In several branches of mathematics, we are interested in finding largest or smallest object with a certain property in a combinatorial search space. For example, in Ramsay theory we are interested in finding the largest graph that does not contain certain subgraphs. Following Ádám Wagner's work, in this project we leverage reinforcement learning to find such constructions. After a brief experimentation with Ramsay theory, we turned our attention to the no-three-in-line problem: selecting the maximum number of grid points on an n×n regular square grid such that no three of the chosen points fall on the same line.

Another project explored a particular flavour of reinforcement learning in widely used benchmark environments. This project explored deep Q-learning, tabular Q-learning, curriculum learning (changing the strength of gravity for the lunar lander). Playing with these environments allowed the group to understand how and why Q-learning works, to gain first-hand experience with the difficulty of neural-network-based reinforcement learning, and to gear up to solving more difficult problems.

To everyone's big surprise, some large language models (LLMs) seem to be capable of mathematical problem-solving, even without explicit training. In this project we processed past problem sets from the Náboj high school mathematics competition, and evaluated publicly available LLMs on their ability to solve these medium difficulty problems. The team found that without creative prompting techniques even the likes of GPT4 are generally unable to do this. Only when using techniques like progressive hint prompting (PHP) did the models perform at reasonable level.

on Day 2 Prof Gábor Csányi visited and gave a presentation about the use of AI techniques in quantum chemistry. He presented retreat participants with a challenge: design a descriptor of a set of points which (A) is invariant under permutations, translations and rotations, (B) varies smoothly with its inputs and (C) is invertible. We discussed why the multiset of atom-to-atom distances does not satisfy the invertibility criterion, and Gábor presented a pair of counterexamples that map to the same representation. He then left the group to think about a better solution, and we did. The result is a genuinely new idea, that solves the problem in a very simple and computationally tractable way.