That classic result was a way to transform any algorithm with a given time budget into a new algorithm with a slightly smaller space budget. Williams saw that a simulation based on squishy pebbles would make the new algorithm’s space usage much smaller—roughly equal to the square root of the original algorithm’s time budget. That new space-efficient algorithm would also be much slower, so the simulation was not likely to have practical applications. But from a theoretical point of view, it was nothing short of revolutionary.
For 50 years, researchers had assumed it was impossible to improve Hopcroft, Paul and Valiant’s universal simulation. Williams’ idea—if it worked—wouldn’t just beat their record—it would demolish it.
“I thought about it, and I was like, ‘Well, that just simply can’t be true,’” Williams said. He set it aside and didn’t come back to it until that fateful day in July, when he tried to find the flaw in the argument and failed. After he realized that there was no flaw, he spent months writing and rewriting the proof to make it as clear as possible.
At the end of February, Williams finally put the finished paper online. Cook and Mertz were as surprised as everyone else. “I had to go take a long walk before doing anything else,” Mertz said.
Valiant got a sneak preview of Williams’ improvement on his decades-old result during his morning commute. For years, he’s taught at Harvard University, just down the road from Williams’ office at MIT. They’d met before, but they didn’t know they lived in the same neighborhood until they bumped into each other on the bus on a snowy February day, a few weeks before the result was public. Williams described his proof to the startled Valiant and promised to send along his paper.
“I was very, very impressed,” Valiant said. “If you get any mathematical result which is the best thing in 50 years, you must be doing something right.”
PSPACE: The Final Frontier
With his new simulation, Williams had proved a positive result about the computational power of space: Algorithms that use relatively little space can solve all problems that require a somewhat larger amount of time. Then, using just a few lines of math, he flipped that around and proved a negative result about the computational power of time: At least a few problems can’t be solved unless you use more time than space. That second, narrower result is in line with what researchers expected. The weird part is how Williams got there, by first proving a result that applies to all algorithms, no matter what problems they solve.
“I still have a hard time believing it,” Williams said. “It just seems too good to be true.”
Williams used Cook and Mertz’s technique to establish a stronger link between space and time—the first progress on that problem in 50 years.Photograph: Katherine Taylor for Quanta Magazine
Phrased in qualitative terms, Williams’ second result may sound like the long-sought solution to the P versus PSPACE problem. The difference is a matter of scale. P and PSPACE are very broad complexity classes, while Williams’ results work at a finer level. He established a quantitative gap between the power of space and the power of time, and to prove that PSPACE is larger than P, researchers will have to make that gap much, much wider.
That’s a daunting challenge, akin to prying apart a sidewalk crack with a crowbar until it’s as wide as the Grand Canyon. But it might be possible to get there by using a modified version of Williams’ simulation procedure that repeats the key step many times, saving a bit of space each time. It’s like a way to repeatedly ratchet up the length of your crowbar—make it big enough, and you can pry open anything. That repeated improvement doesn’t work with the current version of the algorithm, but researchers don’t know whether that’s a fundamental limitation.
“It could be an ultimate bottleneck, or it could be a 50-year bottleneck,” Valiant said. “Or it could be something which maybe someone can solve next week.”
If the problem is solved next week, Williams will be kicking himself. Before he wrote the paper, he spent months trying and failing to extend his result. But even if such an extension is not possible, Williams is confident that more space exploration is bound to lead somewhere interesting—perhaps progress on an entirely different problem.
“I can never prove precisely the things that I want to prove,” he said. “But often, the thing I prove is way better than what I wanted.”
Editor’s note: Scott Aaronson is a member of Quanta Magazine’s advisory board.
Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
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![This former notorious red-light district is now one of the world’s top AI hubs | TechCrunch
What every U.K. AI startup wants to know these days is, how can I get office space in King’s Cross?
The area is so hot that a VC firm allegedly recently won a deal by promising a founder office space in the neighborhood. “We stop at nothing to win deals [for] and to support” founders, “including helping them source office space when needed,” the firm told me when asked about the rumor, declining to confirm or deny any details.
