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Igor M. Nikonov

Igor M. Nikonov

Candidate of Physical and Mathematical Sciences (PhD), Docent

Department of Differential Geometry and Applications
Faculty of Mechanics and Mathematics, Lomonosov Moscow State University
Main Building, 1 Leninskiye Gory, Moscow 119991, GSP-1, Russia

E-mail: nikonov@mech.math.msu.su

Publications Full list of publications

1. Life and career

Igor M. Nikonov was born in 1978 in Noginsk-5, Moscow Region. In 1995 he finished the educational complex “School – Kindergarten No. 345” in Moscow and entered the Faculty of Mechanics and Mathematics of Lomonosov Moscow State University, graduating in 2000.

From 2000 to 2003 he studied in the postgraduate programme of the Division of Mathematics at the Department of Differential Geometry and Applications under the supervision of Professor Yu. P. Solovyov. In 2003 he defended his candidate dissertation “Characteristic classes of approximately finite algebras” in specialty 01.01.04 (geometry and topology).

Since 2003 I. M. Nikonov has worked at the Faculty of Mechanics and Mathematics, first as assistant and since 2012 as docent at the Department of Differential Geometry and Applications, where he has recently taught the course “Mathematical Models of Economics”.

2. Degrees and positions

Born1978, Noginsk-5, Moscow Region.
EducationFaculty of Mechanics and Mathematics, MSU, 1995–2000; postgraduate programme 2000–2003.
Candidate of Physical and Mathematical Sciences (PhD)2003; advisor Professor Yu. P. Solovyov. Thesis: “Characteristic classes of approximately finite algebras”.
PositionsAssistant (since 2003), Docent (since 2012), Department of Differential Geometry and Applications, MSU.
Research fieldsNon-commutative geometry; cyclic homology and cohomology of Hopf algebras; characteristic classes; low-dimensional topology; knot theory, virtual knots and graph-links; parity theory; Khovanov homology.

3. Research

Cyclic homology appeared in the early 1980s in the works of A. Connes and B. Tsygan. It is the central object of non-commutative geometry and plays the role of de Rham cohomology for non-commutative manifolds.

The first constructions of characteristic classes mapping the K-theory of a non-commutative manifold to homology groups of non-commutative differential forms also date from that time. The characteristic classes constructed by Connes take values in cyclic homology; M. Karoubi generalized Connes' classes by introducing the notion of differential calculus on a non-commutative manifold. Another approach was developed by A. S. Mishchenko and Yu. P. Solovyov and their students (Yu. I. Zhuraev, F. Yu. Popelensky, E. V. Korneeva, I. M. Nikonov), who based the construction of non-commutative differential forms on non-commutative vector fields (derivations of the algebra of functions on a non-commutative manifold). It turned out that there is a map taking Karoubi's characteristic classes to the Zhuraev–Mishchenko–Solovyov classes; these classes were computed for a number of algebras (finite-dimensional semisimple algebras, approximately finite C*-algebras, von Neumann algebras).

In 1998 cyclic cohomology of Hopf algebras with coefficients in modular pairs in involution was defined by Connes and Moscovici in the study of the index of foliations; after a series of generalizations, in 2003 cyclic homology of Hopf algebras with coefficients in stable anti-Yetter–Drinfeld modules was constructed. I. M. Nikonov, jointly with G. I. Sharygin, proposed a construction of a pairing in cyclic homology of Hopf algebras with coefficients generalizing the construction of M. Kraynik, and applied it to the computation of Hopf-cyclic cohomology of algebras of smooth functions.

Knot theory and low-dimensional topology

  • Parity theory for knots and virtual knots: weak parities and functorial maps, parity functors, parity on based matrices, universal parity on free two-dimensional knots, virtual index cocycles.
  • Graph-links and their invariants, including Khovanov homology of graph-links; graph-links: the state of the art (with D. P. Ilyutko and V. O. Manturov).
  • Khovanov–Lipshitz–Sarkar homotopy type for links in thickened higher genus surfaces; homotopical Khovanov homology; groups Gnk and braid groups of manifolds.
  • Maximally symmetric height atoms and maximally symmetric cell decompositions of surfaces (with E. A. Kudryavtseva and A. T. Fomenko); description of degenerate two-dimensional singularities with one critical point.
  • Geometry of extremal networks: sets admitting connection by graphs of finite length, the Maxwell formula for the length of an extremal network (with A. O. Ivanov, A. A. Tuzhilin, D. P. Ilyutko, A. G. Bannikova).

