Alexander Kolpakov

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List of Papers ::: Preprints

1. Subspace stabilisers in hyperbolic lattices

   Mikhail Belolipetsky, Nikolay Bogachev, Alexander Kolpakov, Leone Slavich
   arXiv:2105.06897
   Comments: 78 pages, 4 figures. Final version (to appear in J. Assoc. Math. Res.). The paper has been rewritten. What was Theorem 1.9, now becomes Theorem 1.2, and therefore all other theorems in the introduction have shifted: Thm 1.n → 1.(n+1). Sections 2.2, 2.3, 2.4, 3.3, and 5 have been rewritten, and some proofs (e.g. of Theorems 1.8 & 1.9) have been simplified
   Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Group Theory (math.GR); Metric Geometry (math.MG); Number Theory (math.NT)

2. Expansion properties of Whitehead moves on cubic graphs

   Laura Grave de Peralta, Alexander Kolpakov
   arXiv:2303.13923
   Comments: 10 pages, 17 figures
   Journal: Comptes Rendus. Math. (2024)
   Subjects: Combinatorics (math.CO)

3. Burau representation, Squier’s form, and non-Abelian anyons

   Alexander Kolpakov
   arXiv:2510.18186
   Comments: 13 pages, 2 figures
   Subjects: Quantum Physics (quant-ph); Information Theory (cs.IT); Mathematical Physics (math-ph)

4. Benford’s Law from Turing Ensembles and Integer Partitions

   Alexander Kolpakov, Aidan Rocke
   arXiv:2502.16314
   Comments: 10 pages, 2 figures
   Journal: Phys. Rev. E (2025)
   Subjects: Information Theory (cs.IT); Computational Complexity (cs.CC); Discrete Mathematics (cs.DM); Mathematical Physics (math-ph)

5. Lagrangians, Renormalization, and Quantization in Prefix Coding

   Alexander Kolpakov, Aidan Rocke
   arXiv:2506.23447
   Comments: arXiv:2506.23447 is split from arXiv:2502.16314; 15 pages, 2 tables
   Subjects: Information Theory (cs.IT); Mathematical Physics (math-ph)

6. Multiplicative Turing Ensembles, Pareto’s Law, and Creativity

   Alexander Kolpakov, Aidan Rocke
   arXiv:2510.04167
   Comments: 23 pages, 2 figures, 1 table
   Subjects: Information Theory (cs.IT); Computational Complexity (cs.CC); Mathematical Physics (math-ph)

7. Loss-Complexity Landscape and Model Structure Functions

   Alexander Kolpakov
   arXiv:2507.13543
   Comments: 25 pages, 11 figures
   Subjects: Information Theory (cs.IT); Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Mathematical Physics (math-ph)

8. Fast Geometric Embedding for Node Influence Maximization

   Alexander Kolpakov, Igor Rivin
   arXiv:2506.07435
   Comments: 8 pages, 4 figures, 18 tables
   Subjects: Social and Information Networks (cs.SI); Artificial Intelligence (cs.AI); Machine Learning (cs.LG)

9. Dimensionality reduction for homological stability and global structure preservation

   Alexander Kolpakov, Igor Rivin
   arXiv:2503.03156
   Comments: 22 pages, 12 figures
   Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Mathematical Software (cs.MS)

10. A ripple in time: a discontinuity in American history

    Alexander Kolpakov, Igor Rivin
    arXiv:2312.01185
    Comments: 6 pages, 8 figures; to appear in 8th NLPIR Okayama, Japan | December 13–15, 2024 as “Discovering temporal and personality aspects in meager and highly variable text samples”
    Subjects: Computation and Language (cs.CL); Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Social and Information Networks (cs.SI)

11. The Information Geometry of UMAP

    Alexander Kolpakov, Aidan Rocke
    arXiv:2309.01237
    Comments: 12 pages, 2 figures, 3 tables; to appear in Le Matematiche
    Subjects: Computational Geometry (cs.CG); Discrete Mathematics (cs.DM); Information Theory (cs.IT); Geometric Topology (math.GT)

