**Seminario teoría de grupos UAM-ICMAT**

**Miercoles 9 de Octubre, 12:30, Aula 520 UAM **

** Albert Garreta (UPV/EHU) **

** Title: Equations and first-order theory of one-relator monoids and of word-hyperbolic monoids. **

** Abstract: **

** First we will introduce the classes of word-hyperbolic monoids and of one- **

** relator monoids, and we will compare these to their group theoretic counter- **

** parts. We will also provide a broad overview on the topics of first-order theory **

** and Diophantine problems in groups and monoids, paying special attention **

** to their algorithmic aspects. We will then proceed to discuss the decidability **

** of different fragments of the first-order theory in word-hyperbolic monoids **

** and one-relator monoids. In particular we will prove in an informal manner **

** that in many of these one can encode (by means of systems of equations) **

** the longstanding open problem of word equations with length constraints. A **

** consequence of this result is that these monoids have undecidable positive **

** AE-theory. This is joint work with Robert D. Gray.**