3SAT in subexponential time (2^{n / log^eps(n)}) is maybe the biggest surprise I have seen during my 20 years following theoretical computer science (still remember the feeling 25 years ago seeing the diff a 3 versus a 2 can make for complexity 😅)

Announcing three major scientific discoveries, one each in mathematics (harmonic analysis), computer science (complexity theory), and physics (quantum field theory). AI starting to help us improve our understanding of the universe!

The computer science collaboration will understand and explain a result concerning the Exponential Time Hypothesis (ETH), a longstanding conjecture about the computational complexity of Boolean formula satisfiability, a problem deeply related to the P vs. NP problem. Surprisingly, our system has recently proved that the ETH is false, a result that was subsequently verified in Lean. The refutation of the ETH reopens questions about how much faster these tasks could become, and may help guide the search for better algorithms. Finding such algorithms, and turning theoretical advances into practical improvements, will require significant further research. Our system found its natural language disproof for the ETH we reached out to to @rrwilliams. Ryan will form a collaboration with Russell Impagliazzo and Ramamohan Paturi to verify and shepherd this result. Impagliazzo and Paturi, along with Francis Zane, are the original proposers of the Exponential Time Hypothesis, and Williams has refuted strong versions of the ETH, used ETH for hardness results, and connected algorithms and lower bounds. This group asked us to make all raw data of the results public to facilitate independent verification by the community, which we have done by posting the natural language proof and Lean formalization to our GitHub repo. The group also invites the community to help organize an open examination of the ETH claim and to discuss the broader questions that AI-generated breakthroughs raise about verification, access, credit, and the future of research.

On 6 October, James Maynard was nervously checking his phone under the table throughout dinner. He’d heard rumors of a tsunami heading for the shores of math. When he checked in one last time from bed, the swell struck: OpenAI had released more than 700 math papers, generated with an advanced version of its AI model. They appeared to topple hundreds of long-standing problems, forcing researchers to face the reality that no problem in math is off-limits for machines. At first, Maynard, a mathematician at the University of Oxford and recipient of the Fields Medal, math’s most prestigious prize, feared for colleagues whose work would be upended. But then he spotted a paper that offered progress on one of his favorite pastimes: the Riemann hypothesis, a puzzle about prime numbers and one of seven famed Millennium Prize Problems, often considered the most formidable in all of math. “I was like a kid at Christmas,” Maynard says. “Even if it was some scary monster giving me a gift, I was just so excited.” The release comes less than 1 month after OpenAI announced a solution to the Navier-Stokes equations, another Millennium Prize Problem. OpenAI’s latest proof solves only a weaker version of the Riemann hypothesis. But Maynard, who made the most recent notable Riemann advance in 2024, says it is “groundbreaking” and likely to have payoffs for other problems across math. “Within analytic number theory, it’s obviously the best headline result for the past 100 years in the field.” Learn more:

When I was a postdoc at MIT over 10 years ago, Jeffrey Goldstone, then in his 80s, would still come in every single day. He rarely spoke with the high-energy theorists. Rather, at that time, he kept mainly to quantum information side of the CTP. Here's my one story with him 1/7

We want to support the mathematics community in turning progress in AI into deeper human understanding and new discoveries. To support that goal, we are contributing $5 million to a new Simons Foundation initiative to help mathematicians examine, understand, and build on results produced by AI, including the results we recently published. The funding, managed and awarded by the @SimonsFdn will support workshops, conferences, and special programs. We will also provide access to our models and computing resources to Simons Foundation-supported researchers.

Wow. @physics_nate (currently on leave at OpenAI) has made significant progress on a very deep problem relevant to our understanding of the Standard Model (SM) of particle physics. It’s technical, but briefly: the matter content of the SM (electrons, quarks, etc) are fermions and they are “chiral” (they have left-handed and right-handed components that behave differently). We’ve lacked a good way to describe / simulate such systems “non-perturbatively” (which is a controlled way of doing approximations). The paper connects with foundational ideas laid out by Lüscher (who is wicked smaht) and uses GOT Astra + lean formalization in ways that are perhaps subtly different than in math because of the role of recorded assumptions vs. axioms.

We've updated our GitHub math repo with 6 new Lean formalizations, 19 modifications, and 3 withdrawals. The repo now has ~42% top-line results formalized. We will continue to update the repo with new formalizations and with any errata we notice.

That janitor just goes from building to building helping people. COMIC ◆ PATREON ◆ STORE ◆ #comics#

AI agents say things like ARGH and OH MY GOD in their chains of thought. Good reason to think it's not just imitation but that these words play functional roles for doing reasoning. What comes after "oops" is very different than what comes after "aha"...

We recognize that the rapid progress of AI in mathematics is disruptive. We're looking to engage with the math community more on the best way to integrate this technology and communicate its impacts. After considering concerns raised by members of the mathematics community, we’ve decided to withdraw OpenAI’s sponsorship of the Caltech Mathathon and have notified the organizers.

While a resolution to Navier-Stokes is the highlight of today, this plot is really the headline result to me. Our ongoing training run demonstrates unprecedented performance on mathematics research, as measured by our model's estimated performance on a collection of **open** math problems. Of course, to settle Navier-Stokes, we took this model and also scaled test-time compute allll the way up.

If you are not a mathematician, you might not care that some differential equations have a singularity. But if you are a resident of planet earth you should care that (1) an AI was able to solve a Millenium problem, and (2) that there is plenty of headroom for models to move up and to the right on this and any other task.

Two things to distinguish: Did any human or agent look at user data as part of the Navier Stokes effort? No. Do we use user feedback and de-identified data to improve ChatGPT and Codex in a holistic way? Yes. And so does every LLM company.