szegő assistant professor of mathematics @Stanford
Joined July 2024
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Notably, Bombieri-Stepanov used the polynomial method to prove the Riemann hypothesis for curves over finite fields
One of my long-time dreams is to see the polynomial method proof generalized to the Riemann hypothesis for varieties...
Moreover, this proof may be viewed as an example of the polynomial method.
The polynomial method itself has a remarkably similar story, with the finite field Kakeya conjecture getting resolved in a surprisingly compact proof after many years
xitter.cf/jdlichtman/status/2000…
Now that I've had time to sit down and absorb this new proof:
The exact writing is a bit too compressed for my liking, but the argument is quite a gem.
Jared Duker Lichtman retweeted
The flood of results building on OpenAI's Quasi-Riemann Hypothesis breakthrough keeps coming!
Today, we have a preprint from Baiying Liu showing that all finite-order Hecke L-functions are zero-free to the right of 0.8749!
This improves on 31/32 (= 0.96875), established only a few days ago by Cureton and Thorner.
Some fascinating developments following OpenAI's shock breakthrough proof of the Quasi-Riemann Hypothesis, which was released publicly on October 6:
1. Building on OpenAI's proof, a recent Hexagon preprint (released on October 12) by Cureton and Thorner establishes that ALL finite-order Hecke L-functions are zero-free to the right of 1 - 1/6000.
2. And now, today, an arXiv preprint by Frei, Somos and Widmer uses this result to prove unconditionally the 2007 Ellenberg–Venkatesh bounds on ℓ-torsion in class groups of number fields.
I was amused to see the AI declaration in the second paper: "No artificial intelligence tools were used in this work."
Moreover, this proof may be viewed as an example of the polynomial method.
The polynomial method itself has a remarkably similar story, with the finite field Kakeya conjecture getting resolved in a surprisingly compact proof after many years
xitter.cf/jdlichtman/status/2000…
Jared Duker Lichtman retweeted
A Fields Medalist says AI can now understand mathematical proofs even when his notes are almost unreadable to other mathematicians.
John Pardon explains that until recently, AI was little more than a spelling checker for his mathematical writing.
Now, he says, it can examine highly compressed, incomplete research notes and identify subtle mathematical mistakes.
For example, a missing condition such as finite or sifted in a statement about colimits can change its meaning. According to Pardon, AI can recognize such omissions by interpreting the broader mathematical context.
His assessment is:
“The AI understands. 100%.”
Pardon also reports that AI has proved several nontrivial lemmas for him. However, he says it has not yet produced a major breakthrough in his own research.
Pardon believes AI is changing mathematical research, but he also emphasizes that students must work through easier problems themselves to develop the understanding needed for harder ones.
He finds value in deliberately approaching unfamiliar problems, even when AI could immediately explain the answers.
Full Interview: futureclass.substack.com/p/j…
Someone should make a website called Proofs from the Book with the most elegant proofs across many problems, and continually updated
Jared Duker Lichtman retweeted
The Riemann ζ function connects integers, prime numbers and complex numbers in a mysterious and powerful way. In this story, I report how analytic number theorists are coming to terms with a claimed proof of the quasi-Riemann hypothesis by OpenAI:
quantamagazine.org/ai-proved…
Jared Duker Lichtman retweeted
Among the 377 results that OpenAI announced two weeks ago, one stood out. It claimed to settle the “quasi-Riemann hypothesis” — a weaker version of the Riemann hypothesis, which is arguably the biggest open question in math. @kkakaes reports: quantamagazine.org/ai-proved…
Jared Duker Lichtman retweeted
We don't yet have robust mathematical foundations of physics and other sciences. Not only is the Hilbert's 6th problem (axiomatization of physics) still unsolved, we don't even have a logic where experimental verifiability is a core primitive.
This has suddenly become one of the key bottlenecks towards building trustworthy verifiers for scientific reasoning, that would power AI x Science.
We recently found one approach to make concrete progress towards this. Very excited! But there's more work to do.
We're looking for collaborators who have spent time thinking about the Hilbert's 6th problem. DM me if you're that person.
@FryRsquared But I think the analogy to pixels is the reverse:
Pixels give a continuous-to-discrete approximation of an image,
whereas Navier-Stokes is giving a discrete-to-continous approximation of fluid molecules
I prefer \epsilon to denote standard small quantities tending to zero, often as a helper variable inside proofs
I only break out \varepsilon on special occasions, for the main theorem
It may surprise those outside number theory to learn that, for decades, people had written several papers that assumed a Siegel zero exists:
Famously, in 1982 Heath-Brown proved the twin prime conjecture in this hypothetical world.
Now in light of the Siegel zero disproof, a collection of the literature has now become something of a historical curiosity.
londmathsoc.onlinelibrary.wi…
The new paper “is actually remarkably well-written,” Maynard says
science.org/content/article/…
"Just let AI generate new Math," some people say.
AI doesn't live in our world and we already struggle to read some of the proofs it writes. For instance, look at OpenAI's 166-page Navier-Stokes proof. Experts believe it's correct, yet Fields Medalist James Maynard said it's been "very difficult to really extract any human understanding" from it.
So, if AI discovers the next big idea and no human can understand it, who is that math for?
"OpenAI's math drop didn't contain much cryptography"
Some people on X realised that OpenAI's math drop didn't contain much cryptography. @dpaleka notes that a few more fields are under-represented: largely the ones that are useful for AI research
