheme to tackle fuel poverty and pollution had led to low quality installations and even potential fraud.

It blamed incompetent subcontractors and weak monitoring and government oversight for the issues, which in extreme cases could cause fires.

Fuel poverty campaigners warned the system had "let cowboys through the front door", saying it must be fixed to bring down energy bills and keep people warm.

Almost all homes - some 98%, affecting 22,000 to 23,000 properties - fitted with external wall insulation under the ECO are affected, the NAO said.

A further 29% of homes with internal wall insulation - around 9,000 to 13,000 dwellings - also face major issues that need fixing.

A small percentage of homes - 6% with external insulation and 2% with internal - put people in immediate danger, such as poor ventilation that could cause carbon monoxide poisoning, and electrical safety issues that could start fires.

ECO is a scheme that obliges energy companies to pay for energy efficiency measures in vulnerable households out of consumer bills.

Gareth Davies, head of the NAO, said ECO is "important to help reduce fuel poverty and meet the government's ambitions for energy efficiency".

But "clear failures in the design and set-up" had led to "poor-quality installations, as well as suspected fraud", he added.

'Gaming the system'

The report says the reason things had gone so badly wrong could be down to work being subcontracted to individuals and firms who are not competent or certified, uncertainty over standards, and businesses "cutting corners" or "gaming the system".

The energy regulator Ofgem last year estimated businesses had falsified claims for ECO installations in between 5,600 and 16,500 homes.

That means they could have claimed between £56m and £165m from energy suppliers - ultimately paid for by bill-payers.

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Martin McCluskey, the government minister for energy consumers, criticised the "unacceptable, systemic failings" that had affected thousands of families.

He added: "We are fixing the broken system the last government left by introducing comprehensive reforms to make this process clear and straightforward, and in the rare cases where things go wrong, there will be clear lines of accountability, so consumers are guaranteed to get any problems fixed quickly."

The government urged households to take up the free audit that will be offered in a forthcoming letter, and said installers would be forced to remedy the issues free of charge.

However, insulation has the potential to vastly improve homes, analysts pointed out.

Jess Ralston from energy think tank ECIU said: "The majority of households that have benefitted from insulation schemes have lower bills and warmer, healthy homes, particularly during the early years of the gas crisis when the UK's poor quality housing stock was one of the reasons we were so badly hit compared to other European countries."

Simon Francis, co-ordinator of the End Fuel Poverty Coalition, said: "Insulation and ventilation, when done properly, are among the safest and most effective ways to bring down energy bills and keep people warm."

But the report had revealed a "system that has let cowboys through the front door, leaving thousands of victims living in misery and undermining public trust".

Sue Davies, Which? head of consumer protection policy, called it "a damning indictment of a failed scheme, where poor oversight has allowed rogue traders to cause huge damage to people's homes and lives".

She said the government must take swift action to rectify the damage, as well as ensuring "there is no repeat of this scandal by putting in place robust consumer protections and effective oversight".