Robert Gray is speaking in the Algebra Seminar this week. We will go out for lunch after the talk. When: Friday 12 May, 12-1pm Where: Carslaw 173 Title: One-relator groups, monoids and inverse monoids Abstract: It is a classical result of Magnus proved in the 1930s that the word problem is decidable for one-relator groups. In contrast, it remains a longstanding open problem whether the word problem is decidable for one-relator monoids. A natural class of algebraic structures lying between monoids and groups is that of inverse monoids. An inverse monoid is called special if it is defined by a presentation where all the defining relations are of the form w=1. There is strong motivation for studying this class coming from results of Ivanov, Margolis and Meakin (2001) who showed that if all special one-relator inverse monoids with defining relator w=1, where w is a reduced word, have decidable word problem then this would answer positively the open problem of whether all one-relator monoids have decidable word problem. In this talk I will speak about some recent results on the algebraic and algorithmic properties of special and one-relator inverse monoids. I will explain some of the methods used in this area including the theory of Schutzenberger graphs. I’ll also explain the connections with some problems about one-relator groups including the submonoid membership problem, coherence, and the question of which right-angled Artin groups embed as subgroups.