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Millionshchikov, Dmitry Vladimirovich

Total publications: 34 (31)
in MathSciNet: 26 (25)
in zbMATH: 23 (22)
in Web of Science: 21 (20)
in Scopus: 20 (20)
Cited articles: 19
Citations in Math-Net.Ru: 55
Citations in Web of Science: 48
Citations in Scopus: 64
Presentations: 19

Number of views:
This page:3975
Abstract pages:5624
Full texts:1825
References:643
Millionshchikov, Dmitry Vladimirovich
Associate professor
Doctor of physico-mathematical sciences (2019)
Speciality: 01.01.04 (Geometry and topology)
E-mail:
Keywords: graded Lie algebra, Lie algebra cohomology, nilmanifolds, symplectic structure, complex structure, Lie algebra deformation, pro-nilptent Lie algebra, Virasoro algebra, Klein-Gordon equation, minimal model of the rational homotopy type.
UDC: 515.179, 512.81, 512.818.4, 512.664.3, 512.664.8

Subject:

Algebraic Topology, deformations of Lie algebras, Lie algebra cohomology, infinite-dimensional Lie algebra growth, invariant geometrical structures on smooth manifolds, Lie algebra of vector fields, hyperbolic PDE

Biography

In 1981 he entered the Department of Mechanics and Mathematics of Lomonosov Moscow State University and in 1986 graduated with honors. In 1994, under the leadership of Academician S.P. Novikova defended his thesis on the topic “Sullivans minimal models and smooth differential forms of manifolds”. Since 1989, he has been working at the Chair of Higher Geometry and Topology of the Department of Mechanics and Mathematics of Lomonosov Moscow State University.

The main scientific results:

Narrow in the sense of Zelmanov and Shalev naturally graded Lie algebras of width 3/2 were classified. The problem is solved both in the finite-dimensional and infinite-dimensional cases for the field of reals and for the field of complex numbers.

It is proved that the commutants of the characteristic Lie algebras of the sine-Gordon and Tzitzeika equations are narrow naturally graded Lie algebras of width 3/2 and they are isomorphic to the positive parts of affine Lie algebras of rank two. Thus, both characteristic Lie algebras have slow linear growth.

It is proved that for a compact solvmanifold with a completely solvable Lie group G, the Morse – Novikov cohomology of the solvmanifold G /,, i.e. the cohomology of a de Rham complex with a deformed differential d + w, where w is a closed 1-form, is nontrivial if and only if the cohomology class [w] belongs to some finite subset in the one-dimensional cohomology of the solvmanifold G / Γ defined in terms of the tangent Lie algebra of the group Lee G.

Explicit formulas are found for all the differentials of the Rocha-Caridi-Wallach-Feigin-Fuchs resolution. These formulas are given in terms of the special vectors of Verma modules over the Witt algebra. Explicit formulas are found for an important series of singular vectors. Explicit formulas for the differentials of such a resolvent allow one to calculate the cohomology of the positive part of the Witt algebra with coefficients in its arbitrary module.

Buchstabers hypothesis is proved that cohomology with trivial coefficients of the algebra of vector fields on the line (the positive part of the Witt algebra) is generated by iterated nontrivial higher Massey products of two one-dimensional cohomology classes.

The cohomology of two infinite-dimensional Vergne algebras were calculated. These two algebras are from the Zelmanov-Shalev-Fialovsky classification list of narrow positively graded Lie algebras of maximal class.

In the framework of the theory of filtered deformations of positively graded Lie algebras over a field of characteristic zero: 1) equations are found explicitly defining the variety of Lie algebras of the maximum class (the variety of filiform Lie algebras); 2) it is shown that there exists a bijection between the moduli deformation moduli space of the finite-dimensional factor algebra of the positive part of the Witt algebra and the weighted projective space of dimension four.

Author and organizer of the exhibitions "Geometric configurations" and "The diverse world of geometry" at the 6th and 7th Science Festivals at Moscow State University M.V. Lomonosov (central platform). As part of the team of authors, he prepared a "Collection of Problems in Analytical Geometry and Linear Algebra (edited by Yu. M. Smirnov). Moscow, Publishing House of Physics and Mathematics. 2000, 2nd Edition, Publishing House of Moscow State University, 2005; 3rd ed., ICMMO, 2016.

