President Trump says he wants a quick cease-fire in Ukraine. But President Vladimir V. Putin of Russia appears to be in no rush, and the blowup on Friday between Mr. Trump and Ukraine’s president may give Russia’s leader the kind of ammunition he needs to prolong the fight.

With the American alliance with Ukraine suffering a dramatic, public rupture, Mr. Putin now seems even more likely to hold out for a deal on his terms — and he could even be tempted to expand his push on the battlefield.

The extraordinary scene in Washington — in which Mr. Trump lambasted President Volodymyr Zelensky of Ukraine — was broadcast as the top story on state television in Russia on Saturday morning. It played into three years of Kremlin propaganda casting Mr. Zelensky as a foolhardy ruler who would sooner or later exhaust the patience of his Western backers.

For the Kremlin, perhaps the most important message came in later remarks by Mr. Trump, who suggested that if Ukraine did not agree to a “cease-fire now,” the war-torn country would have to “fight it out” without American help.

That could set up an outcome that Mr. Putin has long sought, at the cost of tens of thousands of Russian lives: a dominant position over Ukraine and wide-ranging concessions from the West.

In fact, Mr. Trump’s professed attempts to end the war quickly could intensify and prolong it, experts warned. If the United States is really ready to abandon Ukraine, Mr. Putin could try to seize more Ukrainian territory and end up with more leverage if and when peace talks ultimately take place.

“Russia will be willing to keep fighting for longer, and more bitterly,” said Konstantin Remchukov, a Moscow newspaper editor with Kremlin ties, describing the consequences of Mr. Trump’s public break with Mr. Zelensky. “If Zelensky says the Ukrainian people are ready to keep on fighting, Moscow will say, ‘Sure, let’s keep fighting.’”

If Friday’s angry encounter in Washington leads to a further drop in U.S. military support for Ukraine, Mr. Remchukov said in a phone interview, the consequences could be profound, possibly even encouraging Mr. Putin to return to the broader territorial aims he pursued when he began his invasion in 2022.

“I wouldn’t be surprised if Moscow decided to go further, to Odesa or Mykolaiv,” Mr. Remchukov said, referring to key Black Sea ports that remain under Ukrainian control. “It could change the strategic direction of the offensive.”

Despite the striking alignment that has emerged between Mr. Trump and Mr. Putin in recent weeks, many analysts have spotted a key difference in their views. While the American president says he wants to “stop the death” in Ukraine as soon as possible, the Russian leader says he wants to resolve the “root causes” of the war first.

For Mr. Putin, that terminology is code for his desire for a wider deal that would prevent Ukraine from joining NATO, limit the size of its military and grant Russia influence over its domestic politics — along with a broader pullback of the NATO alliance across Eastern and Central Europe.

Such a deal, of course, would take months to negotiate, which is why Mr. Putin has appeared resistant to the idea of a quick cease-fire. The spat in the White House on Friday appeared to play into the Kremlin’s hands because it may convince Mr. Trump that Mr. Zelensky, rather than Mr. Putin, is the more recalcitrant of the two leaders.

“You tell us, ‘I don’t want a cease-fire,’” Mr. Trump told Mr. Zelensky in the Oval Office. “I want a cease-fire because you’ll get a cease-fire faster than an agreement.”

Mr. Zelensky on Saturday reiterated his opposition to a quick cease-fire with Mr. Putin, saying that the Russian leader could not be trusted to uphold one. Instead, he said, Ukraine needed security guarantees from the West to deter future Russian attacks.

But Mr. Zelensky also signaled that he had not completely given up hope on repairing the relationship with Mr. Trump. And since the Friday meeting, he has publicly expressed thanks for American support, after Vice President JD Vance accused him of not being grateful enough.

A Moscow foreign-policy analyst who is close to the Kremlin said on Saturday that any delay to peace talks was likely to benefit Russia because there was no deal in sight at present that would satisfy Mr. Putin. The analyst insisted on anonymity because of the sensitivities in Moscow of speaking to Western journalists.

