Celebratio Mathematica

A. M. Vinogradov, B. Delone, and D. Fuks: “Rational approximations to irrational numbers with bounded partial quotients,” Dokl. Akad. Nauk SSSR (N.S.) 118 : 5 (1958), pp. 862–865. MR 101227 Zbl 0080.26502 article

Boris Nikolaevich Delone	Related
Alexandre Mikhailovich Vinogradov	Related

D. B. Fuks and A. S. Švarc: “Cyclic products of a polyhedron and the imbedding problem,” Dokl. Akad. Nauk SSSR 125 (1959), pp. 285–288. MR 106467 Zbl 0087.38301 article

D. B. Fuks: “On homotopy duality,” Dokl. Akad. Nauk SSSR 141 : 4 (1961), pp. 818–821. An English translation was published in Sov. Math., Dokl. 2 (1961). MR 137115 article

D. B. Fuks: “On duality in homotopy theory,” Sov. Math., Dokl. 2 (1961), pp. 1575–1578. English translation of Russian original published in Dokl. Akad. Nauk SSSR 141:4 (1961). Zbl 0113.17002 article

D. B. Fuks and A. S. Švarc: “K gomotopicheskoy teorii funktorov v kategorii topologicheskix prostranstv” [On the homotopy theory of functors in the category of topological spaces], Dokl. Akad. Nauk SSSR 143 : 3 (1962), pp. 543–546. An English translation was published in Sov. Math., Dokl. 3 (1962). MR 137116 article

D. B. Fuks and A. S. Shvarts: “On the homotopy theory of functors in the category of topological spaces,” Sov. Math., Dokl. 3 (1962), pp. 444–447. English translation of Russian original published in Dokl. Akad. Nauk SSSR 143:3 (1962). Zbl 0126.18504 article

D. B. Fuks: “Natural mappings of functors in a category of topological spaces,” Mat. Sb. (N.S.) 62(104) : 2 (1963), pp. 160–179. An English translation was published in Fifteen papers on algebra (1963). MR 165478 Zbl 0149.20003 article

D. B. Fuks: “Natural mappings of functors in the category of topological spaces,” pp. 267–287 in Fifteen papers on algebra. Edited by N. I. Ahiezer, S. D. Berman, K. K. Billevic, I. V. Bogacenko, and S. P. Demuskin. American Mathematical Society Translations. Series 2 50. American Mathematical Society (Providence, RI), 1963. English translation of Russian original published in Mat. Sb. 62(104):2 (1963). Zbl 0178.26103 incollection

Samuïl Davidovich Berman	Related
Naum Ilyich Akhiezer	Related
K. K. Billevic	Related
I. V. Bogacenko	Related
S. P. Demushkin	Related

D. B. Fuks: “Eilenberg–MacLane complexes,” Uspehi Mat. Nauk 21 : 5(131) (1966), pp. 213–215. MR 206949 article

D. B. Fuks: “Spectral sequences of fiberings,” Uspehi Mat. Nauk 21 : 5(131) (1966), pp. 149–180. An English translation was published in Russ. Math. Surv. 21:5 (1966). MR 208600 article

D. B. Fuks: “Eckmann–Hilton duality and theory of functors in the category of topological spaces,” Uspehi Mat. Nauk 21 : 2(128) (1966), pp. 3–40. An English translation was published in Russ. Math. Surv. 21:2 (1966). MR 216498 article

D. B. Fuks: “Eckmann–Hilton duality and the theory of functors in the category of topological spaces,” Russ. Math. Surv. 21 : 2 (1966), pp. 1–33. English translation of Russian original published in Uspehi Mat. Nauk 21:2(128) (1966). Zbl 0153.53203 article

D. B. Fuks: “Eilenberg–MacLane complexes,” Russ. Math. Surv. 21 : 5 (1966), pp. 205–207. English translation of Russian original published in Uspehi Mat. Nauk 21:5(131) (1966). Zbl 0168.21001 article

