How Machines Explore, Conjecture, and Discover Mathematics
In this talk, we illustrate mathematical research developing methodologies that combine optimization, machine learning, and mathematical structure to navigate large, complex, and highly constrained search spaces with a focus on the Hadwiger–Nelson problem: a long-standing open problem in discrete geometry and extremal combinatorics concerning colorings of the plane without monochromatic unit-distance pairs.
Overview
Artificial Intelligence is increasingly becoming a genuine partner in mathematical research, not only as a computational tool, but as a driver of exploration, conjecture generation, and discovery. Under the umbrella of AI4Math, we develop methodologies that combine optimization, machine learning, and mathematical structure to navigate large, complex, and highly constrained search spaces that are inaccessible to traditional approaches.
In this talk, we illustrate this paradigm through a concrete case study: the Hadwiger–Nelson problem, a long-standing open problem in discrete geometry and extremal combinatorics concerning colorings of the plane without monochromatic unit-distance pairs. We show how neural networks can be used as expressive approximators to transform a mixed discrete–continuous geometric problem with hard constraints into a differentiable optimization problem with a probabilistic loss. This enables gradient-based exploration of admissible configurations and directly led to the discovery of two novel six-colorings, yielding the first improvement in thirty years for the off-diagonal variant of the problem.
Presenters
Sebastian Pokutta, Vice President, Zuse Institute Berlin (ZIB); Professor, Technische Universität Berlin (TU Berlin)
Brief Biography
Sebastian Pokutta is Vice President of the Zuse Institute Berlin (ZIB) and Professor of Mathematics at TU Berlin, with a research focus on Artificial Intelligence and Optimization. He holds leadership roles as Chair of the Cluster of Excellence MATH+ and the Research Campus MODAL. He earned his diploma and Ph.D. in mathematics from the University of Duisburg-Essen, followed by postdoctoral work and a visiting lectureship at MIT. He also held positions in industry with IBM ILOG and in consulting. Before returning to Germany, Pokutta was the David M. McKenney Family Associate Professor in the School of Industrial and Systems Engineering and Founding Associate Director of the Machine Learning @ GT Center at Georgia Tech, as well as a Professor at the University of Erlangen-Nürnberg. His contributions have been recognized with several honors, including the Gödel Prize (2023), a STOC Test of Time Award (2022), and the NSF CAREER Award (2015), alongside various best paper and early career awards including the Coca Cola Early Career Professorship (2014) and the David M. McKenney Family Early Career Professorship (2016).