Solving equations is one of the main topics in mathematics. A more general and more difficult problem is to describe which formulas of the first-order logic hold in a given group. Recent works on the Tarski's problems (Kharlampovich, Miasnikov, Sela) opened a new direction of research called now "Algebraic geometry over groups". We are going to discuss some methods and techniques used for the solution of these problems, and developments in the algebraic geometry for groups, Lie Algebras, and other algebraic systems.
Another new direction of research is the theory of fully residually free and residually free groups. Finitely generated fully residually free groups (limit groups) play a crucial role in the theory of equations and first-order formulas over a free group. It is remarkable that these groups, which have been widely studied before, turn out to be the basic objects in newly developing areas of algebraic geometry and model theory of free groups. These groups are exactly the coordinate groups of irreducible algebraic varieties over a free group; they have the same existential theory as a non-abelian free group. They are relatively hyperbolic and have many properties similar to those of free groups. Many algorithmic problems can be solved in this class. Residually free groups are subgroups of direct products of finitely many fully residually free groups. This provides an approach for studying their properties. The activity in this field has been growing incredibly fast, huge progress has been achieved, but a lot of work is yet to be carried out.
The Aisenstadt Chair Alexander Razborov will be giving the Aisenstadt Chair lectures during this workshop.
Equations in solvable groups
N. Romanovskii (Novosibirsk)
Complexity of the diophantine problem in a free group
I. Lysenok (Steklov Institute, Moscow)