# Pure Maths Colloquium: Carl Frederick Nyberg-Brodda

This talk is part of the Pure Maths Colloquium at the University of St Andrews. Check out our upcoming talks at https://theran.lt/pure-colloquium/.

Where: | Zoom 812 1857 5349 |

When: | Oct 14 2021 @ 16.00 |

Video: | Link (internal) |

Speaker: | Carl Frederick Nyberg-Brodda University of East Anglia |

Title: | One-relation monoids and virtually free groups |

The word problem for one-relation monoids has been called one of the most fundamental open problems in all combinatorial algebra. It has been open for over a century, and was described by P. S. Novikov as “containing something transcendental”. In this talk, I will give an overview of this problem (including all definitions!) and some important reduction results proved by S. I. Adian and his students. I will then focus on certain special classes, and show some recent results forming part of a program to understand the formal language theory of the word problem for one-relation monoids. This latter topic connects with the famous Muller-Schupp theorem, which identifies groups with context-free word problem as those which are virtually free. I will show that the Muller-Schupp theorem can be generalised to rather broad classes of one-relation monoids. As one consequence, I will show that it is decidable whether a one-relation monoid containing a non-trivial idempotent has context-free word problem. As another consequence, the difficult rational subset membership problem is solved for many classes of one-relation monoids.