D. B. Fuks: “The spectral sequence of a fibering,” Russ. Math. Surv. 21 : 5 (1966), pp. 141–171. English translation of Russian original published in Uspehi Mat. Nauk 21:5(131) (1966). Zbl 0169.54701 article

D. B. Fuks: “Duality of functors in the category of homotopy types,” Dokl. Akad. Nauk SSSR 175 : 6 (1967), pp. 1232–1235. An English translation was published in Sov. Math., Dokl. 8 (1967). MR 220282 article

D. B. Fuks: “Some remarks on the duality of functors in the category of abelian groups,” Dokl. Akad. Nauk SSSR 176 : 2 (1967), pp. 273–276. MR 220802 Zbl 0201.02402 article

I. M. Gel’fand and D. B. Fuks: “Cohomology of Lie groups with real coefficients,” Dokl. Akad. Nauk SSSR 176 : 1 (1967), pp. 24–27. An English translation was published in Sov. Math., Dokl. 8 (1967). MR 226664 article

I. M. Gel’fand and D. B. Fuks: “Topological invariants of noncompact Lie groups connected with infinite dimensional representations,” Dokl. Akad. Nauk SSSR 177 : 4 (1967), pp. 763–766. An English translation was published in Sov. Math., Dokl. 8 (1967). MR 226665 article

I. M. Gel’fand and D. B. Fuks: “The topology of noncompact Lie groups,” Funkcional. Anal. i Priložen. 1 : 4 (October 1967), pp. 33–45. An English translation was published in Funct. Anal. Appl. 1:4 (1967). MR 226666 article

D. Fuks, A. Fomenko, and V. Gutenmaher: Gomotopicheskaya topologiya [Homotopic topology]. Izdatel’stvo Moskovskogo Universiteta (Moscow), 1967. This and part 2 were combined into a single volume in 1969. MR 254845 book

Anatoly Timofeevich Fomenko	Related
Victor L. Gutenmacher	Related

D. B. Fuks: “Duality of functors in the category of homotopy types,” Sov. Math., Dokl. 8 (1967), pp. 1007–1010. English translation of Russian original published in Dokl. Akad. Nauk SSSR 175:6 (1967). Zbl 0153.53204 article

I. M. Gel’fand and D. B. Fuks: “Topology of noncompact Lie groups,” Funct. Anal. Appl. 1 : 4 (October 1967), pp. 285–295. English translation of Russian original published in Funkcional. Anal. i Priložen. 1:4 (1967). Zbl 0169.54702 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of Lie groups with real coefficients,” Sov. Math., Dokl. 8 (1967), pp. 1031–1034. English translation of Russian original published in Dokl. Akad. Nauk SSSR 176:1 (1967). Zbl 0169.54801 article

I. M. Gel’fand and D. B. Fuks: “Topological invariants of noncompact Lie groups related to infinite- dimensional representations,” Sov. Math., Dokl. 8 (1967), pp. 1483–1486. English translation of Russian original published in Dokl. Akad. Nauk SSSR 177:4 (1967). Zbl 0169.54802 article

D. B. Fuks: “The Maslov–Arnol’d characteristic classes,” Dokl. Akad. Nauk SSSR 178 : 2 (1968), pp. 303–306. An English translation was published in Sov. Math., Dokl. 9 (1968). MR 225340 article

I. M. Gel’fand and D. B. Fuks: “Classifying spaces for principal bundles with Hausdorff bases,” Dokl. Akad. Nauk SSSR 181 : 3 (1968), pp. 515–518. An English translation was published in Sov. Math., Dokl. 9:3 (1968). MR 232391 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of the Lie algebra of vector fields on the circle,” Funkcional. Anal. i Priložen. 2 : 4 (1968), pp. 92–93. An English translation was published in Funct. Anal. Appl. 2:4 (1968). MR 245035 article

