OpenAI Is Close To Solving Another Millennium Prize Math Problem
Remember when OpenAI landed in hot water earlier this month after mathematicians accused the company of trying to steal their thunder for solving an ultra-difficult math problem? (I know it feels like a million years ago given all the news cycles since then.)
Well, it turns out OpenAI is close to solving another math problem from the Millennium Prize Problems, a set of seven famous math challenges including that earlier “Navier-Stokes existence and smoothness problem,” according to a person with knowledge of the solution.
Employees expect the next problem, known as the Hodge Conjecture, to be solved relatively soon, the person said. The Hodge Conjecture asks whether certain geometric features of shapes defined by polynomial equations can always be described using simpler algebraic pieces. It could take the company longer to announce the solution, though, because it’s trying to figure out how to collaborate with the math community to make the announcement without triggering another PR nightmare, the person said.
Why is OpenAI interested in solving these obscure math problems in the first place? After all, they’re not cheap to solve: it likely took millions of dollars to solve the original Navier-Stokes problem. (OpenAI used a variant of its next pre-trained model, codenamed “Doug,” to solve that problem, the person said.)
However, some OpenAI researchers believe that math is the natural next area for its models to tackle after software engineering. Both domains have similar characteristics: they both require step-by-step logical reasoning and produce answers that can often be automatically checked for correctness. Some researchers even believe that the kind of automation that took over software engineering over the last year will happen to mathematics in the next six to nine months.
Additionally, solving hard math problems also furthers AI developers’ efforts to automate machine learning research, which involves a lot of math. Such automation is part of their effort to get to a point where the AI can develop new AI, known as recursive self improvement.
Testing frontier models’ ability to solve never-before-solved math problems also helps researchers understand how far they’ve advanced and their complex reasoning capabilities. If they’re able to solve hard questions like Navier-Stokes and the Hodge Conjecture, it could be a sign that they could solve similarly difficult questions in other fields like biology or chemistry.
And, it doesn’t hurt that such announcements (when they don’t come with outraged mathematicians) give OpenAI shiny new milestones to tout in the race towards superintelligent AI.
Unsolved
The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture is known in certain special cases, e.g., when the solution set has dimension less than four. But in dimension four it is unknown.
In the twentieth century mathematicians discovered powerful ways to investigate the shapes of complicated objects. The basic idea is to ask to what extent we can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. This technique turned out to be so useful that it got generalized in many different ways, eventually leading to powerful tools that enabled mathematicians to make great progress in cataloging the variety of objects they encountered in their investigations. Unfortunately, the geometric origins of the procedure became obscured in this generalization. In some sense it was necessary to add pieces that did not have any geometric interpretation. The Hodge conjecture asserts that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles.