# Alexander G. Melnikov

Office: |
Cotton 323 |

Address: |
School of Mathematics and Statistics, Victoria University of Wellington, Kelburn Parade, Wellington, New Zealand |

Email: |
alexander.g.melnikov@gmail.com |

## Academic CV

Click here for my CV.

## What's new?

Computable topological groups.Preprint.With Koh and Ng.

Separating notions in effective topology.Submitted.With Ng.

Primitive recursive reverse mathematics.Preprint.With Bazhenov, Fiori-Carones, and Liu.

Effectively compact spaces.Submitted.With Downey.

Computably and punctually universal spaces.Submitted.With with Bagaviev, Batyrshin, Bazhenov, Bushtets, Dorzhieva, Koh, Kornev, Ng.

Computably totally disconnected locally compact groups.Preprint.With Nies.

Computable topological abelian groups.Submitted.With Lupini and Nies.

An Arithmetic analysis of compact surfaces.Submitted.With Harrison-Trainor.

Computable Stone spaces.Submitted.With Bazhenov and Harrison-Trainor.

Computable classifications of continuous, transducer, and regular functions.Submitted.With Franklin, Holzl, Ng, and Turetsky.

## PhD dissertations

Classification problems and constructive models. |

Computability and Structure. |

Effective properties of completely decomposable abelian groups. |

## Surveys

Computable analysis and classification problems.Conference on Computability in Europe CiE 2020: Beyond the Horizon of Computability,pp 100-111. With Downey.

Foundations of online structure theory.The Bulletin of Symbolic Logic, 25 (2019), no. 2, 141–181.With Bazhenov, Downey, and Kalimullin.

Foundations of online structure theory II: the operator approach.Logical Methods in Computer Science, July 21, 2021, Volume 17, Issue 3.With Downey and Ng.

Eliminating unbounded search in computable algebra.Lecture Notes in Computer Sciencebook series, volume 10307, Conference on Computability in Europe CiE 2017: Unveiling Dynamics and Complexity, pp 77-87 .

Computable Abelian Groups.The Bulletin of Symbolic Logic,Vol. 20, No. 3 (2014), pp. 315-356.

Computable Model Theory. A book chapter in:Turing's Legacy: Developments from Turing Ideas in Logic, R. Downey, editor, Cambridge University Press (2014), pp. 124-194.With Ekaterina Fokina and Valentina Harizanov.

## Journal papers and papers in refereed conference proceedings

Punctual degrees and lattice embeddings.To appear in proceedings of Aspects of Computation, World-Scientific.With Kalimullin and Zubkov.

New degree spectra of Polish spaces.Sib.Math.J.,volume 62, pages 882–894 (2021).

Relativizing computable categoricity.To appear in Proceedings of the Amer.Math.Soc.With Downey and Harrison-Trainor.

Decomposability and computability.To appear in Algebra i Logika.With Khoussainov.

Punctual definability on structures.To appear in Annals of Pure and Applied Logic.With Kalimullin and Montalban.

Non-density in punctual computability.Proceedings of the Amer.Math.Soc.,Volume 172, Issue 9, October–November 2021, 102985. With Greenberg, Harrison-Trainor, and Turetsky

A note on computable distinguishing colorings.Lobachevskii Journal of Mathematics,42, pages 693–700 (2021). With Bazhenov, Greenberg, Miller, and Ng.

Punctual categoricity relative to a computable oracle.Lobachevskii Journal of Mathematics,42, pages 735-742 (2021). With Kalimullin.

Computability of Polish spaces up to homeomorphism.The Journal of Symbolic Logic,2020, 85(4):1-25. With Harrison-Trainor and Ng.

On the complexity of classifying Lebesgue spaces.The Journal of Symbolic Logic,.85(3): 1254-1288 (2020). With Brown and McNicholl.

Punctual categoricity and universality.The Journal of Symbolic Logic,85(4): 1427-1466 (2020). With Downey, Greenberg, Ng, and Turetsky.

Online presentations of finitely generated structures.Theoretical Computer Science, 2020, 12 Vol. 844, 195-216.With Bazhenov, Kalimullin, and Ng.

Enumerating abelian p-groups.Journal of Algebra,Volume 560, 15 October 2020, Pages 745-790. With Downey and Ng.

A structure of punctual dimension two.Proceedings of the Amer.Math.Soc.,148 (2020), 3113-3128. With Ng.

Graphs are not universal for online computability.Journal of Computer and System Sciences,Volume 112, September 2020, Pages 1-12. With Downey, Harrison-Trainor, Kalimullin, and Turetsky.

Turing reducibility in the fine hierarchy.Annals of Pure and Applied Logic,Volume 171, Issue 7, July 2020, 102766. With Selivanov and Yamaleev.

Decompositions of decidable abelian groups.The International Journal of Algebra and Computation, Vol. 30, No. 01, pp. 49-90 (2020).With Bazhenov and Goncharov.

A Note on Realization of Index Sets in $\Pi_1^0$ Classes.Algebra Logika, 58:5 (2019), 659–663.With Downey.

Punctual copies of algebraic structures.Sibirsk. Mat. Zh., 2019, Volume 60, Number 6, Pages 1271-1285.With Frolov, Kalimullin, and Zubkov.

Random subgroups of the rationals.Proceedings of MFCS 2019; Leibniz International Proceedings in Informatics, Art. No. 25, 14 pp.

Automatic and polynomial-time algebraic structures.The Journal of Symbolic Logic,84 (2019), no. 4, 1630-1669. With Bazhenov, Harrison-Trainor, Kalimullin, and Ng.

Categorical linearly ordered structures.Annals of Pure and Applied Logic,170 (2019), no. 10, 1243–1255. With Downey and Ng.

