Diophantine problem (the problem of algorithmic solvability of equations) is one of the classical problems in mathematics. This problem is decidable in algebraically closed fields, in the fields of real and p-adic numbers. Diophantine problem for integers is undecidable. This was a solution of Hilbert's 10th problem by Matiyasevich based on the work of Davis, Putnam and Robinson. Diophantine problem for rationals is one of the major open problems. In this expository talk I will discuss the modern theory of equations in groups and non-commutative algebras, the decidability of the Diophantine problem in free groups and monoids and undecidability in free group algebras and in free associative and Lie algebras. I will also talk about algebraic geometry in free groups.
Olga Kharlampovich is Mary P. Dolciani Professor of Mathematics at the CUNY Graduate Center and Hunter College. She is the winner of the Krieger-Nelson and Mal'cev Prizes, and a medal from the Soviet Academy of Sciences.