12. On the impossibility of discovering a formula for primes using AI

    Alexander Kolpakov, Aidan Rocke
    arXiv:2308.10817
    Comments: 29 pages; parts of this manuscript are accepted as a separate paper in PLOS ONE
    Subjects: Computational Complexity (cs.CC)

13. Robust affine point matching via quadratic assignment on Grassmannians

    Alexander Kolpakov, Michael Werman
    arXiv:2303.02698
    Comments: 9 pages, 23 figures; added comparison to GrassGraph; notably, GrassGraph quickly loses accuracy on our test examples with noise and occlusion; to appear in Pattern Recognition Letters
    Subjects: Computer Vision and Pattern Recognition (cs.CV)

14. Machine Learning of the Prime Distribution

    Alexander Kolpakov, Aidan Rocke
    arXiv:2403.12588
    Comments: 10 pages; parts of arXiv:2308.10817 reworked and amended; author’s draft; accepted in PLOS ONE
    Subjects: Information Theory (cs.IT); Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Number Theory (math.NT)

15. Kleinian sphere packings, reflection groups, and arithmeticity

    Nikolay Bogachev, Alexander Kolpakov, Alex Kontorovich
    arXiv:2203.01973
    Comments: 20 pages, 5 figures; final version; Math. Comp. 2024, Vol. 93, no. 345, pp. 505–521
    Subjects: Geometric Topology (math.GT); Combinatorics (math.CO); Group Theory (math.GR)

16. Thin hyperbolic reflection groups

    Nikolay Bogachev, Alexander Kolpakov
    arXiv:2112.14642
    Comments: 10 pages, 1 figure, revision and generalization
    Subjects: Group Theory (math.GR); Geometric Topology (math.GT); Number Theory (math.NT)

17. An approach to robust ICP initialization

    Alexander Kolpakov, Michael Werman
    arXiv:2212.05332
    Comments: 9 pages, 18 figures, 1 table
    Subjects: Computer Vision and Pattern Recognition (cs.CV); Computational Geometry (cs.CG); Optimization and Control (math.OC)

18. Embedding non-arithmetic hyperbolic manifolds

    Alexander Kolpakov, Stefano Riolo, Leone Slavich
    arXiv:2003.01707
    Comments: 20 pages, 5 figures. Final version
    Journal: Math. Res. Lett., vol. 29 (2022) n. 1, 247–274
    Subjects: Geometric Topology (math.GT)

19. The signature of cusped hyperbolic 4-manifolds

    Alexander Kolpakov, Stefano Riolo, Steven T. Tschantz
    arXiv:2006.12095
    Comments: 13 pages, 4 figures, 7 tables. To appear in Int. Math. Res. Notices (minor changes wrt the previous version)
    Journal: Int. Math. Res. Not. IMRN 2023:9 (2023), 7961–7975
    Subjects: Geometric Topology (math.GT)

20. Infinitely many quasi-arithmetic maximal reflection groups

    Edoardo Dotti, Alexander Kolpakov
    arXiv:2109.03316
    Comments: 8 pages, 2 figures; to appear in Proc. Amer. Math. Soc. (2022)
    Subjects: Group Theory (math.GR); Algebraic Topology (math.AT); Geometric Topology (math.GT); Number Theory (math.NT)

21. Infinitely many arithmetic hyperbolic rational homology 3-spheres that bound geometrically

    Leonardo Ferrari, Alexander Kolpakov, Alan W. Reid
    arXiv:2203.01997
    Comments: 19 pages, 1 figure, 1 table; improved exposition, some new results added
    Subjects: Geometric Topology (math.GT)

22. Packable hyperbolic surfaces with symmetries

    Maria Dostert, Alexander Kolpakov
    arXiv:2010.12028
    Comments: 11 pages, 4 figures; to appear in Canadian Math. Bulletin (2022)
    Subjects: Geometric Topology (math.GT); Combinatorics (math.CO); Metric Geometry (math.MG)

23. Three-dimensional maps and subgroup growth

    Rémi Bottinelli, Laura Ciobanu, Alexander Kolpakov
    arXiv:1712.01418
    Comments: 17 pages, 6 figures, 1 table
    Journal: Manuscripta Math. (2021)
    Subjects: Group Theory (math.GR); Combinatorics (math.CO); Geometric Topology (math.GT)

