Final Activity Report Summary - LCCRM (Equations in free groups) The topic of the project was the study of equations in free groups from a variety of perspectives: combinatorial, algorithmic, algebraic and geometric. Solving equations in groups and monoids has been explored by many mathematicians and computer scientists. This problem can be tackled algorithmically and expressed via algebraic geometry over groups. Also, solving equations is a key ingredient in the study of the elementary and universal theories of a group, and shows up in theoretical computer science in the context of unification theory.Within this project the fellow has worked on many interconnected aspects of free groups. In the area of decision problems and their complexity we mention the paper 'Polynomial-time complexity for instances of the endomorphism problem in free groups', which has appeared in the International Journal of Algebra and Computation. This paper establishes, among other things, that there is a polynomial-time algorithm for deciding the solvability, in free groups, of two-variable equations in which all the variables occur on one side of the equality and all the constants on the other side. Furthermore, regarding equations and their solutions, the fellow together with Sasa Radomirovic, has produced the paper 'Restricted walks in regular trees, which has appeared in the Electronic Journal of Combinatorics.The analysis of equations and their solutions also played a key role in the study of Galois theory of free groups. The fellow has produced the paper 'Two examples in the Galois theory of free groups' with Prof. Warren Dicks, the scientist in charge, which has appeared in the Journal of Algebra. A related direction of work has been expanding some of the known results for free groups to much wider classes of groups, via group actions on trees and generalisations of trees. In this direction, the fellow, together with N.Brady A. Martino and S. O Rourke, has written the paper 'The equation x^py^q=s^r in groups that act freely on Lambda-trees', which will appear in the Transactions of the American Mathematical Society.The fellow has participated in more than a dozen conferences during the time of her fellowship, and presented her work at most of these events. Also, the fellow has been invited to give talks in numerous seminars in Spain, United States, France, Germany and Switzerland. The fellow will continue to interact with the group theorists in Barcelona: Enric Ventura, Jose Burillo and Armando Martino, with whom she has started joint projects during the stay at the Centre de Recerca Matematica, stay facilitated by the Marie Curie Fellowship.