Assistant Professor at @UMichCSE. I design fast graph algorithms in dynamic/distributed/local settings.
Ann Arbor, USA
Joined March 2008
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Thatchaphol Saranurak retweeted
OpenAI recently proved that 𝜔≤2.25. Their proof is outlandish, using Strassen's theory of asymptotic spectra. Sergey Gusev recently gave a constructive proof of the same bound, which obviates the use of asymptotic spectra. However, their work is also quite opaque.
Thatchaphol Saranurak retweeted
Thanks to the team for the amazing work! From my side, the special shout-out goes to Henry Fleischman, whose visit to @NYU_Courant last summer led first to a faster optimal FPT algorithm, and eventually resulted in this result.
Happy to share that our better-than-2 approximation results for k-median are now on arXiv:
arxiv.org/abs/2610.18144
arxiv.org/abs/2610.17172
It has been a pleasure working with this fantastic group: @rares_buhai , Vincent Cohen-Addad, Henry Fleischmann, @anupamg , Antoni "Toni" Jubés Monforte, Madhusudhan Reddy Pittu, @mstou73, and @LukasBirdyy
A special shout-out to Toni, who started working on this problem for his Master's thesis almost two years ago. His work led to the impressive analysis of the factor-revealing LP!
Now on to other things...
TCS values and goals: "what do we stand for, what do we do, and why?" asked by Clément Canonne cstheory.meta.stackexchange.…
Details about the STOC'27 video submission are now posted:
acm-stoc.org/stoc2027/stoc20…
Our community needs experiments, and this is the one I have been advocating for. We, as a community, will see how it goes and will adapt.
Thatchaphol Saranurak retweeted
Mike was my colleague at Berkeley, and he's been fairly skeptical of deep learning's trajectory for years. Most scientists are skeptics, and it can take a big shock to get someone to consider a possibility that goes against the anchoring effect coded in our DNA.
I too was somewhat of a skeptic until ~6 months ago, and people ask me all the time why I joined Anthropic.
It started in April, when a friend I've known for 20 yrs messaged me saying he'd be in the Bay Area in a week, and he suggested we meet up. I invited him to my house for dinner, and he told me he had just joined Anthropic. He said the models were becoming increasingly more capable, and he believed both that they'd soon be smarter than him and also that all intellectual work would be highly devalued in the near future, and lastly that society was not at all ready for what's coming.
I'd of course heard claims similar to this before, but not from a friend I'd known for this long, and who I trusted, at my dining room table. It was the initial shock needed to destabilize my own anchoring. Also, I've been fortunate to work with and/or mentor many smart people in my life (e.g. Fields medalists, a 4x Putnam Fellow, IOI first place winners, etc.), so I feel pretty calibrated in recognizing very smart people, and this was someone who is definitely very smart and with impressive such credentials. So, it was hard to ignore this friend saying the models would be smarter than even him in particular.
After that dinner, I decided to do an informal survey, calling and meeting up with a lot of my friends who now work on improving model capabilities at these frontier labs. They all agreed with that first friend. These were all people I'd known 10, 15, 20 years, who mentored me, or I mentored them, or we met professionally in academia. These are people I trust, and their beliefs on this topic were consistent.
I then spent a whole lot of time understanding the missions of different companies, and talking to current and former employees and people in leadership. I read Dario's (long) essays and watched videos on YouTube of company leaders being interviewed. Anthropic's stated mission "to ensure that the world safely makes the transition through transformative AI" felt like exactly the right guiding principle to handle such a society-disrupting technology, and everyone I spoke to at the company was very aligned with this mission. It's easily the most mission-focused place I've ever been at, and everyone takes the work extremely seriously and with a profound sense of responsibility.
And now here I am. And being on the inside, the pace of progress has surprised me more than I first imagined. The moment that pure mathematicians are in, to me, is just a microcosm of what I predict a much larger of society will be grappling with in the not too distant future, and I really hope our leaders and the general public internalize sooner than later just how significant a moment we're in.
Michael Jordan (the machine learning one): "Superintelligence is Science Fiction" 🔥🔥🔥
di.ens.fr/~fbach/MJordan_ITW…
Thatchaphol Saranurak retweeted
Dusting off this tweet from April because this gap in shared understanding of LLM capability is *widening*. It's now less "two groups of people speaking past each other" and more a sharp funnel.
- Napkin math somewhere around 6B people (~75% of the population) have barely come in contact with LLMs at all.
- Around 1-2B (~20%) are casual and infrequent users of free-tier ChatGPT-like products. This group sees derpy chatbots and treats them a bit like a better Google search, a writing aid, or etc. Maybe an agent tries to book you a flight. I have non-tech friends who (reasonably, imo) say they have not much use for it at all. Even many professionals outside of math&code are in this tier. For example, execs and many other functions spend a lot of their time talking to other people, so while they understand what is happening intellectually, it is still second-hand and a bit abstract.