The neighborhood’s popularity began back in 2016 when DeepMind — then newly acquired by Google — moved in. Soon after, a flood of AI startups followed, wanting to be around the Google DeepMind magic. Today, they hope to take advantage of the cluster of AI talent there.
This has transformed King’s Cross into one of the world’s top AI hubs, rivaled only by San Francisco and Beijing. Around London, it’s known by the sobriquet “Knowledge Quarter,” as it’s home to names like OpenAI, Meta, Isomorphic Labs, Cusp AI, Wayne, Recursive, and, a little farther down the road, Synthesia and Anthropic. The European Technology Network (ETN) just moved into a glossy new office nearby, while University College London sits around the corner.
Mixed in with the new developments are trendy food spots like Hoppers and BAO. Hop a train from King’s Cross, and founders can be in Cambridge in 45 minutes to source talent or can be in Paris in two hours to strike a deal.
Who would have guessed that a little more than 20 years ago, this was one of the seediest areas in London?
“In the ’80s, crack and heroin made the area a major narcotics market,” Hussein Kanji, an investor at Hoxton Ventures, said, recalling syringes in tree trunks and gangs patrolling the streets. “In 1982, the local church was occupied by the English Collective of Prostitutes for 12 straight days.” Then, in the early 2000s, a real estate developer had a dream and, well, “now it is the AI hotbed of the United Kingdom,” Kanji said. “What a change.” Around 18 months ago, his portfolio company BioCorteX moved from the neighborhood Holborn to the Jellicoe building in King’s Cross, hoping to be near the action. “Lots going on in London right now,” Nik Sharma, co-founder of BioCorteX, told me. “Lots of hyperscalers moving in.” That includes, reportedly, Jeff Bezos’ AI company Prometheus, which is also said to be in talks to move into the Jellicoe.
There are around 3,600 AI startups in London, which, together, have raised around .1 billion out of the .8 billion raised in the city since late July, according to Dealroom. Since the start of June, AI-related startups have leased more than 1 million square feet of office space in London, according to the real estate firm Knight Frank. With that, prime rents in King’s Cross have risen 18% over the past three years, Chris Dunn, a commercial insight associate at the firm, told me. That percentage represents only the largest leases encompassing at least 10,000 square feet, like the ones OpenAI and Prometheus are signing. The shorter deals go for even more, he said, and now the vacancy rate for conventional office space is just 0.9%. “Demand has outstripped supply,” he continued.
Today, one of the big topics of the area is sovereignty. It was a wake-up call for many when Anthropic shut off access to Mythos and Fable this summer, leaving some in the ecosystem to conclude: “We’d better look after ourselves,” Saul Klein, co-founder of the VC firm Phoenix Court, told me.
Phoenix Court is located in the King’s Cross area and has three portfolio companies in the vicinity, including Olix (which just announced a .3 billion valuation), Early Health and CoMind. Robin Klein, co-founder of the firm, said the shutdown of Fable and Mythos access was a “small but sharp reminder that Europe can’t simply rent its AI capabilities and capacity; it needs to build and hold some of its own.” King’s Cross, he said, is where much of this building is actually happening.
“The bigger question,” he continued, “is whether the U.K. builds the infrastructure, compute, energy, capital, to make this self-reliance durable, rather than just hosting outposts of U.S. labs.”
Image Credits:Phoenix Court
Top founders want to stay
Simon Kohl, founder of Latent Labs, has offices in King’s Cross and San Francisco. The London office, at the moment, is growing faster, and he’s more bullish than ever on the ecosystem, he said. “The mood right now feels less like London trying to catch up and more like London becoming one of the default places to start a serious AI company,” he said. Look around and you are likely to see Wayve testing its autonomous cars. Founded in 2017 by co-founder Alex Kendall, the unicorn is one of London’s biggest success stories.