24. On free semigroups of affine maps on the real line

    Alexander Kolpakov, Alexey Talambutsa
    arXiv:2105.09387
    Comments: 7 pages, 1 figure
    Subjects: Combinatorics (math.CO); Dynamical Systems (math.DS); Number Theory (math.NT)

25. Euclidean volumes of hyperbolic knots

    Nikolay Abrosimov, Alexander Kolpakov, Alexander Mednykh
    arXiv:2107.03275
    Comments: 11 pages, 2 figures
    Subjects: Geometric Topology (math.GT); Number Theory (math.NT)

26. Cusps of hyperbolic 4-manifolds and rational homology spheres

    Leonardo Ferrari, Alexander Kolpakov, Leone Slavich
    arXiv:2009.09995
    Comments: 15 pages, 1 figure, 1 table
    Journal: Proceedings of the London Mathematical Society 123, 636–648 (2021)
    Subjects: Geometric Topology (math.GT)

27. Kissing number in non-Euclidean spaces of constant sectional curvature

    Maria Dostert, Alexander Kolpakov
    arXiv:2003.05547
    Comments: 17 pages, 2 figures, 4 tables; ancillary files available on GitHub. This is a merger of arXiv:1907.00255 and arXiv:1910.02715 (both withdrawn)
    Journal: Math. Comp. (2021)
    Subjects: Combinatorics (math.CO); Metric Geometry (math.MG); Optimization and Control (math.OC)

28. Growth rates of Coxeter groups and Perron numbers

    Alexander Kolpakov, Alexey Talambutsa
    arXiv:1912.05608
    Comments: 21 pages, 2 figures; to appear in IMRN
    Subjects: Group Theory (math.GR); Combinatorics (math.CO); Geometric Topology (math.GT); Number Theory (math.NT)

29. Space vectors forming rational angles

    Kiran S. Kedlaya, Alexander Kolpakov, Bjorn Poonen, Michael Rubinstein
    arXiv:2011.14232
    Comments: 30 pages
    Subjects: Metric Geometry (math.MG); Algebraic Geometry (math.AG); Number Theory (math.NT)

30. On faces of quasi-arithmetic Coxeter polytopes

    Nikolay Bogachev, Alexander Kolpakov
    arXiv:2002.11445
    Comments: 14 pages, 3 figures; ancillary files available
    Journal: Int. Math. Res. Notices (2020)
    Subjects: Geometric Topology (math.GT); Group Theory (math.GR); Metric Geometry (math.MG); Number Theory (math.NT)

31. Symmetries of hyperbolic 4-manifolds

    Alexander Kolpakov, Leone Slavich
    arXiv:1409.1910
    Comments: 32 pages, 10 figures; IMRN (2015); SAGE worksheet available; minor correction noted in arXiv:1710.07534
    Journal: International Mathematics Research Notices (2016) 9, 2677–2716
    Subjects: Geometric Topology (math.GT); Combinatorics (math.CO); Metric Geometry (math.MG)

32. Hyperbolic four-manifolds, colourings and mutations

    Alexander Kolpakov, Leone Slavich
    arXiv:1507.02747
    Comments: 24 pages, 11 figures; classification in Proposition 3.2 is incomplete: see arXiv:2009.09995 for correction
    Journal: Proc. London Math. Soc. (2016), 113 (2), 163–184
    Subjects: Geometric Topology (math.GT); Combinatorics (math.CO); Differential Geometry (math.DG)

33. Some hyperbolic three-manifolds that bound geometrically

    Alexander Kolpakov, Bruno Martelli, Steven T. Tschantz
    arXiv:1311.2993
    Comments: 10 pages, 1 figure; ancillary Mathematica notebook available
    Subjects: Geometric Topology (math.GT)

34. A hyperbolic counterpart to Rokhlin’s cobordism theorem

    Michelle Chu, Alexander Kolpakov
    arXiv:1905.04774
    Comments: 16 pages, 4 figures, 3 tables; to appear in IMRN; ancillary file available
    Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Group Theory (math.GR)

35. Semidefinite programming bounds for the average kissing number

    Maria Dostert, Alexander Kolpakov, Fernando Mário de Oliveira Filho
    arXiv:2003.11832
    Subjects: Metric Geometry (math.MG); Optimization and Control (math.OC)

