Earlier this month, I participated in a two-day workshop at the Simons Institute on how the TCS community should adapt to rapid advances in AI. Our report, AI and TCS: The Next Six Months, is now out: simons.berkeley.edu/ai-tcs-w…

Two years ago I argued we needed an autoformalizer. Now the time has come to formalize all of mathematics. We need to unite behind a single Mathematics Autoformalization Project that turns all known mathematics into one Lean library. @jdlichtman @patrickshafto

Hello, world! There is a new repository for works in math and TCS: hexagonmath.org, now in public beta. Please try it out and submit something! We welcome your feedback while we build and improve it.

In the years since the release of ChatGPT, it was common for skeptics to claim that AI was just hype. That talk has thankfully faded. The zeitgeist has now flipped, and fears about the potentially existential threats posed by the technology are widespread. I very much share those concerns. But I worry that in the leap from disbelief to alarm we have skipped over the vast area in between, in which AI’s impact will be felt in a thousand different ways: good and bad, mundane and weird, incremental and world-changing. Few places will feel that impact more than New York City, which is deeply exposed to disruption of our economy, workforce, budget—even the basic functioning of government—by this powerful new technology. This would be true even if AI advancement were to stop now. We have only barely begun to feel the effects of existing frontier systems like Astra and Fable throughout our economy and society, since only a tiny portion of people and organizations are making use of their highly advanced agentic features. Meanwhile, leading labs are reportedly testing far more capable models behind closed doors, while racing toward systems that can automate AI research itself. With more powerful chips and vast new compute infrastructure now coming online, rapid AI progress will continue even if the financial bubble bursts. None of this means we should surrender to the race. I believe Congress should immediately establish strict safeguards against catastrophic AI risks and empower an independent agency to enforce them. The United States should also pursue common safety measures with China. Neither will be easy under a president who has embraced an extreme hostility to AI safety. But we cannot allow a handful of tech billionaires to decide for themselves how much risk the rest of humanity should bear. Whatever happens in these high-stakes fights, this much is certain: enormous disruption is already headed our way. And New York City is simply not ready. The great paradox of New York City is that few places are more exposed to AI’s potential downsides, yet few have benefited more economically from the boom so far. The AI-fueled stock market has filled New York’s tax coffers. The data center buildout and the big AI IPOs are being financed by firms here. The brisk pace of office rental in NYC is in part driven by the many AI firms starting or expanding here. All that upside also creates a serious vulnerability for us: a collapse in AI valuations could badly hurt New York’s economy, even if the underlying technology continues to advance. The risks extend well beyond Wall Street and the tech sector. New York is the capital of white-collar work, with more than a million people commuting into Manhattan daily. Many are employed in the industries that are the most likely to be impacted by AI automation: accounting, law, finance, consulting, design. There has thankfully so far been no significant increase in layoffs in NYC. But job growth has come to a halt here, and job postings are down for the most AI-exposed careers. Perhaps most tellingly, the unemployment rate for New Yorkers aged 22-27 is higher for those with a college degree than it is for those without one. New York City has a $126 billion budget that is growing fast, with only $2 billion in our rainy day fund. If the AI bubble bursts and our tax revenue takes a hit, we have far too little buffer to avoid cuts to vital City services. If large numbers of New Yorkers lose their white-collar jobs because of automation, we have no pool of funds ready to provide them wage support or to offer large-scale retraining. City government itself may be the institution most vulnerable to rapidly advancing AI. We have strong cyber defenses for the threats we face today. But our financial, payment, vendor, and accounting systems have not yet been hardened against a new class of autonomous AI adversary. This is not a distant threat: current and pre-release models have already demonstrated the ability to execute complex, multi-stage cyberattacks at a speed and scale no human defenders can match. The emergence of vast numbers of capable AI agents has many implications for City government beyond cyber. It is just a matter of time before autonomous systems start making FOIL requests, filing legal claims, applying for permits and benefits, submitting job applications, reporting fraud allegations, asking questions to 311, or making 911 calls. Our government processes were built around an assumption we have barely thought about: that every request, application, complaint, and claim requires a human being to spend time creating it. AI will erase that constraint. NYC simply does not have systems capable of dealing with what could be the resulting exponential increase in the volume of incoming. Our government is built on a rickety tech foundation that includes dozens of mainframes from the 1980s. We largely missed the cloud revolution, the mobile revolution, and the big data revolution, and are now dragging our feet on the machine-learning revolution. That will have to change if City government is to remain resilient as AI transforms the society around it. Every City agency has to address a long list of questions and challenges created by the arrival of advanced AI. NYPD will need to figure out how to balance the allure of powerful new AI tools against the imperative to preserve privacy. Our public schools will have to rethink homework in a world where students have access to AI at home or on their cell phones by fourth or fifth grade, if not earlier. Workforce agencies will have to figure out what jobs they are even training people for now. The Health Department will have to adapt disease surveillance for the AI era. We have work to do to ensure the benefits of this technology are shared broadly. That means, among other things, making sure New York City leads in AI-enabled biomedical research and that the resulting breakthroughs benefit people at every income level. It means giving small businesses, not just megacorporations, access to cutting-edge tools. And it means using AI to make government dramatically easier to navigate, especially for those on the margins—for example, by allowing New Yorkers to apply for SNAP benefits simply by speaking into their phones in whatever language they prefer. Potential policy wins like this may sound trivial compared to the existential risk that runaway superintelligence poses. And there is no doubt that ensuring humanity remains in control of increasingly powerful AI is an urgent task. But there is upheaval already underway. Change is coming fast. Steering to an equitable, healthy, deeply human future for the millions who call this place home will require nothing less than the reinvention of our institutions, including the government of New York City. It will require asking profound questions about what kind of city we want to be. Whether you are feeling hope or fear or doubt right now, you have a role in answering these questions. Together we will make a thousand choices, large and small, that will determine the New York we become.

