This paper proposes to use Artificial Intelligence tools, especially maching learning, to help discover patterns and relations in mathematical objects, so it can assist mathematicians in discovering and proposing conjectures, and also help to prove conjectures.
Computing has already been used heavily in assisting mathematicians in proving theorems. The authors in this paper address another direction, which is using AI to guide people’s mathematical intuition. Basically, patterns or relations in mathematical objects are regarded as implicit functions, and we want to use machine learning tools to approximate and learn these functions. In order to learn, we need lots of training samples, which can also be generated by the help of AI and computing.
This paper gives two examples of applications: topology and representation theory. Both of them are abstract mathematical fields, which are hardly regarded to be related to machine learning and AI. The first application is in low dimensional topology; the authors propose a relation between the geometric invariants and the algebraic invariants of a knot, and machine learning is used to find such a (nonlinear) relation and test it. The second application is the study of relations between KL polynomials and Bruhat intervals in representation theory and combinatorics. Machine learning helps to propose a conjecture on the relation.
Though the framework proposed by the authors is still limited, it helps us to understand better how machines could be used to help on another direction of solving mathematical problems. It has a great potential to contribute to various branches of both pure and applied mathematics.