Founder & CEO of Alembic, CUDA architect, wrangler of GPUs, 1st gen Cuban. I tweet in long delayed bursts as I’m usually too busy building.
San Francisco, CA
Joined June 2008
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More people in mathematics and AI should read literature and watch theatre. I just got to see Tom Stoppard’s Arcadia again at @stoppardtheatre
A play about math, love, intimacy and physics. Some thoughts:
Some thoughts seeing it after a couple decades. My thoughts and pondering after seeing it.
- The deepest mathematical idea in Arcadia is that knowing the laws of a system is not the same as being able to predict it, reverse it, or reconstruct its past.
- Complicated behavior does not require a complicated equation or a random input.
having a closed-form solution does not automatically give you useful long-range prediction.
- more compute can reduce numerical error, but it cannot manufacture measurements you never had.
- A very complicated object can have a short generative description.
- Thomasina can glimpse the procedure without having the computational capacity to explore its consequences at scale
- Mixing happens readily; the reverse does not happen spontaneously on ordinary human timescales. A statistical formulation could use Boltzmann entropy.
How do reversible microscopic laws produce effectively irreversible macroscopic behavior?
- Statistical mechanics provides a powerful explanation involving typical microscopic states, macroscopic descriptions, and low-entropy initial conditions. Fine-grained Gibbs entropy under Hamiltonian evolution need not increase even while the system’s macroscopic Boltzmann entropy does. Those are different descriptions, not a contradiction.
- this kind of makes me wonder why the universe had such a low-entropy past in the first place.
- Chaos and entropy must be kept separate: sensitivity to initial conditions is not the same thing as increasing thermodynamic entropy.
- Mathematical certainty and historical certainty are different achievements.
In the end… Understanding the general process by which things happen does not necessarily let you recover a particular life, conversation, or lost moment.
Most compelling thing to me.
That play is not saying mathematics understanding is futile. It is showing why understanding, prediction, control, and recovery are four different things…. the desire to understand still matters… even with the other three are out of reach.
@ACTSanFrancisco you’re all sleeping on the best play to run against the @AnthropicAI and @OpenAI IPOs.
Now the Hugginface acquisition rumors make sense. Jensen loves himself some kawaii robots.
We have a huge news to share today!
Today we are unveiling the first truly accessible RL robot - welcome Microduck
A 25 cm tiny open-source biped with 15 actuators and packed with sensors (camera, speaker, LiDAR, NFC, bluetooth, wifi, etc) that you train yourself with reinforcement learning.
It's also playable out of the box with more than half a dozen fun and playful pre-trained policies to have it walk, sit, crouch, roller-skate, pick up objects with its articulated beak, and recover on its own.
And all for less than $400.
See all the details, play with the simulator and order it at: pollen-robotics.com/microduc…
(video with sound on 🔊)
A pure speeds and feeds advertisement from Apple. No emotion, just bragging, so we’re now two generations away from Steve Jobs.
I trust @SemiAnalysis_ less and less these days. Starting to feel like a mix of bombastic claims, smaller tests, and insider trading. And the timing of the article drops and tone of the exclusives is eyebrow raising.
Tomás Puig retweeted
This is quite a remarkable result:
Posed in 1939, the Jacobian conjecture is one of the central open problems in algebraic geometry, but was just disproved by Alpoge, Matthew, and Claude Fable 5.
The Jacobian conjecture roughly says that a multivariable polynomial F has an inverse function (made out of polynomials) provided the Jacobian is non-singular (i.e. matrix of partial derivatives has non-zero determinant). This condition is neccessary by the Inverse Function Theorem from multivariable calculus. The hard question is whether it is also sufficient.
Evidently, Fable found that
F(x,y,z) = ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z)
has det(J_F) = -2 non-zero. However F is not invertible, since F sends three different pts (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to the same image (-1/4, 0, 0).
Beyond the disproof itself, it would be value to know if a suitably refined conjecture is recoverable. Per @Acer, GPT5.6 has proposed:
"A constant-Jacobian polynomial local biholomorphism with no loss of sheets at infinity—e.g. a proper Keller map—is an automorphism."
I would be interested to know if any algebraists (e.g. @levent @littmath) have a reaction to this...
A further twist to the story:
Not only was the Jacobian conjecture one of the central open problems in algebraic geometry, it was (a special case of) Yitang Zhang's PhD problem!
The catch was that Zhang's advisor had him solve it, assuming a lemma of his advisor. But that lemma turned out to be false! As a result, Zhang's thesis crumbled and he then struggled to get recommendation letters and a permanent academic position. Despite all this, Zhang went on to prove bounded gaps between primes!
This is one of the most inspiring stories in modern mathematics, and was a motivation for me to work in the same area for my doctorate.
hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final
((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)
Tomás Puig retweeted
hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final
((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)
Tomás Puig retweeted
it's worth noting that due to rising costs of pretraining, the frontier isn't *that* accessible. training a 5T-A100B model on 50T tokens on GB300s costs like $10 million for the hero run. double it for RL. there's only a few thousand companies in the world that could justify it
I’ve found 5.6 is quite good for multi-gpu ML work and complex orchestration code. Much better than Opus 4.8 was.
Tomás Puig retweeted
Collette shouldn’t be calling Dylan anymore. There should be no more privileged information, access, or support given
SemiAnalysis got way too influential way too fast. Not enough life experience, and now you have a bunch of kids effectively pushing around the narratives that our entire stock market revolves around. Platforming them was a mistake
Great now Anthropic is starting to use the word consciousness for LLMs. Why did they not learn over claiming things hurts them in the long term? 🙄
When people discuss that one model, or one algorithm change, is going to move the market I like to remind them of Jevron’s paradox.
“As technological improvements increase the efficiency of a resource, the total consumption of that resource often increases rather than decreases. Because efficiency lowers the relative cost, it unlocks new demand and widespread adoption.”
The problem with the anti open source discourse for AI is key people not understanding many find a liberated fragment of reality more valuable than a larger captive universe.
No matter the choice, open or walled garden, it remains a prisoner of the data that created it.
Tomás Puig retweeted
We consume data we did not create.
We inherit tools we did not invent.
We run on chips we did not make.
But when the commons bears fruit, we fence it.