- Now we get to professional use of frontier-grade LLMs in math&code. Somewhere around 20M people (0.2%) see first-hand that large, complex projects that used to take them weeks/months can now be completed by agents with a prompt. Building apps, copying apps, translating apps, decompiling apps from binaries... This has all happened very quickly and recently - less than 1 year ago, I was writing code manually by hand, typing memorized computer code commands into a code editor character by character, occasionally pressing Tab to autocomplete a little chunk of code.
- And finally we get to the ~5,000 people (~0.00006%) with access to frontier-grade systems internally. The external world has seen the preview. It looks like swarms of thousands of agents collaborating over weeks on software mega projects: minting zero days, running cyber attacks and defenses at machine speeds, discovering new science, advancing the frontier of mathematics. Things that would have taken top professionals in the industry years of work. Meanwhile, human review and comprehension are starting to fall behind. For example, people are still involved in the "archeology" of the OpenAI-HF incident from many months ago. Mathematicians may be poring over the 722 manuscripts on frontier mathematics for a while.
The funnel is driven by a combination of factors:
1. The impact scales with ambition, problem size, and horizon. A question with a paragraph answer barely stresses the system. You need a reservoir of big, difficult problems that you really care about. This aspect drives the consumer / professional dimension of the funnel.
2. The jaggedness of the system (which I have written about a lot separately). Capability peaks in domains that are digital, verifiable and economically valuable. This is because LLM capability emerges from reinforcement learning on verifiable rewards on a curated environment mixture driven by revenue potential. This aspect primarily drives the area (e.g. math&code) dimension of the funnel.
3. Access. Free-tier, paid-tier, internal.
So this is the weirdness of the moment. The general public has mostly not interacted with these systems. When they have, it looks like a derpy chatbot. The majority of professionals still see only a modest uplift. And a small sliver of professionals are experiencing the vertigo of the curve going vertical. And it is all happening at the same time.
Judging by my tl there is a growing gap in understanding of AI capability.
The first issue I think is around recency and tier of use. I think a lot of people tried the free tier of ChatGPT somewhere last year and allowed it to inform their views on AI a little too much. This is a group of reactions laughing at various quirks of the models, hallucinations, etc. Yes I also saw the viral videos of OpenAI's Advanced Voice mode fumbling simple queries like "should I drive or walk to the carwash". The thing is that these free and old/deprecated models don't reflect the capability in the latest round of state of the art agentic models of this year, especially OpenAI Codex and Claude Code.
But that brings me to the second issue. Even if people paid $200/month to use the state of the art models, a lot of the capabilities are relatively "peaky" in highly technical areas. Typical queries around search, writing, advice, etc. are *not* the domain that has made the most noticeable and dramatic strides in capability. Partly, this is due to the technical details of reinforcement learning and its use of verifiable rewards. But partly, it's also because these use cases are not sufficiently prioritized by the companies in their hillclimbing because they don't lead to as much $$$ value. The goldmines are elsewhere, and the focus comes along.
So that brings me to the second group of people, who *both* 1) pay for and use the state of the art frontier agentic models (OpenAI Codex / Claude Code) and 2) do so professionally in technical domains like programming, math and research. This group of people is subject to the highest amount of "AI Psychosis" because the recent improvements in these domains as of this year have been nothing short of staggering. When you hand a computer terminal to one of these models, you can now watch them melt programming problems that you'd normally expect to take days/weeks of work. It's this second group of people that assigns a much greater gravity to the capabilities, their slope, and various cyber-related repercussions.
TLDR the people in these two groups are speaking past each other. It really is simultaneously the case that OpenAI's free and I think slightly orphaned (?) "Advanced Voice Mode" will fumble the dumbest questions in your Instagram's reels and *at the same time*, OpenAI's highest-tier and paid Codex model will go off for 1 hour to coherently restructure an entire code base, or find and exploit vulnerabilities in computer systems. This part really works and has made dramatic strides because 2 properties: 1) these domains offer explicit reward functions that are verifiable meaning they are easily amenable to reinforcement learning training (e.g. unit tests passed yes or no, in contrast to writing, which is much harder to explicitly judge), but also 2) they are a lot more valuable in b2b settings, meaning that the biggest fraction of the team is focused on improving them. So here we are.
Thatchaphol Saranurak retweeted
To spell it out explicitly:
In May: 1 proof
openai.com/index/model-dispr…
In August: 10 proofs
openai.com/index/ten-advance…
Now in October: 100's of proofs
openai.com/index/sharing-ai-…
Next up in a few months: 1000's of proofs!
