# Kyriakos Kalorkoti

*Rewrite Systems, monoids and free resolutions*

A monoid is simply a set of strings over an alphabet with a set of equalities (from which other equalities follow by the obvious notion of equivalence). It is a very old result that the word problem for monoids has no algorithmic solution in general. However if we have a finite convergent rewrite system for a monoid then it has solvable word problem and (assuming a finite alphabet) a finite presentation. For a long time the question of whether the converse is true was open (i.e., if a monoid has a finite presentation and solvable word problem does it have a finite convergent rewrite system?). In the late 1980's and early 1990's it was realized that homological methods could be used to resolve the question. In this talk I will give an overview of the problem and the methods used, keeping to general description rather than technical details.