(Israeli soldiers on the Gaza border.Credit…Tsafrir Abayov/Associated Press)

It was the pictures of Palestinians swimming and sunning at a Gaza beach that rubbed Yehuda Shlezinger, an Israeli journalist, the wrong way. Stylish in round red glasses and a faint scruff of beard, Mr. Shlezinger unloaded his revulsion at the “disturbing” pictures while appearing on Israel’s Channel 12.

“These people there deserve death, a hard death, an agonizing death, and instead we see them enjoying on the beach and having fun,” complained Mr. Shlezinger, the religious affairs correspondent for the widely circulated right-wing Israel Hayom newspaper. “We should have seen a lot more revenge there,” Mr. Shlezinger unrepentantly added. “A lot more rivers of Gazans’ blood.”

It would be nice to think that Mr. Shlezinger is a fringe figure or that Israelis would be shocked by his bloody fantasies. But he’s not, and many wouldn’t be.

Israel has hardened, and the signs of it are in plain view. Dehumanizing language and promises of annihilation from military and political leaders. Polls that found wide support for the policies that have wreaked devastation and starvation in Gaza. Selfies of Israeli soldiers preening proudly in bomb-crushed Palestinian neighborhoods. A crackdown on even mild forms of dissent among Israelis.

The Israeli left — the factions that criticize the occupation of Palestinian lands and favor negotiations and peace instead — is now a withered stump of a once-vigorous movement. In recent years, the attitudes of many Israelis toward the “Palestinian problem” have ranged largely from detached fatigue to the hard-line belief that driving Palestinians off their land and into submission is God’s work.

(Israeli soldiers on the Gaza border.Credit…Tsafrir Abayov/Associated Press)

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magine you had a friend who gave different answers to the same question, depending on how you asked it. “What’s the capital of Peru?” would get one answer, and “Is Lima the capital of Peru?” would get another. You’d probably be a little worried about your friend’s mental faculties, and you’d almost certainly find it hard to trust any answer they gave.

That’s exactly what’s happening with many large language models (LLMs), the ultra-powerful machine learning tools that power ChatGPT and other marvels of artificial intelligence. A generative question, which is open-ended, yields one answer, and a discriminative question, which involves having to choose between options, often yields a different one. “There is a disconnect when the same question is phrased differently,” said Athul Paul Jacob, a doctoral student at the Massachusetts Institute of Technology.

To make a language model’s answers more consistent — and make the model more reliable overall — Jacob and his colleagues devised a game where the model’s two modes are driven toward finding an answer they can agree on. Dubbed the consensus game, this simple procedure pits an LLM against itself, using the tools of game theory to improve the model’s accuracy and internal consistency.

“Research exploring self-consistency within these models has been very limited,” said Shayegan Omidshafiei, chief scientific officer of the robotics company Field AI. “This paper is one of the first that tackles this, in a clever and systematic way, by creating a game for the language model to play with itself.”

“It’s really exciting work,” added Ahmad Beirami, a research scientist at Google Research. For decades, he said, language models have generated responses to prompts in the same way. “With their novel idea of bringing a game into this process, the MIT researchers have introduced a totally different paradigm, which can potentially lead to a flurry of new applications.”

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Mathematicians have disproved a major conjecture about the relationship between curvature and shape.

n an old Indian parable, six blind men each touch a different part of an elephant. They disagree about what the elephant must look like: Is it smooth or rough? Is it like a snake (so thinks the man touching the trunk) or a fan (as the man touching the ear proposes)? If the blind men had combined their insights, they might have been able to give a correct account of the nature of the elephant. Instead, they end up fighting.

For decades, topologists have hoped to avoid falling into a similar trap. They thought they could characterize mathematical shapes by synthesizing numerous local measurements. But newly discovered, paradoxically curved spaces show that this isn’t always possible. “Things can be much more wild than what we thought,” said Elia Bruè of Bocconi University in Italy, who worked with two other mathematicians to demonstrate this.

Topologists stretch and compress the shapes they study. An infinitely thin rubber band, from a topological perspective, is equivalent to a circle, because you can easily deform it into a circular shape. Topologists tend to characterize shapes according to their global properties: Do they have holes, like a doughnut? Do they go on forever, like an infinite plane, or are they “compact” like the surface of a sphere? Do their “straight” lines go on indefinitely — making them what mathematicians call “complete” — or are there dead ends?

But as with the elephant in the parable, it can be hard to directly perceive the global nature of topological shapes. And so mathematicians want to understand their relationship to local geometric properties, like curvature. What can you say about a shape’s global topology, given information about how it curves at every point?

In 1968, John Milnor, a renowned mathematician then at Princeton University, conjectured that an average sense of a complete shape’s curvature was enough to tell us that it couldn’t have infinitely many holes. For the next 50 years, many results supported his claim. “You were tempted to believe it was true, because it was true in so many realistic cases,” said Jeff Cheeger of the Courant Institute of Mathematical Sciences at New York University. “And how in God’s name could you construct a counterexample to it?”

In this area of mathematics, said Vitali Kapovitch of the University of Toronto, “the Milnor conjecture was probably the biggest open problem.”

And so in 2020, Bruè and two colleagues set out to prove it. They ended up finding a counterexample instead — and built an entirely new kind of topological shape in the process. “It’s fantastic work,” Cheeger said. “A landmark.”

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(Israeli Prime Minister Benjamin Netanyahu, left, and National Security Minister, Itamar Ben-Gvir in Israel’s parliament last year. Credit…Abir Sultan/EPA, via Shutterstock)

Radical forces in Israeli society have moved from the fringes to the mainstream and put Israel’s democracy in peril. Here are the takeaways from our investigation.

For decades, most Israelis have considered Palestinian terrorism the country’s biggest security concern. But there is another threat that may be even more destabilizing for Israel’s future as a democracy: Jewish terrorism and violence, and the failure to enforce the law against it.

Our yearslong investigation reveals how violent factions within the Israeli settler movement, protected and sometimes abetted by the government, have come to pose a grave threat to Palestinians in the occupied territories and to the State of Israel itself. Piecing together new documents, videos and over 100 interviews, we found a government shaken by an internal war — burying reports it commissioned, neutering investigations it assigned and silencing whistle-blowers, some of them senior officials.

It is a blunt account, told in some cases for the first time by Israeli officials, of how the occupation came to threaten the integrity of the country’s democracy.

Settlers Pursuing a Theocratic State Have Become Lawmakers

Officials told us that once fringe, sometimes criminal groups of settlers bent on pursuing a theocratic state have been allowed for decades to operate with few restraints. Since Prime Minister Benjamin Netanyahu’s coalition government came to power in 2022, elements of that faction have taken power — driving the country’s policies, including in the war in Gaza.

The lawbreakers have become the law.

Bezalel Smotrich, the finance minister and the official in Netanyahu’s government with oversight over the West Bank, was arrested in 2005 by the Shin Bet domestic security service for plotting road blockages to halt the Israeli withdrawal from Gaza. He was released with no charges. Itamar Ben-Gvir, Israel’s national security minister, had been convicted multiple times for supporting terrorist organizations and, in front of television cameras in 1995, vaguely threatened the life of Prime Minister Yitzhak Rabin, who was murdered weeks later by an Israeli student.

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