SAIR competition: Andrews–Curtis challenge

Summary

[This is a guest post by Lucas Fagan . This blog post was initially written in a different file format and converted using AI. — T.] I am excited to announce the Andrews–Curtis Conjecture Challenge , which opens today. This challenge is a collaboration between the SAIR Foundation and the Math-AI group at Caltech , organized by Sergei Gukov, Terence Tao, and myself. The Andrews–Curtis conjecture is one of the most prominent open problems in combinatorial group theory and also has deep connections to low-dimensional topology. Its potential counterexamples are relevant to the search for exotic smooth four-spheres and the smooth four-dimensional Poincaré conjecture , as well as the Generalized Property R conjecture about surgery on links . Yet unlike many open problems at its level, Andrews–Curtis can be formulated as a combinatorial search problem with easily checkable solutions, making it ideal for a challenge of this form. At Caltech, we have been developing reinforcement learning and combinatorial search methods for this problem to resolve potential counterexamples (see What makes math problems hard for reinforcement learning: a case study and The Two-Hump Problem ). However, many important cases have resisted all our efforts. We hope that this challenge will lead to resolving these and more (or disproving the conjecture), especially given the recent progress in AI. We give more details about the different tracks of the competition below. To briefly introduce the problem: the Andrews–Curtis conjecture says that any balanced presentation of the trivial group can be transformed to the trivial presentation using the following moves (which do not change the underlying group): (AC1) Invert a relator: . (AC2) Multiply a relator by another: for some . (AC3) Conjugate a relator by a generator or its inverse: for some . Two presentations connected by these moves are called AC-equivalent ; a presentation that is AC-equivalent to the trivial presentation is called AC-trivial . As a simple example, is AC-trivial: It is generally suspected that the conjecture is false ; there are many simple potential counterexamples in which all computational efforts have failed to find a path to the trivial presentation. The most notable of these is the Akbulut–Kirby family whose AC-triviality is open for . Indeed, , with total relator length , is the shortest possible counterexample…