Ilya Kazachkov

Equations in free groups and applications

Hilbert's tenth problem asks to provide an algorithm to solve polynomial equations over the integers. In 1970 Matiyasevich, relying on the work of Davis, Putnam and Robinson, showed that in fact such an algorithm does not exist.

One of the approaches to the undecidability of Hilbert's tenth for the integers was via the study of equations over free groups. Many prominent mathematicians, such as Mal'cev, Lyndon, Appel, Culler etc, studied particular classes of equations over free groups, but some inherent problems in tackling the general problem convinced many of its undecidability. In his seminal work in the 80's, Makanin presented an algorithm for solving equations in a free group. This algorithm, known nowadays as the Makanin-Razborov process, is ``basic'' and appears in other branches of mathematics.

In this talk, I will give an overview of Hilbert's tenth problem over different structures with an emphasis on Makanin's algorithm for solving equations in the free group and its applications in different areas.

No knowledge of group theory is assumed.