As thousands and thousands of individuals were coming down from the thrill of the FIFA World Cup Final at the beginning of last week, a distinct sort of excitement was constructing throughout the mathematical community.
Levent Alpöge, a mathematician working at the factitious intelligence company Anthropic, made a really casual announcement on X that he had found a counterexample to the Jacobian conjecture, a really old and well-known problem in a field of mathematics called algebraic geometry. He had done this using Anthropic’s large language model Claude Fable 5, released to most people only just a few weeks ago.
That is just the newest of many striking mathematical breakthroughs made by mathematicians working with large language models. But this one feels a bit of different to those who have come before.
What Is the Jacobian Conjecture?
First, what’s a conjecture? It’s an concept that some mathematicians consider is true but no person has been capable of prove or disprove.
Now to the Jacobian conjecture. It’s fairly abstract but not too difficult to explain.
The conjecture involves functions, that are like little machines which you place a number of numbers into and out pop other numbers based on some rule or equation. On this case, the functions use what are called polynomials.
Specifically, it’s about situations where the numbers represent points in an area, like coordinates on a map. So we are able to imagine that when the function takes in some numbers and puts out another numbers, it’s moving the points in space.
You may test how “nicely” a function moves every thing around in space by calculating something called the Jacobian determinant. If the Jacobian determinant is at all times a continuing number that will not be zero, then the function never folds or crushes space around a specific point.
The Jacobian conjecture states that when the Jacobian determinant is a non-zero constant, there should at all times exist one other function, also made up of polynomials, that reverses the unique one. This may return all of the points to their starting positions.
Not every function is reversible. For instance, if our starting function moves two of the unique points onto a single point, then we cannot reverse it. Once the points have been merged, we cannot distinguish between them to send them back to the appropriate positions.
A Long History of Attempts—and Failures
The 2-dimensional version of the Jacobian conjecture was stated by Czech mathematician Ludwig Kraus in 1884. It was generalized to any variety of dimensions by German mathematician Ott-Heinrich Keller in 1939.
It was considered so compelling that Fields Medalist Stephen Smale included it in his 1998 list of Mathematical Problems for the Next Century.
During its long history, the Jacobian conjecture has been the topic of many claimed proofs, including by Beniamino Segre and Wolfgang Gröbner, two famed Twentieth-century mathematicians. Nevertheless, in each case, subtle errors were found that invalidated the arguments.
Despite this, there have also been various valid efforts showing the conjecture is true with various restrictions. Computational results have also shown it’s true in two dimensions for polynomials as much as degree 100 (that’s, including powers of the variables as much as 100).
But no person had proved the final case—or found an example showing the conjecture was improper.
A Deceptively Easy Answer
Considered one of the important thing reasons the Jacobian conjecture is so intriguing is that, in theory, it must be easy to seek out a counterexample. It is easy to provide you with examples of functions that merge points, and in addition examples of polynomial mappings which have a continuing Jacobian determinant.
Nevertheless, finding a polynomial mapping with each properties is the challenge. Indeed, as one Math Stack Exchange user noted in a post from 2017, “for all what we all know, some smart undergraduate can simply write a formula […] that will probably be a counter-example to this conjecture.”
Indeed, this did become the case for Alpöge’s function, which is brief enough to suit right into a single X post. He found an example of a function in three dimensions which has a continuing Jacobian determinant of -2, and which moves multiple input points to the identical output point, so it will not be reversible.
It shows the conjecture is fake for each dimension larger than 2, with the unique conjecture in two dimensions remaining open. The brevity of the counterexample made it easy for other mathematicians to confirm.
The Latest Advance in a Growing Series
Alpöge’s discovery is the newest in a string of high-profile mathematical breakthroughs made by large language models. Recent examples include OpenAI’s disproof of the unit distance conjecture, and the proof of Erdős’ problem 1196 by Liam Price, a 23-year-old amateur mathematician.
Each examples illustrate one of the crucial striking strengths of AI models. They’ll draw on ideas from different areas of mathematics, combining them in a novel strategy to prove astonishing results.
On the time of writing, details haven’t been made public regarding exactly how Alpöge prompted the AI model to supply the Jacobian conjecture counterexample and what its output looked like. Nevertheless, thus far this result appears to be of a distinct nature.
Unlike many other recent AI-assisted breakthroughs, the counterexample itself is remarkably easy. The issue to find it seems to have lain not in an intricate construction or a lengthy proof, but reasonably to find a very good way of navigating an unlimited search space of possible polynomial mappings to seek out one with the appropriate properties.
This implies AI may prove to be just as helpful for locating unexpected mathematical objects because it is for constructing proofs. What this implies for the longer term of mathematics—and human mathematicians—stays to be seen.