So, this happened.
A 31-year-old conjecture in geometry has a new proof.
Zhao Song and Xinzhi Zhang have posted a proof of the Kannan-Lovász-Simonovits (KLS) conjecture, a problem dating back to 1995. Microsoft Research announced the result on October 21.
The conjecture concerns high-dimensional geometry and probability. In simple terms, it deals with how difficult it is to divide certain high-dimensional shapes into two large pieces using a relatively small cut. The conjecture says that a simple flat cut can approximate the best possible division within a fixed constant, regardless of how many dimensions we’re working with.
The result has applications in probability, optimization, and algorithms that sample from high-dimensional distributions.
AI was used during the research, and another group of mathematicians independently developed a proof. The papers are still preprints, but if the results hold up, a problem that’s been open for more than three decades is settled.
There has also been some progress on the quasi-Riemann hypothesis.
Among the hundreds of mathematical manuscripts OpenAI released earlier this month were two claimed proofs of the quasi-Riemann hypothesis.
The Riemann hypothesis itself says that every nontrivial zero of the Riemann zeta function has real part 1/2. The quasi-Riemann hypothesis asks for something weaker. Rather than requiring all those zeros to lie on one vertical line, it asks whether they can be confined to a narrower strip than the one we already know.
One of OpenAI’s arguments claims that the real parts of the zeros lie between 1/8 and 7/8. Another gives the weaker bounds of 1/12 and 11/12.
According to an October 19 report in Quanta Magazine, several number theorists have examined the shorter proof and largely agree that it is correct. Work is continuing on understanding and checking the arguments.
None of this proves the Riemann hypothesis. But establishing either of these narrower bounds would still be a substantial result in number theory.
Terence Tao has started an AI-assisted mathematics challenge.
On October 20, Tao announced an experimental project involving Landau-Siegel zeros and the class number one problem.
The class number one problem concerns imaginary quadratic number fields. One way to think about it is that these are number systems where we can ask whether factorization into prime elements behaves as nicely as it does with ordinary integers.
The problem was solved in 1952, but Tao is interested in whether AI can help produce a different proof.
Participants are being asked to extract explicit numerical bounds from existing arguments involving Siegel zeros. Some of these proofs establish that a constant exists without telling us exactly what the constant is. Tao wants to see whether AI can help work out those missing numerical details.
The project is open and collaborative, with an emphasis on checking results and documenting how AI contributes to the mathematics.
There is also a new proof concerning matrix multiplication.
Rafael Pass posted a paper on October 20 establishing that the matrix multiplication exponent, written ω, is at most 2.25.
The exponent measures how efficiently large matrices can be multiplied. If we multiply two n × n matrices using the standard algorithm, we need roughly n³ arithmetic operations. Mathematicians have known for decades that we can do better.
The question is how much better.
The new proof establishes the bound ω ≤ 2.25, a result OpenAI had recently claimed in its collection of AI-generated mathematics papers.
Pass’s argument is constructive, meaning it gives an explicit way to achieve the claimed bound rather than just showing that such a method must exist.
The paper is still a preprint, but it provides another approach to the result.
Finally, another mathematician has taken a closer look at one of OpenAI’s proofs.
Jenia Tevelev, an algebraic geometer at the University of Massachusetts Amherst, posted a 19-page reconstruction of OpenAI’s proof of Nagata’s conjecture on October 21.
Nagata’s conjecture dates to 1959 and concerns algebraic curves in the projective plane. More specifically, it asks how the degree of a curve is constrained when the curve must pass through a collection of points with specified multiplicities.
OpenAI included a claimed proof among the hundreds of manuscripts it released on October 6.
Tevelev has now rewritten and reorganized that argument, producing a shorter explanation of the mathematics while crediting OpenAI with the original result.
This follows several weeks of discussion about how these AI-generated results should be presented and checked. Releasing a proof is one thing, but mathematicians still need to work through the arguments and establish exactly what has been proved.
Tevelev’s paper is another step in that process, although it remains a preprint.
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