“Ten years ago, building a frontier AI company from London felt like an unusual choice,” Kendall told me. “Now it feels like an obvious one.” Wayve moved into King’s Cross in 2018 looking for a space that could double as a garage — “a rare combination in Central London,” Kendall said. He has watched the ecosystem mature around him — and it’s now evident that a startup can stay in London, raise serious capital, hire world-class AI talent, and remain globally competitive, he said. Down the street from Anthropic’s new 158,000-square-foot office is the AI agent builder Sierra and the AI video platform Synthesia.
Laura Gonzalez Florez, Synthesia’s chief of staff and head of people, says the company moved into its glossy new office building a year ago to accommodate its growing team. They were drawn to the area for the same reason as everyone else: “It’s very close to the airport … very close to where a lot of investors are,” she said.
Image Credits:Synthesia
Around two-thirds of Synthesia’s engineers are remote, Gonzalez Florez said, letting the company tap into an affordable, international, and diverse talent pool and helping it scale faster. “From London, we can hire and work, without any problem, people from anywhere, from Slovenia to Portugal,” she said.
Unsurprisingly, London’s AI boom is also causing a talent war.U.K. AI job postings have skyrocketed in the past few years, per data from PwC. When Anthropic announced it moved into town earlier this year, it listed, for example, a salary range of £260,000 to £630,000 for a machine learning research engineer when the average salary in London for the same role is around £102,000. Some founders in the U.K., like those in Silicon Valley, are being forced to raise more and bigger rounds to keep up.
“The real test is whether more globally significant AI companies are founded, funded, and scaled from the U.K., while continuing to attract the world’s best talent to build them here,” Zain Ali, founder of the King’s Cross-based AI legal firm Centuro, told me. “If that continues to happen, King’s Cross won’t just be an AI hub. It’ll become one of the U.K.’s most important strategic assets.”
When you purchase through links in our articles, we may earn a small commission. This doesn’t affect our editorial independence.#Thisformernotorious #redlight #districtis #nowone #worlds #top #hubs #TechCrunchUK This former notorious red-light district is now one of the world’s top AI hubs | TechCrunch
What every U.K. AI startup wants to know these days is, how can I get office space in King’s Cross?
The area is so hot that a VC firm allegedly recently won a deal by promising a founder office space in the neighborhood. “We stop at nothing to win deals [for] and to support” founders, “including helping them source office space when needed,” the firm told me when asked about the rumor, declining to confirm or deny any details.
The neighborhood’s popularity began back in 2016 when DeepMind — then newly acquired by Google — moved in. Soon after, a flood of AI startups followed, wanting to be around the Google DeepMind magic. Today, they hope to take advantage of the cluster of AI talent there.
This has transformed King’s Cross into one of the world’s top AI hubs, rivaled only by San Francisco and Beijing. Around London, it’s known by the sobriquet “Knowledge Quarter,” as it’s home to names like OpenAI, Meta, Isomorphic Labs, Cusp AI, Wayne, Recursive, and, a little farther down the road, Synthesia and Anthropic. The European Technology Network (ETN) just moved into a glossy new office nearby, while University College London sits around the corner.
Mixed in with the new developments are trendy food spots like Hoppers and BAO. Hop a train from King’s Cross, and founders can be in Cambridge in 45 minutes to source talent or can be in Paris in two hours to strike a deal.
Who would have guessed that a little more than 20 years ago, this was one of the seediest areas in London?
“In the ’80s, crack and heroin made the area a major narcotics market,” Hussein Kanji, an investor at Hoxton Ventures, said, recalling syringes in tree trunks and gangs patrolling the streets. “In 1982, the local church was occupied by the English Collective of Prostitutes for 12 straight days.” Then, in the early 2000s, a real estate developer had a dream and, well, “now it is the AI hotbed of the United Kingdom,” Kanji said. “What a change.” Around 18 months ago, his portfolio company BioCorteX moved from the neighborhood Holborn to the Jellicoe building in King’s Cross, hoping to be near the action. “Lots going on in London right now,” Nik Sharma, co-founder of BioCorteX, told me. “Lots of hyperscalers moving in.” That includes, reportedly, Jeff Bezos’ AI company Prometheus, which is also said to be in talks to move into the Jellicoe.