4. Teaching and seminars

I. M. Nikonov teaches at the Department of Differential Geometry and Applications, including the course “Mathematical Models of Economics”, and takes part in the work of the seminar on knot theory and representations (jointly with V. O. Manturov and D. P. Ilyutko).

5. Publications

Full list of publications — 131 records (department page and ISTINA profile)

  1. An example of a non-trivial characteristic class of the group algebra C[Z3]. Moscow University Mathematics Bulletin, no. 4 (2002), pp. 58–60.
  2. Voronovich differential calculi of dihedral groups. Moscow University Mathematics Bulletin, no. 6 (2002), pp. 10–14.
  3. Characteristic classes of finite-dimensional semisimple algebras. Moscow University Mathematics Bulletin, no. 2 (2004), pp. 3–7.
  4. Karoubi characteristic classes of approximately finite algebras. Matematicheskii Sbornik, vol. 196, no. 2 (2005), pp. 85–96.
  5. A. O. Ivanov, I. M. Nikonov, A. A. Tuzhilin. Sets admitting connection by graphs of finite length. Matematicheskii Sbornik, vol. 196, no. 6 (2005), pp. 71–110; Sbornik: Mathematics, 196:6 (2005), pp. 845–884.
  6. A. O. Ivanov, I. M. Nikonov, A. A. Tuzhilin. Geometry of trees of finite length in metric spaces. Proceedings of the Seminar on Vector and Tensor Analysis, vol. 26, MSU, 2005, pp. 178–200.
  7. I. M. Nikonov, G. I. Sharygin. On the Hopf-type cyclic cohomology with coefficients. In: C*-Algebras and Elliptic Theory, Trends in Mathematics, Birkhauser, 2006, pp. 203–212.
  8. Universal Karoubi characteristic classes of nuclear algebras. Fundamentalnaya i Prikladnaya Matematika, vol. 13, no. 8 (2007), pp. 133–169.
  9. D. A. Mikhalin, I. M. Nikonov. A problem on finding a counterfeit coin. Matematicheskoe Prosveshchenie, ser. 3, no. 11 (2007), pp. 149–158.
  10. E. A. Kudryavtseva, I. M. Nikonov, A. T. Fomenko. Maximally symmetric cell decompositions of surfaces and their coverings. Matematicheskii Sbornik, vol. 199, no. 9 (2008), pp. 3–96.
  11. Universal coefficients of cyclic homology of Hopf algebras. In: Contemporary Problems of Mathematics and Mechanics, vol. 3, MSU, 2009, pp. 98–103.
  12. E. A. Kudryavtseva, I. M. Nikonov, A. T. Fomenko. Symmetric and irreducible abstract polyhedra. In: Mathematics, issue 2: Geometry and Topology, Contemporary Problems of Mathematics and Mechanics, MSU, 2009, pp. 58–97.
  13. I. M. Nikonov, A. T. Fomenko, A. I. Shafarevich. On the works of D. F. Egorov on differential geometry. Istoriko-Matematicheskie Issledovaniya, ser. 2, no. 13 (2009), pp. 49–53.
  14. Universal Karoubi characteristic classes of nuclear C*-algebras. Journal of Mathematical Sciences, vol. 129, no. 6 (2009), pp. 837–862.
  15. I. M. Nikonov, G. I. Sharygin. Pairings in Hopf-cyclic cohomology of algebras and coalgebras with coefficients. Journal of K-Theory, vol. 5, no. 2 (2010), pp. 289–348.
  16. Khovanov homology of graph-links. Matematicheskii Sbornik, vol. 203, no. 8 (2012), pp. 125–140.
  17. Weak parities and functorial maps. Sovremennaya Matematika. Fundamentalnye Napravleniya, vol. 51 (2013), pp. 123–141; Journal of Mathematical Sciences, 214:5 (2016), pp. 699–717.