D. B. Fuks: “Maslov–Arnol’d characteristic classes,” Sov. Math., Dokl. 9 (1968), pp. 96–99. English translation of Russian original published in Dokl. Akad. Nauk SSSR 178:2 (1968). Zbl 0175.20304 article

I. M. Gel’fand and D. B. Fuks: “The cohomologies of the Lie algebra of the vector fields in a circle,” Funct. Anal. Appl. 2 : 4 (October 1968), pp. 342–343. English translation of Russian original published in Funkts. Anal. Prilozh. 2:4 (1968). Zbl 0176.11501 article

I. M. Gel’fand and D. B. Fuks: “On classifying spaces for principal fiberings with Hausdorff bases,” Sov. Math., Dokl. 9 : 3 (1968), pp. 851–854. English translation of Russian original published in Dokl. Akad. Nauk SSSR 181:3 (1968). Zbl 0181.26602 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of the Lie algebra of vector fields on a manifold,” Funkcional. Anal. i Priložen. 3 : 2 (1969), pp. 87. An English translation was published in Funct. Anal. Appl. 3:2 (1969). MR 245036 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of the Lie algebra of tangent vector fields of a smooth manifold,” Funkcional. Anal. i Priložen. 3 : 3 (1969), pp. 32–52. An English translation was published in Funct. Anal. Appl. 3:3 (1969). MR 256411 article

D. B. Fuks, A. T. Fomenko, and V. L. Gutenmaher: Gomotopicheskaya topologiya [Homotopic topology]. Izdatel’stvo Moskovskogo Universiteta (Moscow), 1969. This combines Part 1 (1967) and Part 2 (1968). English translations were published in 1986 and 2016. Zbl 0189.54001 book

Anatoly Timofeevich Fomenko	Related
Victor L. Gutenmacher	Related

I. M. Gel’fand and D. B. Fuks: “Cohomologies of Lie algebras of vector fields on a manifold,” Funct. Anal. Appl. 3 : 2 (April 1969), pp. 155. English translation of Russian original published in Funkts. Anal. Prilozh. 3:2 (1969). article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of Lie algebra of tangential vector fields of a smooth manifold,” Funct. Anal. Appl. 3 : 3 (July 1969), pp. 194–210. English translation of Russian original published in Funkcional. Anal. i Priložen. 3:3 (1969). Zbl 0216.20301 article

D. B. Fuchs: “The arithmetic of binomial coefficients,” Kvant 1970 : 6 (1970), pp. 17–25. An English translation appeared in Kvant selecta: Algebra and analysis, I (1999). article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of the Lie algebra of formal vector fields,” Izv. Akad. Nauk SSSR Ser. Mat. 34 : 2 (1970), pp. 322–337. MR 266195 article

D. B. Fuks: “Cohomology of the braid group \( \operatorname{mod} 2 \),” Funkcional. Anal. i Priložen. 4 : 2 (1970), pp. 62–73. An English translation was published (with a slightly different title) in Funct. Anal. Appl. 4:2 (1970). MR 274463 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of the Lie algebra of smooth vector fields,” Dokl. Akad. Nauk SSSR 190 : 6 (1970), pp. 1267–1270. An English translation was published in Sov. Math., Dokl. 11 (1970). MR 285023 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of the Lie algebra of tangent vector fields of a smooth manifold, II,” Funkcional. Anal. i Priložen. 4 : 2 (1970), pp. 23–31. An English translation was published in Funct. Anal. Appl. 4:2 (1970). MR 285024 article

I. M. Gel’fand and D. B. Fuks: “Upper bounds for the cohomology of infinite-dimensional Lie algebras,” Funkcional. Anal. i Priložen. 4 : 4 (1970), pp. 70–71. An English translation was published in Funct. Anal. Appl. 4:4 (1970). MR 287589 article

I. M. Gel’fand and D. B. Fuks: “Cycles that represent cohomology classes of the Lie algebra of formal vector fields,” Usp. Mat. Nauk 25 : 5(155) (1970), pp. 239–240. MR 293660 Zbl 0216.20401 article