The back-and-forth method and computability without delay.Israel Journal of Mathematics,Volume 234, Issue 2, pp 959-1000, 2019. With Ng.

Computable torsion abelian groups.Advances in Mathematics,325 (2018), 864--907. With Ng.

Effectively closed subgroups of the infinite symmetric group.Proceedings of the Amer.Math.Soc,146 (2018), no. 12, 5421-5435. With Greenberg, Nies, and Turetsky.

Computable topological groups and Pontryagin duality.Transactions of the Amer. Math. Soc.,370 (2018), no. 12, 8709-8737.

Uniform procedures in uncountable structures.The Journal os Symbolic Logic,83 (2018), no. 2, 529-550. With Greenberg, Knight, and Turetsky.

On a question of Kalimullin.Proceedings of the Amer. Math. Soc.,146 (2018), no. 8, 3553-3563. With Downey and Igusa.

Computable distributive lattices.Sib.Math.J.,November 2017, Volume 58, Issue 6, pp 959-970. With Bazhenov, Frolov, and Kalimullin.

Torsion-free abelian groups with optimal Scott families.Journal of Math. Logic,18 (2018), no. 1, 1850002, 47 pp.

Computable Polish group actions.The Journal of Symbolic Logic,, 83 (2018), no. 2, 443-460. With Montalban.

Structures computable without delay.Theoretical Computer Science,674 (2017), 73-98.

A Friedberg enumeration of equivalence structures.Journal of Math. Logic.,17 (2017), no. 2, 1750008, 28 pp. With Downey and Ng.

Different versions of categoricity without delay.Algebra and Logic,May 2017, Volume 56, Issue 2, pp 171-177. With Kalimullin and Ng.

On computable field embeddings and difference closed fields.Canadian Journal of Mathematics,69 (2017), no. 6, 1338-1363. With Miller and Harrison-Trainor.

Computable functors and effective interpretability.The Journal of Symbolic Logic,82 (1) (2017), 77-97. With Harroson-Trainor, Miller, and Montalban.

Proper divisibility in computable rings.Journal of Algebra,474 (2017), 180-212. With Greenberg.

Abelian groups and the halting problem.Annals of Pure and Applied Logic,Volume 167, Issue 11, November 2016, Pages 1123-1138. With Downey and Ng.

Comparing Classes of Finite Sums.Algebra i Logika,January 2016, Volume 54, Issue 6, pp 489-501. With Uri Andrews, Dmitriy Dushenin, Cameron Hill, Julia Knight.

Independence in effective algebra.Journal of Algebra,443 (2015), 441-468. With Montalban and Harrison-Trainor.

On Delta2-categoricity of equivalence relations.Annals of Pure and Applied Logic,2015, 166(9), pp. 851-880. With Downey and Ng.

Computable Structures and Operations on the Space of Continuous Functions.Fundamenta Mathematicae,233(2):1-41 (2015). With Keng Meng Ng

Iterated Effective Embeddings of Abelian p-Groups.International Journal of Algebra and Computation,24, 1055 (2014). With Downey and Ng.

Computable Completely Decomposable Groups.Transactions of the American Mathematical Society,Volume 366, Number 8, August 2014, Pages 4243-4266. With Rod Downey.

New Degree Spectra of Abelian Groups.Notre Dame Journal of Formal Logic,141 (2013), 3275-3289.

Effectively Categorical Abelian Groups.Journal of Algebra,Volume 373, 1 January 2013, Pages 223-248. With Rod Downey.

Limitwise Monotonic Sequences and Degree Spectra of Structures.Proceedings of the American Mathematical Society,141 (2013), 3275-3289. With Iskander Kalimullin and Bakhadyr Khoussainov.

Computably Isometric Spaces.The Journal of Symbolic Logic,Volume 78, Issue 4 (2013), 1025-1346.

The Classification Problem for Compact Computable Metric Spaces. In Paola Bonizzoni, Vasco Brattka, and Benedikt L{\"o}we, editors,CiE 2013, volume 7921 ofLecture Notes in Computer Science, pages 320--328. Springer, 2013. With Andre Nies.

Randomness and K-triviality in Computable Metric Spaces.Proceedings of the American Mathematical Society,141 (2013), 2885-2899. With Andre Nies.

Jump Degrees of Torsion-Free Abelian Groups.The Journal of Symbolic Logic,Volume 77, Issue 4 (2012), 1067-1100. With Brooke Andersen, Asher Kach and D. Reed Solomon.

Ulm type, and Coding Rank-Homogeneous Trees in Other Structures.The Journal of Symbolic Logic,Volume 76, Number 3, September 2011, 846-849. With Ekaterina Fokina, Julia F. Knight, C. Maher, and Sara Quinn.

Transforming Trees into Abelian Groups.New Zealand Journal of Mathematics,Vol. 41, (2011), Pages 75-81.

Decidability and Computability of Certain Torsion-Free Abelian Groups.Notre Dame Journal of Formal Logic,, vol 51, number 1, (2010). With Rod Downey, Sergei S. Goncharov, Asher Kach, Julia F. Knight, Oleg V. Kudinov, and Daniel Turetsky.

Computable Ordered Abelian Groups and Fields. In: Programs, Proofs, Processes,Lecture Notes in Computer Science,2010, Volume 6158/2010, 321-330.

Enumerations and Completely Decomposable Torsion-Free Abelian Groups.Theory Comput. Syst,45(4): 897-916 (2009).

## Other writings

Hrushovski constructions and randomizations in computable model theory. A review published inThe Bulletin of Symbolic Logic,Volume 19, 2013, page 399.