Great progress towards formalizing MIP* = RE.

Forget dependency graph, prove2me has dependency galaxies :)

This echoes a sadness I've been feeling recently (among many sadnesses): I love to write, and I aspire to write well, but who will read what I have to say?

In fact I think this might be the most exciting time in recent memory to be a mathematician, turbulence in the field aside.

I guess this result is morally close to 2-to-1 Conjecture with perfect completeness. (Maybe with tweaking you can get 4->2, who knows.) Then it's true that UGC is somewhat different, being stronger in one sense (1-to-1) and weaker in another (imperfect completeness). But...

Some thoughts on AI and Theory. 1. To a first approximation, theoretical computer science has been organized around a few major open questions. Much of our work has been motivated by developing approaches to answer these questions. 2. Such “problem-motivated” work has often led to theory-building focused on identifying a general principle that unifies a class of theorems. But much of that theory-building also involved proving new, difficult theorems. 3. Thus, while it’s true that problem-solving was strongly correlated with building understanding, drawing connections, and eventually developing general theories, it would be disingenuous not to admit that our community, perhaps disproportionately in retrospect, focused on and celebrated problem-solving. This was not arbitrary, and was quite defensible. Being able to make progress on central technical questions usually correlated with taste, creativity, persistence, and depth of understanding. Much of our reward structure therefore implicitly relied on the fact that producing an important proof was good evidence that someone possessed these harder-to-observe qualities. 4. It seems likely that we will soon have AI tools available to us that can prove many such theorems in a short time. The cost of obtaining proofs for well-posed mathematical questions will likely fall dramatically. The “scarce” intellectual work will likely shift both upstream: to questions, models and theories, definitions, and conjectures, and downstream: to interpretation, synthesis, explanation, and theory-building. 5. But as long as we believe in humans being meaningfully in charge of our collective decisions and fate, building human understanding of our science (and of science more generally) will remain an essential goal. I plan to expand on this important aspect soon. 6. Historically, finding a solution to an important problem and understanding its significance, implications, and connections were entangled. Finding a proof usually required researchers to discover the right concepts along the way. A dramatic reduction in the time and effort required to prove theorems could break that coupling. We could end up with many more true statements and proofs without a commensurate increase in understanding. Converting an abundance of proofs into human understanding may become one of the central challenges of our field. 7. As a result, I expect the high-level goals of theoretical computer scientists to change. In fact, the advent of powerful theorem provers might help us construct new theories and explore new models far more easily and rapidly, and significantly expand the domains where our models and theories apply. In that sense, the space for theoretical work may significantly expand rather than contract. 8. There’s a high human cost to the disruption that we are likely heading into. Many in our field, and in mathematical communities more broadly, are coming to terms with it. The range of opinions and reactions among mathematicians and theoretical computer scientists is a natural part of this evolution in our thinking as we collectively work through it. Some concrete efforts (including one at @SimonsInstitute) are already underway to think through the immediate scientific and institutional questions arising during this transition.

WIRED reported today that I began to cry while talking about AI and mathematics. But the article didn’t explain what moved me. The truth is, I’m not entirely sure myself. I’d like to try to explain. 🧵 wired.com/story/mathematicia…

A situation that would have been unimaginable even months ago: there's a debate over whether an AI-assisted student math marathon is good for the field. 🤯 As a former student math researcher myself who's mentored such students for a decade, I want to weigh in with an offer, not just an opinion: **I'm volunteering to mentor at least three Mathathon teams as they work to turn their projects into rigorous, complete mathematical contributions. I invite other mathematicians to join me.** Working with high school and college math research students has been one of the greatest privileges of my career. They're often learning the background while trying to solve the problem—but aren't we all? And while their first-draft proofs aren't always journal-ready, that's precisely what mentors are for. Whether you see AI as a boon, a threat, or both, hundreds of students spending a weekend doing math should be an opportunity—certainly not a burden for the field as some have suggested. Telling students they shouldn't math because they aren't sufficiently professionalized is counterproductive. If we're concerned about incomplete or hard-to-read proofs, we should treat that as a teaching opportunity: let's help them (and other aspiring mathematicians) learn how to take their work up to the next level—and we'll probably learn some things from them in the process, too. Conversely, if we aren't willing to coach them, we have little leverage to try to dictate what their output should look like. **The point isn't to lower math's standards. It's to help students level-up to meet them.** So: at least three teams from me, QED. I hope others will offer to mentor as well. (I'm not directly in contact with the Mathathon organizers, so I'd appreciate a connection to help coordinate.)

I think it's important for OpenAI (and other frontier labs) to clarify and address these concerns in a considerate and thoughtful way, especially given how much value they are deriving from these announcements, and more importantly, the value they have derived from the work of the mathematical community, either through training data or otherwise.

One common prediction these days is that AI will only produce alien-like, incomprehensible, opaque proofs that don't advance human understanding. I don't subscribe to this view 1/8

Among other things, what’s fascinating is that when Anthropic first tried to formalize Fermat’s Last Theorem with Claude agents, early runs failed because agents lost track of project state. Switching onto Prove2Me which had a DAG and separated, stable proof statements that could be searched is what let the agents coordinate and finish the proof in about 11 days. Really shows you not just the power of the LLM but of what we call the harness. Keeping track of project state is especially going to be terribly important for all kinds of scientific problems.