I. M. Gel’fand and D. B. Fuks: “Cohomolgy of Lie algebras of vector fields with nontrivial coefficients,” Funkcional. Anal. i Priložen. 4 : 3 (1970), pp. 10–25. An English translation was published in Funct. Anal. Appl. 4:3 (1970). MR 298703 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of Lie algebra of tangential vector fields, II,” Funct. Anal. Appl. 4 : 2 (April 1970), pp. 110–116. English translation of Russian original published in Funkts. Anal. Prilozh. 4:2 (1970). Zbl 0208.51401 article

This paper is a continuation of an earlier paper by us which we shall cite as [1969]. We recall that in [1969] we studied the cohomologies of the Lie algebra \( \mathfrak{U}(M) \) of smooth tangential vector fields of a smooth, compact orientable manifold \( M \) with coefficients in a trivial real representation. The main result of [1969] was a theorem about the finite dimensionality of these cohomologies (in every dimension). In the course of the proof we introduced in the standard complex \[ \mathscr{C}(M)= \{C^q(M),d^q\} \] of the Lie algebra \( \mathfrak{U}(M) \) a subcomplex \[ \mathscr{C}_1(M) = \{C_1^q(M),d^q\} ,\] designated “diagonal” by us, constructed from the spectral sequence \[ \mathscr{E} = \{E_r^{u,v}, \delta_r^{u,v}\} = E_r^{u,v} \to E^{u+r,v-r+1} ,\] which converged to the homologies of the diagonal complex, and an expression for its first (second) term was derived.

The present work consists of two parts. In the first a new interpretation of the second term of the spectral sequence \( \mathscr{E} \) is given which permits, in particular, proof of the triviality of some of its differentials. In the second part the relation between the passage to the limit of the spectral sequence \( \mathscr{E} \) (i.e., between the homologies of the diagonal complex) and the cohomologies of the algebra \( \mathfrak{U}(M) \) is discussed. The most complete information is obtained for the case when the spectral sequence \( \mathscr{E} \) is trivial (i.e., when \( E_2 = E_{\infty} \)). Making use of the results of both the parts we obtain, for some of the manifolds, in particular for toruses of any dimension and for all orientable two-dimensional manifolds, a description of the ring, adjoined (with respect to the filtrations introduced in 1.2 of [1969]) to the ring of cohomotogies of the Lie algebra of tangential vector fields. In the present work we shall follow the notation introduced in [1969] and will not repeat the definitions given there. For ease in reading we shall mention at the appropriate places the number of section of [1969] containing the necessary explanations.

The present work was preceded by a short note [1970].

D. B. Fuks: “Cohomologies of the group COS \( \mathrm{mod} 2 \),” Funct. Anal. Appl. 4 : 2 (April 1970), pp. 143–151. English translation (with slightly different title) of Russian original published in Funkcional. Anal. i Priložen. 4:2 (1970). Zbl 0222.57031 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of Lie algebra of vector fields with nontrivial coefficients,” Funct. Anal. Appl. 4 : 3 (July 1970), pp. 181–192. English translation of Russian original published in Funkcional. Anal. i Priložen. 4:3 (1970). Zbl 0222.58001 article

In this paper we continue the investigation which we began in [1969, 1970a, 1970b] of the cohomologies of the Lie algebra of formal as well as smooth vector fields (on a smooth manifold). Specifically we study the cohomologies of the algebras with coefficients in various representations, mainly with coefficients in the spaces of exterior differential forms, formal or smooth, respectively. The results obtained in these papers, involving cohomologies with coefficients in spaces of forms of degree 0 (i.e., in the spaces of smooth functions or of formal power series), were also contained in the work of M. V. Losik [1970]. It is true that theorems about the algebra of formal vector fields were not singled out, but were essentially contained in them (see [Losik 1970, §2]). We shall not rely here on the results of M. V. Losik since we will prove them again. Our proof in the corresponding part is not, in principle, different from the proof of M. V. Losik, although it is considerably shorter. The investigation of cohomologies with coefficients in forms of degree greater than zero encounters a series of new difficulties which are overcome with the application of the results of [1970b].