As a visiting professor and researcher, he held temporary positions at the universities of Strasbourg, Lyon, Nantes, Montpellier, Marseille. The author of about 30 scientific articles.

Member of the Academic Council of the Faculty of Mechanics and Mathematics of Moscow State University M.V. Lomonosov. Since 2009 D.V. Millionshchikov is the deputy head of the Department of Higher Geometry and Topology, Faculty of Mechanics and Mathematics, Moscow State University. M.V. Lomonosov.

   
Main publications:
  1. D.V. Millionshchikov, “Lie algebras of slow growth and Klein-Gordon equation”, Algebras and Representation Theory, 21:5 (2018), 1037-1069  crossref
  2. D. V. Millionschikov, “Kogomologii razreshimykh algebr Li i solvmnogoobrazii”, Matem. zametki, 77:1 (2005), 67-79  mathnet  crossref
  3. D. V. Millionschikov, “Algebra formalnykh vektornykh polei na pryamoi i gipoteza Bukhshtabera”, Funkts. analiz i ego pril., 43:4 (2009), 26-44  mathnet  crossref
  4. Fialowski A., Millionshchikov D.V., “Cohomology of graded Lie algebras of maximal class”, Journal of Algebra, 296:1, 157-176  crossref  mathscinet
  5. Millionschikov D.V., “Graded filiform Lie algebras and symplectic nilmanifolds”, In: Geometry, Topology, and Mathematical Physics: S.P. Novikovs Seminar, 2002-2003,, American Mathematical Society Translations, Series 2, 55 (2004), 259-279  mathscinet

http://www.mathnet.ru/eng/person11172
https://scholar.google.com/citations?user=Ee2IbRMAAAA&hl=en
https://zbmath.org/authors/?q=ai:millionshchikov.dmitry-v
https://mathscinet.ams.org/mathscinet/MRAuthorID/239945
https://elibrary.ru/author_items.asp?authorid=6824
ISTINA http://istina.msu.ru/workers/2728971
http://orcid.org/0000-0002-9149-467X
https://publons.com/researcher/R-6055-2016
http://www.researcherid.com/rid/R-6055-2016
https://www.scopus.com/authid/detail.url?authorId=6506684880
https://www.researchgate.net/profile/Dmitri_Millionschikov
https://arxiv.org/a/0000-0002-9149-467X

Full list of publications:
| scientific publications | by years | by types | by times cited in WoS | by times cited in Scopus | common list |



   2020
1. Chang Xuanhao, Krasnoshchekov Sergey V., Pupyshev Vladimir I., Millionshchikov Dmitry V., “Normal ordering of the su(1, 1) ladder operators for the quasi-number states of the Morse oscillator”, Physics Letters, Section A: General, Atomic and Solid State Physics, 384 (2020), 1–8 (Published online)  crossref  isi  scopus
2. Andrew James Bruce, Katarzyna Grabowska, Dmitry Millionshchikov, Vladimir Salnikov and Alexey Tuzhilin, “Working from Home. 2 Months 8 Months and Still Counting…”, EMS newsletter, 12, December (2020), 65-67 https://www.ems-ph.org/journals/show_abstract.php?issn=1027-488X&vol=12&iss=118&rank=15  crossref

   2019
3. D. V. Millionshchikov, “Naturally graded Lie algebras of slow growth”, Sb. Math., 210:6 (2019), 862–909  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi (cited: 3)  elib  scopus (cited: 5)
4. D. V. Millionshchikov, R. Jimenez, “Geometry of Central Extensions of Nilpotent Lie Algebras”, Proc. Steklov Inst. Math., 305 (2019), 209–231  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  scopus
5. I. A. Dynnikov, D. V. Millionshchikov, Mat. Pros., 23, MCCME, Moscow, 2019, 7–19  mathnet