  18. N. V. Volchanetsky, I. M. Nikonov. Maximally symmetric height atoms. Moscow University Mathematics Bulletin, no. 2 (2013), pp. 3–6.
  19. A. G. Bannikova, D. P. Ilyutko, I. M. Nikonov. The length of an extremal network in a normed space: the Maxwell formula. Sovremennaya Matematika. Fundamentalnye Napravleniya, vol. 51 (2013), pp. 5–20; Journal of Mathematical Sciences, 214:5 (2016), pp. 593–608.
  20. D. P. Ilyutko, V. O. Manturov, I. M. Nikonov. Virtual knot invariants arising from parities. Knots in Poland III, Part I, Banach Center Publications, vol. 100 (2014), pp. 99–130.
  21. A new proof of Vassiliev's conjecture. Journal of Knot Theory and Its Ramifications, vol. 23, no. 7 (2014), 1460005.
  22. I. M. Nikonov, G. I. Sharygin. Structure of Hopf-cyclic (co)homology of algebras of smooth functions. Mathematical Notes, vol. 97, no. 4 (2015), pp. 566–582.
  23. V. O. Manturov, I. M. Nikonov. On braids and groups Gnk. Journal of Knot Theory and Its Ramifications, vol. 24, no. 13 (2015), 1541009.
  24. I. M. Nikonov, G. I. Sharygin. Hopf cyclic cohomology and Chern character of equivariant K-theories. Russian Journal of Mathematical Physics, vol. 22, no. 3 (2015), pp. 379–388.
  25. V. O. Manturov, I. M. Nikonov. Homotopical Khovanov homology. Journal of Knot Theory and Its Ramifications, vol. 24, no. 13 (2015), 1541003.
  26. Height atoms with a vertex-transitive symmetry group. Moscow University Mathematics Bulletin, no. 6 (2016), pp. 17–25.
  27. D. P. Ilyutko, I. M. Nikonov. Extremal networks on the λ-normed plane, where λ = 3, 4, 6. Matematicheskii Sbornik, vol. 208, no. 4 (2017), pp. 17–50.
  28. D. P. Ilyutko, I. M. Nikonov. The diagram approach in knot theory and its applications to graph theory. Moscow University Mathematics Bulletin, no. 3 (2018), pp. 65–71.
  29. Description of degenerate two-dimensional singularities with one critical point. Moscow University Mathematics Bulletin, no. 3 (2019), pp. 5–15.
  30. V. O. Manturov, D. A. Fedoseev, S. Kim, I. M. Nikonov. On groups Gnk and Γnk: a study of manifolds, dynamics and invariants. Bulletin of Mathematical Sciences, vol. 11, no. 2 (2021), 2150004.
  31. D. A. Fedoseev, V. O. Manturov, I. M. Nikonov. Manifolds of triangulations, braid groups of manifolds, and the groups Γnk. Lecture Notes in Computational Science and Engineering, vol. 143, Springer, Cham, 2021.
  32. L. H. Kauffman, I. M. Nikonov, E. Ogasa. Khovanov–Lipshitz–Sarkar homotopy type for links in thickened higher genus surfaces. Journal of Knot Theory and Its Ramifications, vol. 30, no. 8 (2021), 2150052.
  33. D. P. Ilyutko, I. M. Nikonov. Weight systems of framed chord diagrams corresponding to Lie algebras. Moscow University Mathematics Bulletin, no. 6 (2022), pp. 60–64.
  34. Virtual index cocycles and invariants of virtual links. Journal of Knot Theory and Its Ramifications, vol. 31, no. 3 (2022), 2250020.
  35. Parity on based matrices. Journal of Knot Theory and Its Ramifications, vol. 31, no. 2 (2022), 2250007.
  36. Parity functors. Journal of Knot Theory and Its Ramifications, vol. 31, no. 6 (2022), 2250041.
  37. On universal parity on free two-dimensional knots. Journal of Knot Theory and Its Ramifications, vol. 31, no. 10 (2022), 2250059.