I. M. Gel’fand and D. B. Fuks: “Upper bounds for cohomology of infinite-dimensional Lie algebras,” Funct. Anal. Appl. 4 : 4 (October 1970), pp. 323–324. English translation of Russian original published in Funkcional. Anal. i Priložen. 4:4 (1970). Zbl 0224.18013 article

In [1969] we studied cohomology with real coefficients of Lie algebras of smooth vector fields on smooth manifolds. Under certain conditions we proved the finite-dimensionality of this cohomology. One of the intermediate assertions was the assertion of the finite-dimensionality of the complete cohomology space of a Lie algebra of formal vector fields (see [1969, Proposition 6.2]; the treatment of this result via formal vector fields is obtained in §1 of [Gel’fand and Fuks 1970]). The purpose of the present note is the formalization and generalization of the method we used to prove this assertion. With the aid of similar methods, as remarked by B. I. Rosenfield, it may be shown that the cohomology space of the Lie algebra of formal tangent vector fields is finite-dimensional. At the same time, for algebras, for example, this method does not succeed in obtaining a proof of the same finite-dimensionality for Hamiltonian formal vector fields, although a certain reduction of the standard cochain complexes is obtained. We do not repeat here the method in question: it is not necessary for understanding what follows. The reader can observe for himself the direct connection between the material set forth below and the proof of Proposition 6.2 of [1969].

I. M. Gel’fand and D. B. Fuks: “On cohomologies of the Lie algebra of smooth vector fields,” Sov. Math., Dokl. 11 (1970), pp. 268–271. English translation of Russian original published in Dokl. Akad. Nauk SSSR 190:6 (1970). Zbl 0264.17005 article

D. B. Fuchs: “On best approximations, I,” Kvant 1971 : 6 (1971), pp. 1–7. An English translation appeared in Kvant selecta: Algebra and analysis, I (1999). article

D. B. Fuchs: “On best approximations, II,” Kvant 1971 : 11 (1971), pp. 8–15. An English translation appeared in Kvant selecta: Algebra and analysis, I (1999). article

D. B. Fuks: “Gomotopicheskaya topologiya” [Homotopic topology], pp. 71–122 in Algebra. Topologija. Geometrija. 1969 [Algebra. Topology. Geometry. 1969]. Edited by R. V. Gamkrelidze. Algebra. Geometriya. Topologiya. VINITI (Moscow), 1971. An English translation was published in J. Sov. Math. 1:3 (1973). Zbl 0215.24301 incollection

I. M. Gel’fand and D. B. Fuks: “Cohomology of the Lie algebra of formal vector fields,” Math. USSR, Izv. 4 : 2 (April 1971), pp. 327–342. English translation of Russian original published in Izv. Akad. Nauk SSSR Ser. Mat. 34:2. Zbl 0216.20302 article

I. M. Gel’fand, D. A. Kazhdan, and D. B. Fuks: “The actions of infinite-dimensional Lie algebras,” Funkcional. Anal. i Priložen. 6 : 1 (1972), pp. 10–15. An English translation was published in Funct. Anal. Appl. 6:1 (1972). MR 301767 article

David A. Kazhdan	Related
Israïl Moiseevich Gelfand	Related

I. M. Gel’fand, D. I. Kalinin, and D. B. Fuks: “Cohomology of the Lie algebra of Hamiltonian formal vector fields,” Funkcional. Anal. i Priložen. 6 : 3 (1972), pp. 25–29. An English translation was published in Funct. Anal. Appl. 6:3 (1973). MR 312531 article