   2018
6. Dmitry Millionshchikov, “Lie Algebras of Slow Growth and Klein–Gordon PDE”, Algebras and Representation Theory, 21:5 (2018), 1037-1069  mathnet  crossref  isi (cited: 6)  elib  scopus (cited: 6)
7. D. V. Millionshchikov, “Polynomial Lie algebras and growth of their finitely generated Lie subalgebras”, Proc. Steklov Inst. Math., 302 (2018), 298–314  mathnet  crossref  crossref  mathscinet  zmath  isi (cited: 1)  elib  scopus (cited: 1)
8. Dmitry Millionshchikov, “Graded Thread Modules over the Positive Part of the Witt (Virasoro) Algebra”, Recent Developments in Integrable Systems and Related Topics of Mathematical Physics, (Kezenoi-Am, Russia, 2016.), Springer Proceedings in Mathematics & Statistics, 273, eds. Buchstaber, Victor M., Konstantinou-Rizos, Sotiris, Mikhailov, Alexander V., Springer, Berlin, 2018, 154–182  crossref  zmath  elib  scopus
9. D. V. Millionshchikov, Dokl. Math., 98:3 (2018), 626–628  mathnet  crossref  crossref  mathscinet  zmath  isi (cited: 1)  elib  scopus (cited: 1)

   2017
10. D. V. Millionshchikov, “Characteristic Lie algebras of the sinh-Gordon and Tzitzeica equations”, Russian Math. Surveys, 72:6 (2017), 1174–1176  mathnet  crossref  crossref  mathscinet  adsnasa  isi (cited: 4)  elib  scopus (cited: 3)

   2016
11. D. V. Millionshchikov, “Virasoro singular vectors”, Funct. Anal. Appl., 50:3 (2016), 219–224  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus (cited: 1)
12. D. V. Millionshchikov, “Complex structures on nilpotent Lie algebras and descending central series”, Rend. Semin. Mat., Univ. Politec. Torino, 74:1 (2016), 163–182  mathnet  mathscinet  elib  scopus (cited: 1)
13. D. V. Millionschikov, “Yavnye formuly dlya osobykh vektorov modulei Verma nad algebroi Virasoro”, Mezhdunarodnaya molodezhnaya shkola-konferentsiya "Shestaya letnyaya shkola po geometrii i matematicheskoi fizike": sbornik statei i tezisov dokladov. (Krasnovidovo, 24-29 iyunya 2016 g.), eds. B.A. Dubrovin i dr.,, MTsNMO, Moskva, 2016, 43–44 http://www.dubrovinlab.msu.ru/files/Sbornik2016.pdf

   2013
14. A. M. Vershik, A. P. Veselov, A. A. Gaifullin, B. A. Dubrovin, A. B. Zhizhchenko, I. M. Krichever, A. A. Mal'tsev, D. V. Millionshchikov, S. P. Novikov, T. E. Panov, A. G. Sergeev, I. A. Taimanov, “Viktor Matveevich Buchstaber (on his 70th birthday)”, Russian Math. Surveys, 68:3 (2013), 581–590  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib

   2009
15. D. V. Millionshchikov, “Algebra of Formal Vector Fields on the Line and Buchstaber's Conjecture”, Funct. Anal. Appl., 43:4 (2009), 264–278  mathnet  crossref  crossref  mathscinet  zmath  isi (cited: 5)  elib
16. D. V. Millionshchikov, “The Variety of Lie Algebras of Maximal Class”, Proc. Steklov Inst. Math., 266 (2009), 177–194  mathnet  crossref  mathscinet  zmath  isi (cited: 4)  elib  elib  scopus (cited: 5)

   2008
17. D. V. Millionshchikov, “Cohomology of Graded Lie Algebras of Maximal Class with Coefficients in the Adjoint Representation”, Proc. Steklov Inst. Math., 263 (2008), 99–111  mathnet  crossref  mathscinet  zmath  isi (cited: 1)  elib  elib  scopus (cited: 1)

   2007
18. D. V. Millionschikov, “Multi-valued functionals, one-forms and deformed de Rham complex”, Topology in Molecular Biology, Biological and Medical Physics, Biomedical Engineering, eds. M.I.Monastyrsky (ed.), Springer, Berlin, Heidelberg, 2007, 189–207 https://link.springer.com/chapter/10.1007  crossref  adsnasa