Dmitry Suslov, an international relations specialist at the Higher School of Economics in Moscow, said in comments published by the Kommersant newspaper that Mr. Trump would become “even more favorable to Russia’s position on a settlement” after “the fiasco of Zelensky’s negotiations with Trump.”

Mr. Suslov also raised the possibility of Russia’s being able to grab far more than the roughly 20 percent of Ukrainian territory in the country’s south and east that Moscow now holds.

If the United States stops providing weapons and intelligence to the Ukrainian military, Mr. Suslov wrote, “the pace of Kyiv’s defeat on the battlefield will accelerate, with the prospect of a complete collapse of the front within months.”

Friday’s scene was a boon for Moscow in other ways, too. It may have helped advance, in just a matter of minutes, one of Mr. Putin’s longtime goals: the removal of Mr. Zelensky from power in Ukraine.

Immediately after the White House meeting, Senator Lindsey Graham, the South Carolina Republican who has been one of his party’s staunchest backers of Ukraine, said, “I don’t know if we can ever do business with Zelensky again.” He called the Ukrainian leader’s behavior in the Oval Office “disrespectful.”

The public dressing-down of Mr. Zelensky also accomplished another longtime goal of Mr. Putin’s: cleaving the Western military alliance led by Washington that united behind Ukraine after Russia’s 2022 invasion. European leaders immediately came out in support of Ukraine after the meeting, setting up a possible split with the United States, their longtime security backer.

Russian officials could hardly control their glee.

Dmitri A. Medvedev, the former Russian president who is deputy chairman of the country’s security council, cheered Mr. Trump on with a post on X, piling on to denounce Mr. Zelensky as an “insolent pig.”

And Konstantin Kosachev, a senior Russian lawmaker, wrote on the Telegram social network, “Zelensky lost this round in a resounding crash,” adding, “He will have to crawl on his knees to the next one.”

Pro-Kremlin commentators who for years have been hurling invective against the United States could barely believe their change in fortune.

Igor Korotchenko, a military analyst who is a regular on Russian talk shows, wrote that he never thought he would be applauding the president of the United States.

“But tonight I applaud the 47th President of the United States Donald Trump — Zelensky was thrown out of the White House like a garbage alley cat,” Mr. Korotchenko wrote in a post on X.

Yet for all the schadenfreude in Russia, Friday’s bitter meeting in Washington did little to illuminate a pathway toward a settlement. And while Mr. Putin may want to extend the war, he could also suffer if it goes on much longer, given the country’s economic problems and steep battlefield casualties.

“The Russian leadership would like to end the war on its own terms, not just restore ties with the U.S.,” Grigorii Golosov, a professor of political science at the European University in St. Petersburg, said in a phone interview. “The prospects for that are not clearer at all despite what happened yesterday.”

For months leading into the American elections last fall, the prospect of a second Trump presidency deepened uncertainty among Ukrainians over how enduring American support would prove in a war threatening their national survival.

After President Volodymyr Zelensky’s disastrous meeting with President Trump in the White House on Friday, many Ukrainians were moving toward a conclusion that seemed perfectly clear: Mr. Trump has chosen a side, and it is not Ukraine’s.

In one jaw-dropping meeting, the once unthinkable fear that Ukraine would be forced to engage in a long war against a stronger opponent without U.S. support appeared to move exponentially closer to reality.

“For Ukraine, it is clarifying, though not in a great way,” Phillips O’Brien, an international relations professor at the University of St. Andrews in Scotland, said in an interview. “Ukraine can now only count on European states for the support it needs to fight.”

The German president, Frank-Walter Steinmeier, said on Saturday that Europe must stand with Ukraine, and prevent the country “from having to accept subjugation.”

“The scene at the White House yesterday took my breath away,” he told the German press agency onboard a plane. “I would never have believed that we would ever have to defend Ukraine from the United States.”

