Jeremy Corbyn, the sixty-eight-year-old leftist who heads the opposition Labour Party in the U.K., wasn't scheduled to appear at the BBC’s televised general-election debate this week. With Theresa May, the Conservative Prime Minister, and the hot favorite to gain reëlection, skipping the event out of a super-abundance of caution, Corbyn's aides didn't think that it would do him any good to debate the Home Secretary, Amber Rudd, who was standing in for May, and the leaders of five smaller parties: the Liberal Democrats, the U.K. Independence Party, the Greens, the Scottish National Party, and the Welsh nationalist party Plaid Cymru.

At the last minute, however, Corbyn, changed his mind. Buoyed by a series of opinion polls showing Labour gaining ground on the Conservatives, he zipped up to Senate House, at the University of Cambridge, where the debate was being held. Seizing on May's absence, and her reluctance to appear anywhere in public except in front of confirmed Conservative supporters, he performed well. "The Tories have been conducting a stage-managed, arms-length campaign, and have treated the public with contempt," he declared. And he went on, "On June 8, you have a choice. More cuts in services and living standards with the Conservatives, or vote Labour to transform Britain for the many and not the few."

This language was similar to that used by Bernie Sanders, the U.S. senator from Vermont who ran an insurgent campaign for President, and the echoes were quite deliberate. Like Sanders, Corbyn is an outsider tilting at the economic and political élites. He is very popular among the young; he is using social media to outflank the mainstream press. And his agenda is explicitly redistributionist: it features sizable tax hikes on rich households and corporations to finance higher spending on public services, such as health care, education, and social care for the elderly. "For the Many, Not the Few" has been Labour's slogan since Corbyn launched the Party’s campaign, on May 9th.

At that juncture, Labour was trailing the Conservatives by twenty points or more in the opinion polls, and virtually everybody at Westminster regarded the outcome of the election as a foregone conclusion. But after seven years of Conservative austerity policies, which have reduced spending in many areas of government, large numbers of Britons, particularly the young and the traditional Labour voters who had drifted away from the Party, seem to be warming to Corbyn's message. On Friday, poll averages showed that the Tory lead had been whittled down to single figures, and some individual polls have indicated the race to be within five or six points.

Rudd, who spoke first in this week’s debate, said that the "only question to consider is who should be in Number Ten to steer Britain to a brighter future: Jeremy Corbyn, with his money-tree, wish-list manifesto and no plan for Brexit, or Theresa May and her record of delivery, and her clear plan for Brexit?" This framing of the race reflected the Conservatives' thinking when, a few weeks ago, they called a snap election just after the Prime Minister received Parliament’s blessing to start the Brexit process. Merely by invoking Corbyn and Labour's reputation as a spendthrift party, the Tory thinking went, the government would be returned to power with a much-enlarged majority in the House of Commons. (In the six-hundred-and-fifty-seat chamber, its current majority is just seventeen.) Such an outcome would, in turn, would give May more leeway in the upcoming Brexit negotiations.

Corbyn is putting this complacent theory to the test. Since he took over as Labour's leader, in September, 2015, much of the British media has excoriated him as too extreme to be electable. Many Labour M.P.s have complained about his lack of leadership skills: last year, they passed a vote of no-confidence in him. But Corbyn, who is popular at the grass roots of the Party, soldiered on. Since the campaign began, he has travelled across the country, sticking doggedly to his message and ignoring the sleights hurled at him. (The Sun has called him a "wallie." The Daily Telegraph compared him to Hugo Chávez.)

His enemies underestimated him. Although nobody would mistake him for a great orator, he has the capability to stay calm and speak in plain English. He's offering a clear alternative to Conservative policies, and the spending proposals he is putting forward have the merits of being backed up by some detailed cost estimates, something the government can't claim about its manifesto.

"If you put the corporation tax up, you are then in a position to deal with the crisis in social care, the crisis in the National Health Service, and properly fund our schools," Corbyn said in response to Rudd's gibes about Labour's spending plans. Addressing May’s suggestion that Britain was at peace with itself under the Conservatives, Corbyn said, "I would just say: Have you been to a food bank? Have you seen people sleeping in our stations? Have you seen the levels of poverty that exist because of your government's conscious decisions on benefits?"

Corbyn even found a clever way to turn around the many criticisms that have been levelled at him, suggesting that he has learned from them. "Leadership is about understanding the people you represent," he said, when Rudd brought up the vote of confidence that went against him. "It is not about being so high and mighty you can't take advice."

The tightening in the polls has unnerved some Conservative politicians, and no wonder. It represents the biggest swing to either major party during an election campaign since U.K. polling began, in 1945. Not surprisingly, there is a lot of debate about how real the swing is.

Some skeptics believe that the polling firms, some of which had adopted new methods after missing the mark in both the 2015 election and the Brexit referendum, are overstating Labour's surge. As happened in 1992, they may be under-counting so-called shy Tories, who don't like to state their preferences. Or they may be over-counting the left-leaning people under thirty-five, some of whom might not actually vote.

Another theory is that the polls are picking up a genuinely big shift. Even many Conservative-leaning pundits agree that May's campaign performance has been underwhelming, and that Corbyn has outshone her. By appearing frequently on television, and generally coming across well, he has countered some of the Fleet Street characterizations of him as a dangerous radical. His anti-establishment populism, which includes a sharp critique of big business, clearly appeals to many young voters. And, with Labour having reconciled itself to Brexit, some of the Party's traditional working-class voters seem to be returning to the fold.

Even if the polls are accurate, however, they still show the Conservatives ahead. At the British online bookmakers, you can still get odds of six-to-one against Labour winning the most seats in next Thursday's vote. If Britain wakes up next Friday morning and Corbyn is the new Prime Minister, it will be a stunning occurrence.

Corbyn has already greatly exceeded expectations, however, and he is evidently enjoying the battle. Ewen MacAskill, a reporter for the Guardian, found him on his campaign bus dispensing pieces of chocolate cake to his aides. “I think you have got to be relaxed," he said. "Because if I get stressed up and tense it does not do anyone any good, particularly me.” One Labour veteran I spoke with, who is not a Corbyn supporter, said of him, "His personal performances have improved almost beyond recognition." Another Labourite and Corbyn skeptic, who had just returned from canvassing in North London with hundreds of young Corbyn supporters, told me, "Jeremy is having a great election—he loves campaigning."