D. I. Kalinin	Related
Israïl Moiseevich Gelfand	Related

I. M. Gel’fand, D. A. Kazhdan, and D. B. Fuks: “The actions of infinite-dimensional Lie algebras,” Funct. Anal. Appl. 6 : 1 (January 1972), pp. 9–13. English translation of Russian original published in Funkcional. Anal. i Priložen. 6:1 (1972). Zbl 0267.18023 article

David A. Kazhdan	Related
Israïl Moiseevich Gelfand	Related

D. B. Fuchs: “Rational approximations and transcendence,” Kvant 1973 : 12 (1973), pp. 9–11. An English translation appeared in Kvant selecta: Algebra and analysis, I (1999). article

I. M. Gel’fand and D. B. Fuks: “PL-foliations,” Funkcional. Anal. i Priložen. 7 : 4 (1973), pp. 29–37. An English translation was published in Funct. Anal. Appl. 7:4 (1974). MR 339195 article

D. B. Fuks: “Characteristic classes of foliations,” Usp. Mat. Nauk 28 : 2(170) (1973), pp. 3–17. An English translation was published in Russ. Math. Surv. 28:2 (1973). MR 415635 Zbl 0272.57012 article

I. M. Gel’fand, D. I. Kalinin, and D. B. Fuks: “Cohomology of the Lie algebra of Hamiltonian formal vector fields,” Funct. Anal. Appl. 6 : 3 (July 1973), pp. 193–196. Zbl 0259.57023 article

As noted in [Gel’fand and Fuks 1970a], computing the cohomology of the Lie algebra of Hamiltonian formal vector fields is an essentially more difficult problem than, say, computing the cohomology of the Lie algebra of all formed vector fields, which was done in [Gel’fand and Fuks 1970b]. The methods used in [1970b] allow one to find the homology of a certain direct summand of the cochain complex of the algebra of Hamiltonian fields without great difficulty, but they yield no information about the complementary summand. In order to test the hypothesis that this complementary summand is acyclic, we have made some computations on an electronic computer. As a result, the above hypothesis has been rejected: we have discovered new and nontrivial cohomology classes of the algebra of Hamiltonian formal vector fields in \( \mathbb{R}^2 \). The important difference between these classes and the cohomology classes of the algebra of all formal vector fields found in [1970b] is that the former cannot be represented by cocycles depending only on the 2-sets of their arguments (see [Gel’fand and Fuks 1970c]).

This partial result seems interesting to us for two reasons. The first is methodological: it turns out that the difficulties encountered in computing the cohomology of the algebra of Hamiltonian fields are fundamental in origin. The second is as follows. The construction of [Godbillon and Vey 1971] and [Bernshtein, and Rozenfel’d 1972] can be carried over to the Hamiltonian case, making it possible to construct, for each cohomology class of the algebra of Hamiltonian formal vector fields, a characteristic class of Hamiltonian fibers of corresponding codimension. In particular, the classes which we will point out here furnish characteristic classes of Hamiltonian fibers of codimension 2. It would be interesting to determine whether or not these characteristic classes are nontrivial.

D. I. Kalinin	Related
Israïl Moiseevich Gelfand	Related

D. B. Fuks: “Characteristic classes of foliations,” Russ. Math. Surv. 28 : 2 (April 1973), pp. 1–16. English translation of Russian original published in Usp. Mat. Nauk 28:2(170). Zbl 0274.57005 article

D. B. Fuks: “Homotopic topology,” J. Sov. Math. 1 : 3 (1973), pp. 333–362. English translation of Russian original published in Algebra. Topologija. Geometrija. 1969 (1971). Zbl 0286.55006 article

D. B. Fuks: “Quillenization and bordisms,” Funkcional. Anal. i Priložen. 8 : 1 (1974), pp. 36–42. An English translation was published in Funct. Anal. Appl. 8:1 (1974). MR 343301 article

I. M. Gel’fand, B. L. Feĭgin, and D. B. Fuks: “Cohomology of the Lie algebra of formal vector fields with coefficients in its dual space and variations of characteristic classes of foliations,” Funkcional. Anal. i Priložen. 8 : 2 (1974), pp. 13–29. An English translation was published in Funct. Anal. Appl. 8:2 (1974). MR 356082 article

