graded Lie algebra,
Lie algebra cohomology,
Lie algebra deformation,
pro-nilptent Lie algebra,
minimal model of the rational homotopy type.
515.179, 512.81, 512.818.4, 512.664.3, 512.664.8
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
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.
D.V. Millionshchikov, “Lie algebras of slow growth and Klein-Gordon equation”, Algebras and Representation Theory, 21:5 (2018), 1037-1069
D. V. Millionschikov, “Kogomologii razreshimykh algebr Li i solvmnogoobrazii”, Matem. zametki, 77:1 (2005), 67-79
D. V. Millionschikov, “Algebra formalnykh vektornykh polei na pryamoi i gipoteza Bukhshtabera”, Funkts. analiz i ego pril., 43:4 (2009), 26-44
Fialowski A., Millionshchikov D.V., “Cohomology of graded Lie algebras of maximal class”, Journal of Algebra, 296:1, 157-176
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
D. V. Millionshchikov, “Naturally graded Lie algebras of slow growth”, Sb. Math., 210:6 (2019), 862–909 (cited: 1) (cited: 1)
D. V. Millionshchikov, R. Jimenez, “Geometry of Central Extensions of Nilpotent Lie Algebras”, Proc. Steklov Inst. Math., 305 (2019), 209–231
I. A. Dynnikov, D. V. Millionshchikov, Mat. Pros., 23, MCCME, Moscow, 2019, 7–19
Dmitry Millionshchikov, “Lie Algebras of Slow Growth and Klein–Gordon PDE”, Algebras and Representation Theory, 21:5 (2018), 1037-1069 (cited: 2) (cited: 2)
D. V. Millionshchikov, “Polynomial Lie algebras and growth of their finitely generated Lie subalgebras”, Proc. Steklov Inst. Math., 302 (2018), 298–314
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
D. V. Millionshchikov, Dokl. Math., 98:3 (2018), 626–628 (cited: 1) (cited: 1)
D. V. Millionshchikov, “Characteristic Lie algebras of the sinh-Gordon and Tzitzeica equations”, Russian Math. Surveys, 72:6 (2017), 1174–1176 (cited: 3) (cited: 3)
D. V. Millionshchikov, “Virasoro singular vectors”, Funct. Anal. Appl., 50:3 (2016), 219–224 (cited: 1)
D. V. Millionshchikov, “Complex structures on nilpotent Lie algebras and descending central series”, Rend. Semin. Mat., Univ. Politec. Torino, 74:1 (2016), 163–182 (cited: 1)
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
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
D. V. Millionshchikov, “Algebra of Formal Vector Fields on the Line and Buchstaber's Conjecture”, Funct. Anal. Appl., 43:4 (2009), 264–278 (cited: 3)
D. V. Millionshchikov, “The Variety of Lie Algebras of Maximal Class”, Proc. Steklov Inst. Math., 266 (2009), 177–194 (cited: 3) (cited: 3)
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 (cited: 1) (cited: 1)
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
D. V. Millionshchikov, “Deformations of Filiform Lie Algebras and Symplectic Structures”, Proc. Steklov Inst. Math., 252 (2006), 182–204 (cited: 2) (cited: 3)
Fialowski, A., Millionschikov, D.V.,, “Cohomology of graded Lie algebras of maximal class”, Journal of Algebra, 296:1 (2006), 157–176 , arXiv: math/0412325
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
D. V. Millionshchikov, “Cohomology of solvable lie algebras and solvmanifolds”, Math. Notes, 77:1 (2005), 61–71 (cited: 8) (cited: 5)
D. V. Millionshchikov, A. Fialowski, “Cohomology of certain $\mathbb N$-graded Lie algebras”, Russian Math. Surveys, 59:6 (2004), 1210–1211
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/}
D. V. Millionshchikov, “Deformations of graded Lie algebras and symplectic structures”, Russian Math. Surveys, 58:6 (2003), 1206–1207 (cited: 1) (cited: 1)
D. V. Millionshchikov, “Cohomology of solvmanifolds with local coefficients and problems of the Morse–Novikov theory”, Russian Math. Surveys, 57:4 (2002), 813–814 (cited: 6) (cited: 2) (cited: 6)
D. V. Millionshchikov, “Filiform $\mathbb N$-graded Lie algebras”, Russian Math. Surveys, 57:2 (2002), 422–424 (cited: 4) (cited: 6)
D. V. Millionshchikov, “Cohomology of nilmanifolds and Goncharova's theorem”, Russian Math. Surveys, 56:4 (2001), 758–759 (cited: 3)
Dmitri V. Millionschikov,, “Cohomology of Nilmanifolds and Gontcharova's 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), Contemp. Math., 288, eds. Marisa Fernández and Joseph A. Wolf, Editors, American Mathematical Society (AMS), Providence, RI, 2001, 381–385 https://www.ams.org/books/conm/288/}
D. V. Millionschikov, “Krichever-NovikovAlgebras 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
D. V. Millionschikov, “Algebry Krichevera-Novikova i dvukhmernye kotsikly”, Izbrannye voprosy algebry, geometrii i diskretnoi matematiki, MGU, Moskva, 1992, 86-91
D. V. Millionshchikov, “Spectral sequences in analytic homotopy theory”, Math. Notes, 47:5 (1990), 458–464 (cited: 2)
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 (cited: 1)
D. V. Millionshchikov, “Equivariant topology of nematic liquid crystals”, Russian Math. Surveys, 41:5 (1986), 157–158