Diophantine Geometry over Groups

• Diophantine geometry over groups I: Makanin-Razborov diagrams , ps , (Archive) , Publications Mathematiques de l'IHES 93(2001), 31-105.

• Diophantine geometry over groups II: Completions, closures and formal solutions, ps , (Archive) , Israel Jour. of Math. 134(2003), 173-254.

• Diophantine geometry over groups III: Rigid and solid solutions, ps , (Archive) , Israel Jour. of Math. 147(2005), 1-73.

• Diophantine geometry over groups IV: An iterative procedure for validation of a sentence, ps , (Archive) , Israel Jour. of Math. 143(2004), 1-130.

• Diophantine geometry over groups V: Quantifier elimination, pdf1 , pdf2 , (Archive) , divided into two parts - Israel Jour. of Math. 150(2005), 1-197, and GAFA 16(2006), 537-706.

• Diophantine geometry over groups VI: The elementary theory of a free group, pdf , (Archive) , GAFA 16(2006), 707-730.

• Diophantine geometry over groups VII: The elementary theory of a hyperbolic group, Proceedings of the LMS, 99(2009), 217-273.

• Diophantine geometry over groups VIII: Stability, Annals of Math. 177(2013), 787-868.

• Free and Hyperbolic Groups are not Equational

• Makanin-Razborov diagrams over free products (with E. Jaligot), Illinois Journal of Math. 54(2010), 19-68.

• Diophantine geometry over groups IX: Envelopes and Imaginaries

• Diophantine geometry over groups X: The Elementary Theory of Free Products of Groups

• Word equations I: Pairs and their Makanin-Razborov diagram

• Non Commutative Algebraic Geometry I: Monomial Equations with a Single Variable

• The rates of growth in a hyperbolic group (with K. Fujiwara)

• A non-terminating iterative procedure I, II - a demonstration of some inherent difficulties.

• A report on Tarski's decidability problem - mistakes and counterexamples.

• A detailed report on Kharlampovich and Myasnikov's work

• A list of research problems - we intend to update the list periodically.