Boris L. Feigin	Related
Israïl Moiseevich Gelfand	Related

S. G. Gindikin, A. A. Kirillov, and D. B. Fuks: “The work of I. M. Gel’fand on functional analysis, algebra, and topology,” Usp. Mat. Nauk 29 : 1(175) (1974), pp. 195–223. An English translation was published in Russ. Math. Surv. 29:1 (1974). MR 386977 Zbl 0288.01022 article

Simon Grigorevich Gindikin	Related
Alexandre Aleksandrovich Kirillov	Related
Israïl Moiseevich Gelfand	Related

I. M. Gel’fand and D. B. Fuks: “PL-foliations, II,” Funkcional. Anal. i Priložen. 8 : 3 (1974), pp. 7–11. An English translation was published in Funct. Anal. Appl. 8:3 (1974). MR 418115 article

S. G. Gindikin, A. A. Kirillov, and D. B. Fuks: “The work of I. M. Gel’fand on functional analysis, algebra, and topology,” Russ. Math. Surv. 29 : 1 (February 1974), pp. 5–35. English translation of Russian original published in Usp. Mat. Nauk 29:1(175) (1974). Zbl 0294.01031 article

This survey is timed to coincide with the sixtieth birthday of I. M. Gel’fand. The authors have confined themselves to those branches of mathematics in which he has been engaged during the last decade, and in the various branches the chronology of the articles covered by the survey is different.

Gel’fand’s research in the theory of group representations, which has lasted for thirty years, falls into several cycles; the majority of his results are widely known and were dealt with in the survey [Vishik et al. 1964] on the occasion of his fiftieth birthday. For this reason we deal here only with the results of the last ten years.

His first articles on integral geometry appeared more than ten years ago, but this branch of mathematics is still in a formative phase. Because of this we include in the survey an outline of Gel’fand’s basic research in integral geometry, not excluding some that is comparatively old.

Topology is a new branch of his scientific activity; all the work in this field was carried out between 1968 and 1973.

Simon Grigorevich Gindikin	Related
Alexandre Aleksandrovich Kirillov	Related
Israïl Moiseevich Gelfand	Related

I. M. Gel’fand and D. B. Fuks: “PL-foliations,” Funct. Anal. Appl. 7 : 4 (October 1974), pp. 278–284. English translation of Russian original published in Funkcional. Anal. i Priložen. 7:4 (1973). Zbl 0294.57016 article

The present study grew out of an attempt to comprehend the local geometric nature of some recently discovered characteristic classes of foliations. The analogy between smooth and piecewise-linear (PL) topologies suggests that such characteristic classes can be readily constructed for foliations on combinatorial manifolds reducible in every simplex to a family of parallel planes. Such foliations (which we call affine PL foliations) are defined in §1, subsection 2; they have a relatively uncomplicated classifying space (subsections 6 and 9) and are canonically smoothed (subsections 14 and 15). It turns out, however, that even the first of the characteristic classes, namely the Godbillon–Vey class, is identically equal to zero on these foliations (subsection 16). To render it nontrivial we extend the class of PL foliations by allowing the leaves in the simplexes to be planes parallel in the projective sense (“projective PL foliations;” see subsection 3). On these foliations the Godbillon–Vey class is no longer trivial (subsection 17).

Like the affine, PL foliations have a straightforward classifying space (subsections 7 and 10).

The present article is limited to foliations of codimension 1, but some of the results, including all those of §§1 and 2, are easily translated to arbitrary codimensions.