   2006
19. D. V. Millionshchikov, “Deformations of Filiform Lie Algebras and Symplectic Structures”, Proc. Steklov Inst. Math., 252 (2006), 182–204  mathnet  crossref  mathscinet  elib (cited: 2)  scopus (cited: 3)
20. Alice Fialowski, Dmitri Millionschikov,, “Cohomology of graded Lie algebras of maximal class”, Journal of Algebra, 296:1 (2006), 157–176 , arXiv: math/0412325  crossref  mathscinet  zmath  adsnasa  elib  scopus (cited: 15)
21. D. Millionschikov, “Massey products in graded Lie algebra cohomology”, Proceedings of the Conference “Contemporary Geometry and related topics” (Belgrade, June 26–July 2, 2005), eds. Neda Bokan and al., Faculty of Mathematics, Belgrad, 2006, 353–377 https://www.emis.de/proceedings/CGRT2005/Articles/cgrt23.pdf  mathscinet  zmath

   2005
22. D. V. Millionshchikov, “Cohomology of solvable lie algebras and solvmanifolds”, Math. Notes, 77:1 (2005), 61–71  mathnet  crossref  crossref  mathscinet  zmath  isi (cited: 9)  elib  elib  scopus (cited: 5)

   2004
23. D. V. Millionshchikov, A. Fialowski, “Cohomology of certain $\mathbb N$-graded Lie algebras”, Russian Math. Surveys, 59:6 (2004), 1210–1211  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  scopus
24. Dmitri V. Millionschikov, “Graded filiform Lie algebras and symplectic nilmanifolds”, Geometry, Topology, and Mathematical Physics, Amer. Math. Soc. Transl.- Series 2., 212, eds. V. M. Buchstaber, I. M. Krichever editors, AMS, Providence RI, 2004, 259–279 http://www.ams.org/books/trans2/212/}  crossref  mathscinet  zmath

   2003
25. D. V. Millionshchikov, “Deformations of graded Lie algebras and symplectic structures”, Russian Math. Surveys, 58:6 (2003), 1206–1207  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib (cited: 1)  scopus (cited: 1)

   2002
26. D. V. Millionshchikov, “Cohomology of solvmanifolds with local coefficients and problems of the Morse–Novikov theory”, Russian Math. Surveys, 57:4 (2002), 813–814  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi (cited: 7)  elib (cited: 2)  scopus (cited: 7)
27. D. V. Millionshchikov, “Filiform $\mathbb N$-graded Lie algebras”, Russian Math. Surveys, 57:2 (2002), 422–424  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi (cited: 4)  scopus (cited: 6)

   2001
28. D. V. Millionshchikov, “Cohomology of nilmanifolds and Goncharova's theorem”, Russian Math. Surveys, 56:4 (2001), 758–759  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  scopus (cited: 3)
29. Dmitri V. Millionschikov,, “Cohomology of Nilmanifolds and Gontcharovas Theorem”, Global differential geometry: the mathematical legacy of Alfred Gray. Proceedings of the international congress on differential geometry held in memory of Professor Alfred Gray. (Bilbao, Spain, September 18-23, 2000), Contemporary Mathematics, 288, eds. Marisa Fernandez and Joseph A. Wolf,, American Mathematical Society (AMS), Providence, RI, 2001, 381–385 https://www.ams.org/books/conm/288/}  crossref  mathscinet  zmath

   1997
30. D. V. Millionschikov, “Krichever-Novikov Algebras and the Cohomologies of Algebra of Meromorphic Vector Fields”, "Solitons, Geometry, and Topology: On the Crossroad, Amer. Math. Soc. Transl.- Series 2, 179, eds. V.M.Buchstaber, S.P.Novikov editors, AMS, Providence RI, 1997, 101–108 http://www.ams.org/books/trans2/179  crossref  mathscinet

   1992
31. D. V. Millionschikov, “Algebry Krichevera-Novikova i dvukhmernye kotsikly”, Izbrannye voprosy algebry, geometrii i diskretnoi matematiki. Pod red. O.B. Lupanova i A.I. Kostrikina., ISBN 5-211-02901-1, Izd-vo MGU, Moskva, 1992, 86-91

   1990
32. D. V. Millionshchikov, “Spectral sequences in analytic homotopy theory”, Math. Notes, 47:5 (1990), 458–464  mathnet  crossref  mathscinet  zmath  isi (cited: 2)  elib

   1988
33. D. V. Millionshchikov, “Embeddings of a minimal model of $k$-homotopy type in the algebra of smooth forms $\Lambda^*(M)$”, Russian Math. Surveys, 43:2 (1988), 179–180  mathnet  crossref  mathscinet  zmath  adsnasa  isi (cited: 1)  elib

