Artificial intelligence and computer science are driving developments in many areas of society – including in scientific research. This has prompted the Max Planck Society and the Alexander von Humboldt Foundation to honour outstanding achievements in the use of algorithms in mathematics, microscopy and climate research in 2024: The Max Planck-Humboldt Research Award, endowed with 1.5 million euros, goes to Geordie Williamson, who was Advanced Researcher at the Max Planck Institute for Mathematics from 2011-2016, and is now Professor at the University of Sydney. Williamson uses artificial intelligence (AI) for his fundamental work in mathematics.

Scientists today use artificial intelligence in many areas, especially in the natural sciences, for tasks such as analysing data or images. In theoretical mathematics, on the other hand, AI has barely been used thus far. Now Geordie Williamson is aiming to change that. In his previous work he has already used artificial neural networks, which can guide mathematical intuition by drawing attention to previously unrecognised relationships in a large number of mathematical objects. Artificial intelligence can also help to generate examples or counterexamples that prove or disprove mathematical assumptions. Although artificial neural networks can recognise patterns in large data sets very efficiently and effectively, they know nothing about mathematics. It therefore remains the task of mathematicians to filter out the sensible proposals from AI, to interpret them and, in the case of new assumptions about mathematical relationships, to prove or disprove them. Geordie Williamson wants to optimise the possibilities of using AI in theoretical mathematics in the collaboration made possible by the Max Planck-Humboldt Research Award. To this end, he will work closely with researchers from the University of Bonn and the Max Planck Institute for Mathematics in Bonn, where he will also spend two periods of several months each.

**Connecting the countable with geometry**

Geordie Williamson's previous research work was characterised, among other things, by the fact that he brought together different fields such as combinatorics and geometry. In simple terms, combinatorics can be understood as the branch of mathematics that is dedicated to everything that can be counted; it includes subjects such as graph theory and discrete mathematics. Geometry is about objects in spaces, i.e. straight lines, surfaces, and solids, just like in school maths. Both sub-areas come together in a simple example when the intersection points of a curve and a surface are to be counted. Geordie Williamson has now opened up ways of solving combinatorics problems with geometric tools, for which purpose he first had to develop a kind of common mathematical language for the two fields so that combinatorial problems could be worked on in geometry, but geometry could also be translated into combinatorics. With this approach, Geordie Williamson has proved or disproved various assumptions that mathematicians have been working on intensively, but to no avail, for a long time.

For example, Williamson in collaboration with Ben Elias from the University of Oregon provided a general proof of an important conjecture in mathematics relating to Kazhdan-Lusztig polynomials. The work of David Kazhdan and George Lusztig provided precise recipes for building up certain mathematical objects out of constituent pieces. Imagine a recipe that contains a list of ingredients and instructions on what to do with them, but the recipe does not specify the quantities. Kazhdan and Lusztig hypothesised that there are polynomials in mathematics for such cases, from which the quantities for the recipe can be determined. Polynomials are formulae that are familiar to us in their simple form from the binomial formulae we study in school. Geordie Williamson has proven this assumption, for which evidence had previously been sought in vain for a long time. His methods, borrowed from geometry, also make it much easier to solve the polynomials that provide the unknown data and to analyse them in greater depth.

**Solving knot theory problems with the help of AI**

As part of the collaboration with researchers from the University of Bonn and the Max Planck Institute for Mathematics, all possible as a result of the award, Williamson will tackle various mathematical problems with the help of artificial intelligence. Amongst the problems that they will tackle is a problem in knot theory. In simple terms, this can be explained by the fact that it is often impossible to recognise whether knotted structures, such as in a string, are actually knotted. What this means is: does the knot remain intact when you pull on the ends of the cord or does it unravel? One aim of the project is to identify these cases in a simple way so that these uninteresting cases can be quickly filtered out and the researchers can focus on the real knots. AI is set to provide support here and assistance in gaining new mathematical insights. Geordie Williamson studied at the University of Sydney and received his doctorate from the University of Freiburg in 2008. He then conducted research at Oxford University until 2011 and headed a research group at the Max Planck Institute for Mathematics until 2016. After other shorter stints at the Hausdorff Centre for Mathematics in Bonn and at the Institute for Advanced Study, Princeton he was appointed Professor at the University of Sydney in 2017. He serves as the founding Director of the Sydney Mathematical Research Institute. Geordie Williamson is a Fellow of the British Royal Society and the Australian Academy of Science.

**About the award**

The Max Planck Society and the Alexander von Humboldt Foundation present the Max Planck-Humboldt Research Award, along with 1.5 million euros in prize money, to a researcher from abroad. 80,000 euros in personal prize money is also awarded. The focus here is on personalities whose work is characterised by outstanding potential for the future. The prize is intended to attract particularly innovative scientists working abroad to spend a fixed period of time at a German higher education institution or research facility. The Federal Ministry of Education and Research provides the funding for the award. The focus of the award alternates each year between natural and engineering sciences, life sciences, humanities and social sciences.

© MPI f. Mathematik, Bonn | Impressum & Datenschutz |