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Bags of fragrant spices, crates of dates, frozen chicken and fresh produce. Food and other goods that were scarce during the war have returned to the shops and street markets of Gaza in time for the holy fasting month of Ramadan. And the Israeli bombs have fallen silent.

But the shadow of the war hangs heavy over what was once one of the most joyous seasons in the territory, and life in Gaza has not even begun to return to normal. Street vendors have refrained from playing the special songs they normally would during Ramadan and even if there is more food in the shops, many struggle to afford it.

The first phase of a cease-fire between Israel and Hamas has lasted a month and a half, but it was set to elapse on Saturday, which coincides with the first day of Ramadan fasting from dawn to dusk. It could be extended but so far, there has been little progress toward doing so.

Maisa Arafa, 29, who said that her brother had been killed during the war, has been living in a tent with other relatives as they clear away rubble from their devastated home in northern Gaza in hopes of moving into one room that is still intact.

“More than anything, I wish my brother could come back. That would be the only thing to make Ramadan feel like it used to,” Ms. Arafa said as she shopped in downtown Gaza City. “This is not the Ramadan we knew, or even the life we knew.”

Before the war, Ramadan was one of the most joyful festivals in Gaza. Crowds flocked to the mosques, and streets were festooned with colorful lanterns typical of the Ramadan period.

But an enormous gap stretches between the happy holiday memories of a seemingly irrecoverable past and the desolation and grief left by the 15-month war in Gaza. Many Palestinians in the territory see little to celebrate.

Since the Israel-Hamas cease-fire went into effect in mid-January, hundreds of truckloads per day of food and other supplies have been entering Gaza, offering a degree of relief from the intense hunger many suffered during the war. The constant bombardment that haunted civilians’ lives every day for more than a year has ceased.

Farah Irshi, 21, described how the previous Ramadan felt during the fighting between Israel and Hamas. There was little food and about 25 displaced people crowded into their home amid constant bombardment, she said.

“Now there’s more food in the local market as more aid seems to be entering Gaza, but people, including us, have no money at all,” she lamented. “So it’s as if there isn’t anything in the markets, anyway.”

Abdelhalim Awad, who oversees a bakery and supermarket in central Gaza, said that prices had dropped since the worst days of the war, when a 55-pound sack of flour could cost hundreds of dollars.

Many goods — like frozen chicken and cooking gas — are now in shops and street markets, although others, like chocolate, are still scarce, he said. But they are still expensive and many people already burned through their savings during the war to buy hard-to-find, overpriced food.

“The goods are now available, but people are still only able to buy what they really need,” Mr. Awad said as he watched holiday shoppers come and go, buying what they could for communal meals to break the fast at night.

The war began after the Hamas-led assault on southern Israel on Oct. 7, 2023, killed about 1,200 people and saw some 250 others taken back to Gaza as hostages. The subsequent Israeli military campaign laid waste to large swaths of the Gaza Strip.

Many residents are still displaced or have returned to their homes only to find them ruined by the fighting. Some have returned to the camps for the displaced where they spent much of the last year, while others have pitched tents on the rubble where their houses once stood.

The Israeli campaign killed more than 48,000 Palestinians, including thousands of children, according to local health officials who do not distinguish between civilians and combatants. The Israeli military said it had “eliminated” nearly 20,000 Hamas operatives, without providing detailed evidence to back up that claim.

This week, Gazans walked through local markets in central and northern Gaza, looking for whatever they could afford. One vendor showcased heaps of green and black olives, piles of dates and other goods.

Muhanned Hamad, an accountant from Gaza City, stood in front of a toy vendor’s stall in what was historically a major downtown market. He said he was looking for a holiday lantern to give to his neighbors, a mother and son who had lost their immediate family during the war.

“This Ramadan is nothing like the ones before,” said Mr. Hamad, 39. “The war has drained it of meaning,” he added. “Even with the cease-fire, nothing here feels worthy of celebration.”