36. Kissing number in hyperbolic space

    Maria Dostert, Alexander Kolpakov
    arXiv:1907.00255
    Comments: Will be merged with arXiv:1910.02715
    Subjects: Metric Geometry (math.MG); Combinatorics (math.CO); Optimization and Control (math.OC)

37. Kissing number in spherical space

    Maria Dostert, Alexander Kolpakov
    arXiv:1910.02715
    Comments: Will be merged with arXiv:1907.00255
    Subjects: Metric Geometry (math.MG); Combinatorics (math.CO); Optimization and Control (math.OC)

38. Geodesic growth of right-angled Coxeter groups based on trees

    Laura Ciobanu, Alexander Kolpakov
    arXiv:1504.02774
    Comments: 14 pages, 4 figures; supplementary material and a SAGE worksheet available
    Subjects: Group Theory (math.GR); Combinatorics (math.CO)

39. Telescopic groups and symmetries of combinatorial maps

    Rémi Bottinelli, Laura Grave de Peralta, Alexander Kolpakov
    arXiv:1901.05710
    Comments: 29 pages, 7 figures; final version to appear in Algebraic Combinatorics
    Subjects: Combinatorics (math.CO); Group Theory (math.GR); Geometric Topology (math.GT)

40. Spherical and geodesic growth rates of right-angled Coxeter and Artin groups are Perron numbers

    Alexander Kolpakov, Alexey Talambutsa
    arXiv:1809.09591
    Comments: 12 pages, 2 figures; to appear in Discrete Mathematics
    Subjects: Group Theory (math.GR); Combinatorics (math.CO); Dynamical Systems (math.DS)

41. Volume of a doubly truncated hyperbolic tetrahedron

    Alexander Kolpakov, Jun Murakami
    arXiv:1203.1061
    Comments: Several misprints in the proof of Theorem 1 corrected
    Journal: Aequat. Math. 85 (2013), 449–463
    Subjects: Metric Geometry (math.MG); Complex Variables (math.CV); Differential Geometry (math.DG)

42. Spherical tetrahedra with rational volume, and spherical Pythagorean triples

    Alexander Kolpakov, Sinai Robins
    arXiv:1811.06598
    Comments: 14 pages, 3 figures, 5 tables; to appear in Math. of Comp.; supplementary SageMath worksheets available
    Subjects: Metric Geometry (math.MG); Number Theory (math.NT)

43. Counting cusped hyperbolic 3-manifolds that bound geometrically

    Alexander Kolpakov, Stefano Riolo
    arXiv:1808.05681
    Comments: 17 pages, 7 figures; to appear in Transactions AMS
    Journal: Trans. Amer. Math. Soc. 373:1 (2020), 229–247
    Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Group Theory (math.GR); Metric Geometry (math.MG)

44. On the formulas for effective stiffnesses in a paper by Zheng Ye, Victor L. Berdichevsky and Wenbin Yu

    A. A. Kolpakov, A. G. Kolpakov
    arXiv:1902.05453
    Comments: 2 pages, extended references, final version
    Subjects: Classical Physics (physics.class-ph)

45. Many cusped hyperbolic 3-manifolds do not bound geometrically

    Alexander Kolpakov, Alan W. Reid, Stefano Riolo
    arXiv:1811.05509
    Comments: 11 pages, 3 figures; to appear in Proceedings AMS
    Journal: Proc. Amer. Math. Soc. 18:5 (2020), 2223–2243
    Subjects: Geometric Topology (math.GT); Combinatorics (math.CO); Metric Geometry (math.MG)

46. Free subgroups of free products and combinatorial hypermaps

    Laura Ciobanu, Alexander Kolpakov
    arXiv:1708.03842
    Comments: 27 pages, 3 figures; supplementary SAGE worksheets available
    Journal: Discrete Mathematics, 2019
    Subjects: Combinatorics (math.CO); Group Theory (math.GR); Geometric Topology (math.GT)

47. Computation of homogenized stiffnesses of thin corrugated plates

    A. A. Kolpakov, A. G. Kolpakov
    arXiv:1811.01718
    Comments: 21 page, 5 figures, 1 table
    Subjects: Classical Physics (physics.class-ph)