There are around 3,600 AI startups in London, which, together, have raised around .1 billion out of the .8 billion raised in the city since late July, according to Dealroom. Since the start of June, AI-related startups have leased more than 1 million square feet of office space in London, according to the real estate firm Knight Frank. With that, prime rents in King’s Cross have risen 18% over the past three years, Chris Dunn, a commercial insight associate at the firm, told me. That percentage represents only the largest leases encompassing at least 10,000 square feet, like the ones OpenAI and Prometheus are signing. The shorter deals go for even more, he said, and now the vacancy rate for conventional office space is just 0.9%. “Demand has outstripped supply,” he continued.
Today, one of the big topics of the area is sovereignty. It was a wake-up call for many when Anthropic shut off access to Mythos and Fable this summer, leaving some in the ecosystem to conclude: “We’d better look after ourselves,” Saul Klein, co-founder of the VC firm Phoenix Court, told me.
Phoenix Court is located in the King’s Cross area and has three portfolio companies in the vicinity, including Olix (which just announced a .3 billion valuation), Early Health and CoMind. Robin Klein, co-founder of the firm, said the shutdown of Fable and Mythos access was a “small but sharp reminder that Europe can’t simply rent its AI capabilities and capacity; it needs to build and hold some of its own.” King’s Cross, he said, is where much of this building is actually happening.
“The bigger question,” he continued, “is whether the U.K. builds the infrastructure, compute, energy, capital, to make this self-reliance durable, rather than just hosting outposts of U.S. labs.”
Image Credits:Phoenix Court
Top founders want to stay
Simon Kohl, founder of Latent Labs, has offices in King’s Cross and San Francisco. The London office, at the moment, is growing faster, and he’s more bullish than ever on the ecosystem, he said. “The mood right now feels less like London trying to catch up and more like London becoming one of the default places to start a serious AI company,” he said. Look around and you are likely to see Wayve testing its autonomous cars. Founded in 2017 by co-founder Alex Kendall, the unicorn is one of London’s biggest success stories.
“Ten years ago, building a frontier AI company from London felt like an unusual choice,” Kendall told me. “Now it feels like an obvious one.” Wayve moved into King’s Cross in 2018 looking for a space that could double as a garage — “a rare combination in Central London,” Kendall said. He has watched the ecosystem mature around him — and it’s now evident that a startup can stay in London, raise serious capital, hire world-class AI talent, and remain globally competitive, he said. Down the street from Anthropic’s new 158,000-square-foot office is the AI agent builder Sierra and the AI video platform Synthesia.
Laura Gonzalez Florez, Synthesia’s chief of staff and head of people, says the company moved into its glossy new office building a year ago to accommodate its growing team. They were drawn to the area for the same reason as everyone else: “It’s very close to the airport … very close to where a lot of investors are,” she said.
Image Credits:Synthesia
Around two-thirds of Synthesia’s engineers are remote, Gonzalez Florez said, letting the company tap into an affordable, international, and diverse talent pool and helping it scale faster. “From London, we can hire and work, without any problem, people from anywhere, from Slovenia to Portugal,” she said.
Unsurprisingly, London’s AI boom is also causing a talent war.U.K. AI job postings have skyrocketed in the past few years, per data from PwC. When Anthropic announced it moved into town earlier this year, it listed, for example, a salary range of £260,000 to £630,000 for a machine learning research engineer when the average salary in London for the same role is around £102,000. Some founders in the U.K., like those in Silicon Valley, are being forced to raise more and bigger rounds to keep up.
“The real test is whether more globally significant AI companies are founded, funded, and scaled from the U.K., while continuing to attract the world’s best talent to build them here,” Zain Ali, founder of the King’s Cross-based AI legal firm Centuro, told me. “If that continues to happen, King’s Cross won’t just be an AI hub. It’ll become one of the U.K.’s most important strategic assets.”
When you purchase through links in our articles, we may earn a small commission. This doesn’t affect our editorial independence.#Thisformernotorious #redlight #districtis #nowone #worlds #top #hubs #TechCrunchUK](https://techcrunch.com/wp-content/uploads/2026/08/DM9A2852.jpg?w=680)

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