In this project we propose to expand our methods (Makanin-Razborov's machinery, algebraic geometry over groups, free actions on non-standard trees, structure of limit groups, etc.) developed in the proof of the Tarski's conjectures on first-order theories of free groups into several directions: model theory and algebraic geometry in the presence of negative curvature and for right-angled Artin groups; group actions on Lambda-trees, Lambda-hyperbolic geodesic spaces, and cube complexes; algorithmic problems for groups and elimination processes; equations in a wide variety of groups and algebras.
Solving equations is one of the main themes in mathematics from ancient times. Equations give a universal language to describe scientific problems in all their variety. Over the years (centuries, in fact) equations and their solutions became very complex, so to understand their hidden structure one has to use very elaborate techniques from algebra and geometry. In particular, groups are used to describe symmetries of mathematical objects, they play a fundamental role in studying solutions of equations. Nowadays, when mathematics gets ever more complex, equations alone cannot describe subtleties of scientific phenomena; new problems require much more powerful means and more powerful languages. Most of the essential properties of objects that occur in everyday mathematical practice can be described in a very particular but universal language, so called first-order logic. In this project we study properties of groups, in all their entirety, that can be described in the first-order logic. Along the way we hope to shed some light on several fundamental open problems in group theory and algebra.