48. Embedding arithmetic hyperbolic manifolds

    Alexander Kolpakov, Alan W. Reid, Leone Slavich
    arXiv:1703.10561
    Comments: 20 pages; revised version, typos corrected; Mathematical Research Letters vol. 25, no. 4
    Journal: Math. Res. Lett. 25 (2018), 1305–1328
    Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Group Theory (math.GR)

49. The dual Jacobian of a generalised tetrahedron, and volumes of prisms

    Alexander Kolpakov, Jun Murakami
    arXiv:1409.3355
    Comments: 26 pages, 11 figures, 1 table; to appear in Tokyo Journal of Mathematics
    Subjects: Metric Geometry (math.MG); Complex Variables (math.CV); Differential Geometry (math.DG)

50. Combinatorial decompositions, Kirillov-Reshetikhin invariants and the Volume Conjecture for hyperbolic polyhedra

    Alexander Kolpakov, Jun Murakami
    arXiv:1603.02380
    Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Metric Geometry (math.MG)

51. Hyperbolic four-manifolds with one cusp

    Alexander Kolpakov, Bruno Martelli
    arXiv:1303.6122
    Comments: 24 pages, 15 figures, typos corrected; Geom. and Funct. Anal., 2013
    Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Metric Geometry (math.MG)

52. Interaction of “rigid” quantum systems

    A. A. Kolpakov, A. G. Kolpakov
    arXiv:1212.3702
    Comments: 15 pages; typos corrected
    Subjects: Mathematical Physics (math-ph)

53. On the optimality of the ideal right-angled 24-cell

    Alexander Kolpakov
    arXiv:1211.2944
    Comments: 24 pages, 22 figures, minor corrections
    Journal: Algebraic & Geometric Topology 12 (2012) 1941–1960
    Subjects: Metric Geometry (math.MG); Combinatorics (math.CO); Geometric Topology (math.GT)

54. Examples of rigid and flexible Seifert fibred cone-manifolds

    Alexander Kolpakov
    arXiv:1004.2376
    Comments: 24 pages, 6 figures
    Journal: Glasgow Math. Journal, 2012
    Subjects: Geometric Topology (math.GT); Metric Geometry (math.MG)

55. Complementarity problems for electro-neutral charged bodies

    A. A. Kolpakov, A. G. Kolpakov
    arXiv:1207.5142
    Comments: 8 pages, 1 figure
    Subjects: Mathematical Physics (math-ph)

56. Complementarity problems for two pairs of charged bodies

    A. A. Kolpakov, A. G. Kolpakov
    arXiv:1205.5157
    Comments: typos corrected
    Subjects: Mathematical Physics (math-ph)

57. Deformation of finite-volume hyperbolic Coxeter polyhedra, limiting growth rates and Pisot numbers

    Alexander Kolpakov
    arXiv:1105.6267
    Comments: 26 pages, 16 figures, 4 data tables; minor corrections; European Journal of Combinatorics, 2012
    Subjects: Metric Geometry (math.MG); Combinatorics (math.CO)

58. Volume formula for a ℤ2-symmetric spherical tetrahedron through its edge lengths

    Alexander Kolpakov, Alexander Mednykh, Marina Pashkevich
    arXiv:1007.3948
    Comments: 27 pages, 2 figures; enhanced and improved exposition, typos corrected; Arkiv foer Matematik, 2011
    Subjects: Metric Geometry (math.MG); Complex Variables (math.CV); Differential Geometry (math.DG)

59. Spherical structures on torus knots and links

    Alexander Kolpakov, Alexander Mednykh
    arXiv:1008.0312
    Comments: 17 pages, 5 figures; typos
    Journal: Siberian Math. J., 50(5), 856–866 (2009)
    Subjects: Geometric Topology (math.GT); Metric Geometry (math.MG)

60. Addendum to “Spherical structures on torus knots and links”

    Alexander Kolpakov
    arXiv:1101.1620
    Comments: Addendum to “Spherical structures on torus knots and links”, arXiv:1008.0312
    Subjects: Geometric Topology (math.GT)