Prof. Dr Jürgen Klüners, a mathematician at Paderborn University, together with Prof. Dr Gunter Malle from the Rhineland-Palatinate Technical University (RPTU) in Kaiserslautern, has won the international ‘IGP24 Competition’ organised by the ‘Foundation for Science and AI Research’ (SAIR). The foundation is committed to driving breakthroughs in the natural sciences with the aid of artificial intelligence (AI). The competition was therefore designed to have a mathematical problem set by leading researchers from around the world solved in collaboration with AI. Nevertheless, the team from Paderborn and Kaiserslautern managed to win the competition without using AI. The competition’s organisers include renowned mathematicians such as Fields Medallist Terence Tao; the SAIR board also features Nobel Prize in Physics laureate Barry Barish, as well as Turing Award winners Jeffrey Ullman and Richard Sutton from the field of Computer Science.
“AI has often achieved superhuman feats in Mathematics in recent months. Almost all the other participants relied on the technology, but this time our experience was still enough to win the competition,” says Prof. Klüners delightedly. The two mathematicians from Paderborn and Kaiserslautern only used AI to submit their results, not for the calculations themselves. “Nevertheless, I am very impressed by what AI is capable of. Initiatives such as the SAIR Foundation ensure that it is used responsibly in a scientific context.”
The inverse problem of Galois theory
‘IGP24’ is the abbreviation for ‘Inverse Galois Problem 24’, and so the focus was on a particularly challenging problem from algebra: the inverse problem of Galois theory for degree 24. Mathematicians frequently work with multi-term expressions such asx³ – 2, known as ‘polynomials’. The IGP24 competition centred on polynomials of degree 24 in whichx²⁴ appears as the highest power. Galois groups describe the symmetries of the zeroes of polynomials and thus provide the key to determining whether a polynomial equation can be solved using root expressions (such as in the quadratic formula familiar from school). Such solution formulas always exist up to degree 4; from degree 5 onwards, this is only possible if the Galois group is a so-called solvable group. In classical Galois theory, mathematicians consider a polynomial and then determine the corresponding group. In the inverse problem of Galois theory, the perspective changes; the question is: for every finite group, is there a polynomial with precisely this Galois group?
The challenge lay not only in the theory, but also in the sheer volume of data. For degree 24, this question encompasses 25,000 groups and 165,836 different signatures. Here, the more precise question is whether each group can also be realised with a (reasonably) specified number of real zeros. At the start of the competition, the LMFDB mathematical database contained concrete polynomials for only 286 of these groups and 622 of these signatures. Participants from around the world submitted millions of integer polynomials of degree 24. By July, all 25,000 groups – and thus the Inverse Galois Problem for degree 24 – had been completely solved.
Team from Paderborn and Kaiserslautern comes out on top
The international competition was won by the team led by Prof. Klüners and Prof. Malle. The two mathematicians discovered more than 143,000 polynomials relevant to the problem – entirely without the use of AI. They have been researching the inverse problem of Galois theory for more than 20 years and, over time, have built up a database of polynomials up to degree 23. “During the competition, our database crashed due to being overloaded by the sheer volume of queries,” explains the scientist from the Paderborn Institute of Mathematics.
Unlike the other teams, the team did not use AI for the actual calculation of the polynomials. “However, as we had generated thousands of new polynomials, we needed efficient processes for submitting them during the competition. With the help of AI, we were able to develop efficient scripts that made it easier to submit the polynomials,” said Prof. Klüners.
This text was translated automatically.