Ameera Haroudacontributed reporting.

My grandfather’s idea of an Easter egg hunt involved hiding money in colorful plastic eggs sprinkled around his house on Long Island. Most held coins, but there was always one with a crisp, new $100 bill.

My cousin, Billy-O, and I were the only players. We were usually playful partners in mayhem but as competitors, we took on every hunt with gusto, flipping over cushions, throwing open cabinets, knocking each other aside until, without fail, Billy-O found the $100.

The first time he won, I fought back tears. But after a few years of losses, I exploded.

“It’s just not fair,” I yelled.

“Life’s unfair,” my grandfather told us. “You win or you lose.”

This is what’s called zero-sum thinking — the belief that life is a battle over finite rewards where gains for one mean losses for another. And these days, that notion seems to be everywhere. It’s how we view college admissions, as a cutthroat contest for groups defined by race or privilege. It’s there in our love for “Squid Game.” It’s Silicon Valley’s winner-take-all ethos, and it’s at the core of many popular opinions: that immigrants steal jobs from Americans; that the wealthy get rich at others’ expense; that men lose power and status when women gain.

But nowhere is the rise of our zero-sum era more pronounced than on the world stage, where President Trump has been demolishing decades of collaborative foreign policy with threats of protectionist tariffs and demands for Greenland, Gaza, the Panama Canal and mineral rights in Ukraine. Since taking office, he has often channeled the age he most admires — the imperial 19th century.

And in his own past, zero-sum thinking was deeply ingrained. His biographers tell us he learned from his father that you were either a winner or loser in life, and that there was nothing worse than being a sucker. In Trumpworld, it’s kill or be killed; he who is not a hammer must be an anvil.

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The Kurdish guerrilla group that has been fighting a long-running insurgency against Turkey declared a cease-fire on Saturday, days after a call from its jailed leader to disarm and disband the organization raised hopes of ending a conflict that has killed tens of thousands of people over four decades.

The Kurdistan Workers’ Party, or P.K.K., said the cease-fire would begin immediately. But it also called for Abdullah Ocalan, the P.K.K.’s founder and leader who has been in a Turkish prison for a quarter-century, to be freed so he can oversee the group’s dissolution.

If the P.K.K. does disband, it would resolve a major domestic security threat and mark a political victory for Turkey’s president, Recep Tayyip Erdogan. If negotiations proceed with Mr. Ocalan, it could usher in a new era of peace across the region where Kurds have pursued an armed struggle in a mountainous area that intersects parts of Iraq, Syria and Turkey.

But there are still many unanswered questions.

“This is just the first sentence,” Asli Aydintasbas, a senior fellow at the Brookings Institution, said of Mr. Ocalan’s call to all groups to disarm.

It is not clear whether Turkey will cease armed operations against the P.K.K., who would monitor any truce or what would happen to fighters who do lay down their arms. There is also the question of whether the government has offered the Kurdish fighters anything in return.

But a cease-fire would allow Kurds to start internal consultations and hold local congresses to forge a democratic way forward, something Kurds in Turkey and Syria have said they want to do.

The P.K.K. announcement came two days after Mr. Ocalan said that the group had outlived its life-span and should dissolve itself, a rare message from a leader with broad influence over Kurdish fighters in Turkey, but also around the region, including in Syria and Iraq.

The P.K.K. statement, carried by Firat News Agency, a P.K.K.-linked news site, said “none of our forces will take armed action unless attacked.”

In recent years, Turkey’s military has degraded the P.K.K.’s fighting abilities, which analysts say may have contributed to its willingness to discuss an end to its fight.

Fighters of the P.K.K. revere their leader, Mr. Ocalan, and are expected to heed his call, but the group’s conditional statement suggests it will continue to use its leverage in the bargaining process.

“For these types of organizations, cease-fires are a means to buy time, overcome military setbacks and smooth over cracks among members,” said Oytun Orhan, an analyst at the Center for Middle Eastern Studies based in the Turkish capital, Ankara.

The Turkish government did not immediately comment on the P.K.K. statement or on the group’s call for Mr. Ocalan to be released.

But on Friday, Mr. Erdogan welcomed the appeal by Mr. Ocalan, which came after a series of talks that included Turkish officials; Mr. Ocalan himself; and members of Turkey’s main pro-Kurdish party, the People’s Equality and Democracy Party, or D.E.M.

“We have a historic opportunity to take a step toward demolishing the wall of terror” between Turks and Kurds, he said. He added that Turkish officials would follow up on working to end the conflict, without elaborating on what that would entail.

Mr. Erdogan said in January that the government had offered the P.K.K. no concessions.

The P.K.K. began as a secessionist group that sought to create an independent state for Turkey’s Kurdish minority, but more recently it has said it was seeking greater rights for Kurds inside Turkey.

Turkey, the United States and other countries classify Mr. Ocalan as a terrorist and the P.K.K. as terror group for its attacks that have killed Turkish security forces and civilians. Many Turks see Mr. Ocalan, who was convicted in 1999 of leading an armed terrorist group, as one of the country’s biggest enemies.

Turkey and the P.K.K. have tried over the years to resolve the conflict, most recently through peace talks that started in 2011. Negotiations broke down in 2015, ushering in a deadly new phase.

But last October, a powerful political ally of Mr. Erdogan made a surprising public call to Mr. Ocalan, asking him to tell his fighters to lay down their arms and end the conflict. Doing so, the politician said, could open a pathway for his life sentence to be ended.

The Catholic world is gripped with uncertainty as Pope Francis, 88, remains in a Rome hospital because of a complex lung infection and other serious ailments.

The Vatican said on Saturday that the pope remained stable after a bronchial spasm a day earlier had required him to undergo noninvasive mechanical ventilation. On Saturday, the Vatican said he was alternating that ventilation with long periods of high-flow oxygen therapy.

He had no fever and his blood work “remained stable.” Francis had spent about 20 minutes praying in the chapel attached to his room, the Vatican added.

The prognosis remained guarded, the Vatican said, and doctors still say that the pope is not out of danger.

The previous night’s potentially serious setback had followed several days of improvements. On Thursday, his clinical condition had improved enough for him to move out of critical condition.

Here’s what we know about his condition, and about how things would go should he die.

Francis’ Condition

Francis’ initial respiratory tract infection developed into pneumonia in both lungs, and other complications have emerged during his hospital stay.

Because of an earlier respiratory crisis, he was given high flows of supplemental oxygen. On Feb. 23, the Vatican announced that he was suffering from “initial, mild kidney failure,” which was “under control.”

It had said two days earlier that Francis would not lead the annual service to open the Christian season of Lent.

He is being treated with various drugs, and his doctors have said that calibrating them has been a challenge.

Given his age and history of lung disease, doctors have been guarded about his prognosis.

Upon Death

While Francis’ fate is still unclear, what is more certain are the time-honored rituals that were drafted and refined over the centuries to ensure secrecy and an orderly transition. The carefully choreographed pageantry provides order to a church in its most dramatic moment of flux.

A pope’s death is immediately confirmed by the head of the Vatican’s health department and the cardinal chamberlain of the Holy Roman Church, who becomes the Vatican’s de facto administrator. The pope’s body is dressed in a white cassock and brought to the pope’s private chapel.

The cardinal chamberlain, camerlengo in Italian, is a position now held by Cardinal Kevin Joseph Farrell, a 77-year-old American of Irish origin. He and other officials, along with members of the pope’s family, congregate in the chapel for a ceremony. The body is placed in a coffin made of wood and lined with zinc. The pope is dressed in red, his miter and pallium placed next to him.

