Fields of research interest: theory of algorithms, model theory, algebra, and applications in theoretical computer science. Basing on the result on the existence of non-autostable models with finite algorithmic dimension, the theory of algorithmic dimension is developed. New methods to prove infinite algorithmic dimension are found. These methods made possible a solution to the problem of characterization of spectra of algorithmic dimension for series of concrete classes of models. A criterion of decidability for homogeneous models is established. Basing on it, solutions to problems of M. Morley and of Peretyatkin–Denisov were obtained. A solution to the Malcev problem on the characterization of classes of axioms with strong epimorphisms and strong homomorphisms is given. A series of results on constructive Boolean algebras is obtained. Nilpotent groups of finite algorithmic dimension were studied. A characterization is obtained for autostability of nilpotent torsion-free groups of finite rank as well as for Abelian $p$-groups. A new method to construct computable numbering is suggested that made possible solutions to a series of problems on Friedberg numberings, on families with unique positive numbering, etc. A solution to the problem of autostability of finitary constant expansions was obtained together with American scientists R. Shore, B. Khoussainov, and P. Cholak. A solution to the problem of two-element spectrum with recursive $T$-degree was obtained together with B.Khoussainov. A problem on a family with trivial Rogers semilattice but nontrivial inclusion was solved together with S. A. Badaev. A problem on the existence of strong constructive homogeneous extensions is solved. Together with an Italian logician A. Sorbi, the Rogers semilattice of computable numberings of arithmetical sets was investigated. Together with J. Knight (USA), the structure theory of computable classes of models is suggested, a problem of characterization of $\Sigma_1^1$-relations on computable models is solved. More than 160 papers were published; among them are two manuals: "Lectures in model theory" and "Introduction to the logic and methodology of science" (with Yu. L. Ershov and K. F. Samokhvalov), monographs: "Countable Boolean algebras" (1988) and "Countable Boolean algebras and decidability" (1996, English translation 1997), "Constructive models" (1999, with Yu. L. Ershov, English translation 2000) and "Handbook of recursive mathematics" (1999, S. S. Goncharov, Yu. L. Ershov, A. Nerode, J. Remmel, V. Marek editors).
Main publications:
Goncharov S. S. Countable Boolean algebras and decidability. Siberian School of Algebra and Logic, New-York, NY: Plenum. xii, 1997, 318 p.
Goncharov S. S., Ershov Yu. L. Constructive models. Kluwer Academic / Plenum Press Consultants Burean, xii, New-York, 2000, 293 p.
Goncharov S. S., Khoussainov B. On the spectrum of degrees of decidable relations // Dokl. Math., 1997, 55(1), 55–57.
Goncharov S. S., Khoussainov B., Cholak P., Shore R. Computably categorical structures and expansions by constants // J. Symb. Log., 1999, 64(1), 13–37.
Goncharov S. S. (ed.), Ershov Yu. L. (ed.), Nerode A. (ed.), Remmel J. B. (ed.), Marek V. (ed.). Handbook of Recursive Mathematics. Studies in Logic and the Foundations of Mathematics. 138, 139. Amsterdam: Elsevier. xlvi, 1998, 1372 p.