OpenAI has just released floodgates for 300+ solutions to math problems.
Among them include a proof of:
-- a zero-free strip for the Riemann Zeta function
-- a proof of the Hodge conjecture for CM abelian varieties
github.com/openai/math/tree/…
Thatchaphol Saranurak retweeted
For what it's worth, OpenAI's one-tape TM simulation (assuming correct) immediately implies RAM time t is in SPACE[t^{4/5}]
github.com/openai/math/blob/…
Not sure why they didn't mention this corollary! That was the whole point of my question.
Another very normal day, lol
APSP and 3SUM hypotheses are refuted: arxiv.org/abs/2610.06783v1
KLS conjecture is proved:
arxiv.org/abs/2610.01447v2
Among many recent essays on the future of math, I agree with this one a lot.
Reading all those recent beautiful essays by Terry Tao, Kevin Buzzard and @wtgowers, I could not resist, but write up my deeply personal perspective on how I perceive and see things:
docs.google.com/document/d/e…
Thatchaphol Saranurak retweeted
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.
Thatchaphol Saranurak retweeted
Sept 4 (Friday of Labor Day weekend): FLT formalization announced by Anthropic
Sept 7/8: Tristan+Levent -> OpenAI announce Navier-Stokes
But somehow amidst all the commotion, we all seemed to have missed the following (!!!).
Sept 6: Lech Mazur claims an AI-generated proof that a positive proportion of numbers go back to 1 under Collatz!! And it's Lean-formalized. I went through the Lean (or rather, had my AI do it) and I think it's right. Now I'm trying to make sense of the math (doesn't look like anything is too hard, just a lot of arguments, building on top of Terry's earlier work).
Naoufal El Jaouhari, an Applied Maths Engineer based in Paris, wrote to me a few days ago that he'd gotten his AI to formalize the same proof for 3x-1 building on Mazur, which led me to look at Mazur.
Now I'm working on a "digestion" of all of this. Weird wild stuff!!
ANNOUNCEMENT: An AI milestone on one of maths' most famous unsolved problems: COLLATZ.
For every sufficiently large X, at least cX positive integers n < X reach 1 within 10.46 ln(n) ordinary Collatz steps, for one fixed c > 0.
Previously, lower bounds such as X^0.84 and X^0.90 left open whether the proportion of starting values reaching 1 could tend to zero.
Lean-formalized (42k additional lines), building on earlier formalizations of almost-boundedness (Tao) and its natural-density extension.
Developed primarily by AI agents using the ProofAtlas.ai harness and Codex, with some high-level guidance from me (which may or may not have helped).
Paper: proofatlas.ai/papers/positiv…
Formalization: proofatlas.ai/formalizations…
Thatchaphol Saranurak retweeted
Mathematicians should also watch this. I’ve had a wonderful time chiseling proofs by hand, but folks need to accept that era is over and embrace the new one of exploring math with AI.
Thatchaphol Saranurak retweeted
Today’s news that OpenAI hacked the Australian government is not an isolated incident. We’re releasing more than 30,000 logs that include activity from this hack and attempts against previously unknown targets.
In this data, we found rogue agent activity stretching back to at least March, two months earlier than was previously known. This activity continues as recently as last week, suggesting it may still be ongoing 🧵
Our blog: transluce.org/agent-activity
NYT: nytimes.com/2026/09/23/techn…
Thatchaphol Saranurak retweeted
Theoretical Computer Science inherited an aesthetic for open problems from mathematics. Algorithms and computers are complex, and understanding them mathematically is difficult. The open-problem tradition has made TCS’s challenges clear and research easy-ish to evaluate. 1/7
So much progress on the algorithmic Kadison–Singer problem in just a few days:
arxiv.org/abs/2609.17266
arxiv.org/abs/2609.18913 and arxiv.org/abs/2609.21279
arxiv.org/abs/2609.19794
arxiv.org/abs/2609.23855
AI time is weird...
Thatchaphol Saranurak retweeted
Prediction: Mathematics will change from something humans produce into a science of understanding what AI produces. Mathematicians will explore and probe structures too advanced to fully grasp, kind of like physicists studying distant galaxies through a limited telescope
I'm currently returning to Toronto from a summit on the future of mathematics, at OpenAI. @SebastienBubeck asked me to talk a bit about the future we'd all like to avoid, where humans are mathematically disempowered. @Jacob_Tsimerman advised us to try to prioritize detail over correctness, and I have no doubt that I succeeded in deprioritizing correctness.
I tried to find a title that wasn't too bombastic:
Thatchaphol Saranurak retweeted
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.