   1986
34. D. V. Millionshchikov, “Equivariant topology of nematic liquid crystals”, Russian Math. Surveys, 41:5 (1986), 157–158  mathnet  crossref  mathscinet  zmath  adsnasa  isi  elib

Presentations in Math-Net.Ru
1. Открытие — к 70-летию со дня рождения Б.А. Дубровина
D. V. Millionshchikov
The 7th Summer School on Geometrical Methods in Mathematical Physics
July 7, 2020 09:50   
2. The affine variety of nilpotent Lie groups
D. V. Millionshchikov
Algebraic topology and its applications. Postnikov memorial seminar
April 28, 2020 16:45
3. Narrow Lie algebras: classification and applications
D. V. Millionshchikov
St. Petersburg Seminar on Representation Theory and Dynamical Systems
April 22, 2020 17:00
4. Представления нильпотентных алгебр Ли и произведения Масси
D. V. Millionshchikov
VIII Workshop and Conference on Lie Algebras, Algebraic Groups, and Invariant Theory
January 29, 2020 15:50
5. Инвариантные комплексные структуры на нильмногообразиях
D. V. Millionshchikov
International conference "8th Russian-Armenian Workshop on Mathematical Physics, Complex Analysis and Related Topics"
September 20, 2019 14:00   
6. On geometry of central extensions of nilpotent Lie algebras
D. V. Millionshchikov
Algebraic topology and its applications. Postnikov memorial seminar
April 16, 2019 16:45
7. Когомологии бесконечномерных положительно градуированных алгебр Ли и их приложения
D. V. Millionshchikov
Seminar of the Department of Geometry and Topology "Geometry, Topology and Mathematical Physics", Steklov Mathematical Institute of RAS
March 27, 2019 18:30
8. Naturally graded Lie algebras and geometric structures
D. V. Millionshchikov
Complex analysis and mathematical physics
March 5, 2019 16:00
9. Положительно градуированные алгебры Ли в геометрии, топологии и математической физике
D. V. Millionshchikov
December readings in Tomsk
December 14, 2018 15:10   
10. Рост алгебр Ли и интегрируемость гиперболических УРЧП
D. V. Millionshchikov
Research Seminar of the Department of Higher Algebra MSU
October 15, 2018 16:45
11. Polynomial Lie algebras and growth rates of their subalgebras
Dmitry V. Millionshchikov
International conference "Algebraic Topology, Combinatorics, and Mathematical Physics" on occasion of Victor Buchstaber's 75th birthday
May 25, 2018 15:50   
12. Алгебры Ли-Рейнхарта, полиномиальные алгебры Ли и рост их подалгебр Ли
D. V. Millionshchikov
Seminar of the Department of Geometry and Topology "Geometry, Topology and Mathematical Physics", Steklov Mathematical Institute of RAS
March 21, 2018 18:30
13. Characteristic Lie algebras of the Klein–Gordon PDE
D. Millionshchikov
Dynamics in Siberia - 2018
February 27, 2018 14:30
14. Positively graded Lie algebras of slow growth
D. V. Millionshchikov
Lie groups and invariant theory
September 27, 2017 16:45
15. Алгебры Ли медленного роста и уравнение Клейна-Гордона
D. V. Millionshchikov
Seminar of the Department of Geometry and Topology "Geometry, Topology and Mathematical Physics", Steklov Mathematical Institute of RAS
September 13, 2017 18:30
16. Естественно градуированные про-нильпотентные алгебры Ли ширины два
D. V. Millionshchikov
VI Workshop and Conference on Lie Algebras, Algebraic Groups, and Invariant Theory
February 3, 2017 15:50
17. Алгебра формальных векторных полей на прямой и гипотеза Бухштабера
D. V. Millionshchikov
Seminar of the Department of Geometry and Topology "Geometry, Topology and Mathematical Physics", Steklov Mathematical Institute of RAS
December 16, 2009 18:30
18. Алгебра формальных векторных полей на прямой и гипотеза Бухштабера
D. V. Millionshchikov
Lie groups and invariant theory
October 8, 2008
19. Произведения Масси в когомологиях градуированных алгебр Ли
D. V. Millionshchikov
Seminar of the Department of Geometry and Topology "Geometry, Topology and Mathematical Physics", Steklov Mathematical Institute of RAS
April 26, 2006

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