After the ceremony, the camerlengo drafts a document authenticating the pope’s death, affixing the doctor’s report. He secures the pontiff’s private papers and seals his apartments, which in the case of Francis are a large section of the second floor at the Casa Santa Marta, the Vatican City guesthouse used by visiting cardinals, where Francis has lived throughout his papacy.

He also arranges for the destruction of the so-called fisherman’s ring, used by the pope to seal documents, with a ceremonial hammer, to prevent forgeries.

Paying Respects

Francis has brought an unpretentious style to the church — rejecting elaborate outfits and the ornate papal apartments. His plans keep that up to the end, scaling back some of the funerary pomp and ceremony.

In rewriting the funeral rites in 2024, Francis simplified several elements. Since the 13th century, the bodies of popes have gone on public view, their embalmed bodies placed on a raised pedestal. When John Paul II died in 2005, his body was first brought to the Apostolic Palace within hours for a private viewing for cardinals, bishops and other members of the church hierarchy, as well as prominent Italian officials.

Francis has done away with that sort of viewing. Instead, a public viewing will take place directly in St. Peter’s Basilica, where thousands have thronged to pay respects to popes in the past. But his body will remain in the coffin, which will not be on a raised pedestal. “Francis decided to highlight humility over glorification,” said Agostino Paravicini Bagliani, a church historian.

The College of Cardinals decides on the day and hour that the pope’s body will be brought to St. Peter’s Basilica, in a procession led by the camerlengo, and on when the viewing will commence.

The Funeral

The funeral and burial of the pope are supposed to take place four to six days after his death, and funeral rites in various churches in Rome will last nine days.

Past popes were put in three nested coffins, one of cypress, one of zinc and one of elm. But as part of Francis’ rule change, he decreed that he would be buried in a single coffin, made of wood and lined in zinc.

The coffin is closed the night before the funeral. The pope’s face is covered by a white silk veil, and he is buried with a bag containing coins minted during his papacy and a canister with a “rogito,” or deed, briefly listing details of his life and papacy. The rogito is read aloud before the coffin is closed.

The new rules also allow for a pope to be buried in a church other than St. Peter’s Basilica. Francis has asked to be buried instead in the Basilica of St. Mary Major, a church dear to him and one he often visited to pray in front of an icon of the Virgin Mary.

The Conclave

Within 15 to 20 days of the pope’s death, the dean of the College of Cardinals, Cardinal Giovanni Battista Re, 91, will summon the cardinals to Rome for what is known as a conclave to elect Francis’ successor.

The period between the death of a pope and the election of a new one is called the sede vacante, or “the seat is vacant.” During that time, the College of Cardinals keeps general oversight of the church, but it is barred from making any major decisions.

When the cardinals meet, they gather in the Sistine Chapel. All cardinal electors must swear an oath of secrecy and vote by secret ballot. Only cardinals under age 80 are eligible to vote. A two-thirds majority is needed to elect a new pope, and politicking is part of the process.

Cardinals are not allowed to leave the conclave except in rare cases. The word conclave — from the Latin “with key” — refers to the isolation imposed on them, which is meant to keep the electoral process from dragging on.

During the conclave, the cardinals live in Casa Santa Marta, which was built on John Paul II’s orders to replace the improvised rooming arrangements in the papal palace that had previously housed them.

The White Smoke

The cardinals cast repeated votes until a two-thirds majority emerges. After every vote, the ballot papers are burned in a stove, along with an additive that produces a color, and the smoke is released through a chimney that can be seen from St. Peter’s Square, where crowds typically form to watch and wait. If a vote ends without a two-thirds majority, the smoke is black.

When a decision is reached, the smoke is white.

Inside the Vatican, the dean of the college asks the chosen successor whether he accepts the job. After getting the presumed yes, the dean asks him for the name he wishes to be called as pope.

In the sacristy of the chapel, the new pontiff is dressed in a white cassock. After greeting the cardinals, he proceeds to a balcony of St. Peter’s Basilica, where a senior cardinal proclaims, in Latin, “Habemus papam” or “We have a pope.”