Sela does not have an objective view of Kharlampovich, Myasnikov's work. We will post a paper
dismissing his statements about "fatal mistakes". It just takes time. There are some typos and inessential errors that were fixed in later works. Sela himself has many such mistakes.
The objects in the two works (Sela's and our work) are similar but not identical. They are not amenable to a crude direct interpretation; some of our statements would not be true if interpreted via a ``computer translation'' into his language (and vice versa). One example is

Sela's wrong Theorem 7 from his paper 6 on Diophantine Geometry. This theorem describes groups elementarily equivalent to a non-abelian free groups. Sela claims that our Theorem 41 is wrong too. But our theorem is stated using our concept of regular NTQ groups and is correct. This shows that regular NTQ groups are not completely identical to hyperbolic $\omega$-residually free towers. Many of his critical comments resulted from such an exact translation of our concepts into his language. Additional misrepresentations result from not remembering that some statement was made two pages before (such as that we only consider fundamental sequences satisfying first and second restrictions).

The decidability of the elementary theory of a free group is used in the proof of the decidability of the theory of a torsion free hyperbolic group (our recent preprint in the arxiv) and to make quantifier elimination algorithmic. One can use it to approach the theory of a free product of groups with decidable elementary theories (Malcev's problem). One can also use it to deal with Right Angled Artin groups.

The algorithm to find the abelian JSJ decomposition of a limit group was constructed in our paper "Effective JSJ decompositions" that appeared before, it is used in the proof of the decidability of the theory. Actually many Algebraic Geometry over free groups questions are solved algorithmically (see references on pages 508-514), finding irreducible components of finite systems of equations, analogs of elimination and parametrization theorems in classical Algebraic Geometry etc
Olga Kharlampovich

undecidability of the elementary theory of free groups.' $\endgroup$14more comments