Systems of equations in groups

The Diophantine problem in a ring or group G is concerned with algorithmically solving systems of equations in G. In this talk I will first provide general background on this topic, discussing its theoretical and practical relevance, and I will survey some of the most fundamental results attained so far, as well as some long-standing open problems. Afterwards I will proceed to discuss some of my work on the Diophantine problem in different families of groups and monoids, focusing specially on several classes of solvable groups, where one eventually encounters equations in rings of algebraic integers. I will also mention some results and ongoing work regarding the problem of solving systems of equations, in different groups and monoids, together with different types of length relations between the variables.

This is joint work with Laura Ciobanu; with Robert D. Gray, and with Alexei Miasnikov and Denis Ovchinnikov.