Research

My research area is computable model theory. I am interested in effective properties of mathematical structures and relations on them.

The complete list of my current and past projects:

Peer-reviewed papers

  • Computability-theoretic categoricity and Scott families (with V. Harizanov and D. Turetsky), to appear in Ann. Pure Applied Logic.
  • Degrees of bi-embeddable categoricity of equivalence structures (with N. Bazhenov, D. Rossegger and L. San Mauro), to appear in Arch. Math. Logic.
  • Computable bi-embeddable categoricity (with N. Bazhenov, D. Rossegger and L. San Mauro), to appear in Algebra Logic.
  • Bi-embeddability spectra and bases of spectra (with D. Rossegger and L. San Mauro), to appear in Math. Logic Quarterly.
  • Measuring the complexity of reductions between equivalence relations (with D. Rossegger and L. San Mauro), to appear in Computability.
  • Degree spectra with respect to equivalence relations (with P. Semukhin and D. Turetsky), to appear in Algebra Logic.
  • Computable bi-embeddable categoricity of equivalence structures (with N. Bazhenov,D. Rossegger and L. San Mauro), Proceedings of the 11th Panhellenic Logic Symposium, 2017, 126–132.
  • Bi-embeddability spectra of structures (with D. Rossegger and L. San Mauro), Proceedings of the 11th Panhellenic Logic Symposium, 2017, 32–38.
  • Reducibility and bi-reducibility spectra of equivalence relations (with D. Rossegger and L. San Mauro), Proceedings of the 11th Panhellenic Logic Symposium, 2017, 83–89.
  • Linear orders realized by c.e. equivalence relations (with B. Khoussainov, P. Semukhin and D. Turetsky), J. Symb. Logic, 81 (2016), 463–482.
  • Categoricity spectra for rigid structures (with A. Frolov and I. Kalimullin), Notre Dame J. Formal Logic, 57 (2016), 45–57.
  • Index sets for n-decidable structures categorical relative to m-decidable presentations (with S. Goncharov, V. Harizanov, O. Kudinov, and D. Turetsky), Algebra Logic, 54 (2015), 520–528 (in Russian), 336–341 (English translation).
  • Classes of structures with universe a subset of \omega_1 (with S. Friedman, J. Knight, and R. Miller), J. Logic Computation 23 (2013) 6, 1249-1265.
  • Equivalence relations that are \Sigma^0_3 complete for computable reducibility (with S. D. Friedman and A. Nies), Lecture Notes in Computer Science 7456 (2012) 26-33.
  • On \Sigma^1_1 equivalence relations over the natural numbers (with S. D. Friedman), Math. Logic Quarterly, 58 (2012), 113-124.
  • Isomorphism relations on computable structures (with S. D. Friedman, V. Harizanov, J. Knight, C. McCoy, A. Montalbán, J. Symbolic Logic, 77 (2012), 122-132.
  • Computable embedding problem (with J. Carson, V. Harizanov, J. F. Knight, S. Quinn, C. Safranski, J. Wallbaum), Algebra Logic, 50 (2012), 478-493.
  • Classes of Ulm type and coding rank-homogeneous trees in other structures (with J. F. Knight, A. Melnikov, S. Quinn, C. Safranski), J. Symbolic Logic 76 (2011), 846-869.
  • The effective theory of Borel equivalence relations (with S. D. Friedman, A. Törnquist), Ann. Pure Applied Logic, 161 (2010), 837--850.
  • Degrees of categoricity of computable structures (with I. Kalimullin and R. Miller), Arch. Math. Logic, 49 (2010), 51--67.
  • Equivalence relations on classes of computable structures(with S. D. Friedman), the Proceedings of the Conference of Computability in Europe 2009, Heidelberg, Germany, July 2009, LNSC 5635 (2009), 198-207.
  • Intrinsic bounds on complexity and definability at limit levels (with J. Chisholm, S. S. Goncharov, V.S. Harizanov, J. F. Knight, S. Miller), to appear in J. Symbolic Logic.
  • Index sets for some classes of structures, Ann. Pure Applied Logic, 157 (2009), 139--147.
  • Algorithmic properties of structures for the languages with two unary functional symbols, Logic and Theory of Algorithms, Local Proceedings of CiE'08, 2008, pp. 127-136.
  • Algorithmic properties of structures for the languages with two unary functional symbols, Vestnik NGU, 8 (2008), no. 1, pp. 90-101 (in Russian).
  • Index sets for classes of high rank structures (with W. Calvert, S. S. Goncharov, J. F. Knight, O. Kudinov, A. S. Morozov, V. Puzarenko), Journal of Symbolic Logic, 72 (2007), no. 4, pp. 1418-1446.
  • Index Sets of Computable Structures with Decidable Theories, Computation and Logic in the Real World - Third Conference of Computability in Europe, CiE 2007, Siena, Italy, June 2007, Proceedings, LNCS, v. 4497, pp. 290-296.
  • Index sets of decidable structures, Siberian Mathematical Journal, 48 (2007), no. 5, pp. 939-948.
  • Arithmetical Turing degrees and categorical theories of computable models, Mathematical Logic in Asia. Proceedings of the 9th Asian Logic Conference, 2006, pp. 58-69. 
  • On spectra of computable models, Vestnik NGU, 6 (2006), no.6, pp. 69-73 (in Russian).
  • On complexity of categorical theories with computable models, Vestnik NGU, 5 (2005), no.2, pp. 78-86 (in Russian).

Non-peer-reviewed papers and editorship

  • Editor of the Special Issue Mind Mechanism Mathematics, Mathematical Structures in Computer Science, March 2018.
  • Preface to the Special Issue Mind Mechanism Mathematics, Mathematical Structures in Computer Science, March 2018, 338–339.
  • Computable model theory (with V. Harizanov and A. Melnikov), Turing's Legacy: Developments from Turing's ideas in Logic (Edited by R. Downey), ASL Lecture Notes in Logic 42 (2014), 124-194.

My list of publications in pdf.