I. M. Gel’fand, B. L. Feigin, and D. B. Fuks: “Cohomologies of the Lie algebra of formal vector fields with coefficients in its adjoint space and variations of characteristic classes of foliations,” Funct. Anal. Appl. 8 : 2 (April 1974), pp. 99–112. English translation of Russian original published in Funkcional. Anal. i Priložen. 8:2 (1974). Zbl 0298.57011 article

Boris L. Feigin	Related
Israïl Moiseevich Gelfand	Related

I. M. Gel’fand and D. B. Fuks: “PL-foliations, II,” Funct. Anal. Appl. 8 : 3 (July 1974), pp. 197–200. English translation of Russian original published in Funkcional. Anal. i Priložen. 8:3 (1974). Zbl 0316.57010 article

In the first part of this work [1974] we defined the PL-analog of Haefliger structures (foliations with singularities) of codimension 1. We now investigate PL-foliations of any codimension. The fundamental results pertain to the corresponding classifying spaces, their homologies, and the relationship with the Haefliger classifying spaces \( B\Gamma^q \).

Our constructions refer to the case of oriented foliations. The theory can be readily extended, however, to nonoriented foliations and to foliations with additional transversal structures: nondivergent, simplectic, etc.

This part of the study can for the most part be read independently of [1974], although the latter contains the geometrical background of the formal constructions that follow.

D. B. Fuks: “Quillenization and bordisms,” Funct. Anal. Appl. 8 : 1 (January 1974), pp. 31–36. English translation of Russian original published in Funkcional. Anal. i Priložen. 8:1 (1974). Zbl 0324.57024 article

D. B. Fuks: “Finite-dimensional Lie algebras of formal vector fields and characteristic classes of homogeneous foliations,” Math. USSR, Izv. 10 : 1 (February 1976), pp. 55–62. English translation of Russian original published in Izv. Akad. Nauk SSSR, Ser. Mat. 40:1 (1976). MR 413125 Zbl 0348.57008 article

In [1970], I. M. Gel’fand and the author computed the cohomology of the Lie algebra \( W_n \) of formal vector fields in \( n \)-dimensional space. The present article is devoted to the study of homomorphisms \[ H^*(W_n;\mathbf{R})\to H^*(\mathfrak{g},\mathbf{R}) \] induced by imbeddings of finite-dimensional subalgebras in \( W_n \). We show that there exist elements of \( H^*(W_n;\mathbf{R}) \) which are annihilated by any such homomorphism. On the other hand, we show that the image of the cohomology homomorphism induced by the well-known embedding \[ \mathfrak{gl}(n{+}1,\mathbf{R})\to W_n \] has dimension \( 2^{n-1} + 1 \). The results are applied to characteristic classes of foliations.

D. B. Fuchs, A. M. Gabrielov, and I. M. Gel’fand: “The Gauss–Bonnet theorem and the Atiyah–Patodi–Singer functionals for the characteristic classes of foliations,” Topology 15 : 2 (1976), pp. 165–188. MR 431199 Zbl 0347.57009 article

We prove in this article some formulae of Gauss–Bonnet kind for the characteristic classes of foliations (see [Bernstein and Rosenfeld 1972; Bott and Haefliger 1972]). Namely, fixing a Riemannian metric on a manifold with a foliation allows one to determine explicitly for each of these characteristic classes a representing differential form (see, for example, [Bott 1972]). If a domain \( X \) with a piecewise smooth boundary transversal to the foliation is given in the manifold, then the integral of such a form over \( X \), corrected by adding the integrals over faces of different dimensions of certain forms depending on the foliation and the metric near \( \partial X \), depends only on the induced metric on \( \partial X \) (and, of course, on the foliation on \( X \)). By using these formulae one can, in particular, extend the definition of the characteristic classes of foliations to the piecewise smooth case.

Andrei M. Gabrielov	Related
Israïl Moiseevich Gelfand	Related

D. B. Fuks: “Finite-dimensional Lie algebras of formal vector fields and characteristic classes of homogeneous foliations,” Izv. Akad. Nauk SSSR, Ser. Mat. 40 : 1 (1976), pp. 57–64. An English translation was published in Math. USSR, Izv. 10:1 (1976). Zbl 0336.57016 article

D. Fuchs: “Non-trivialite des classes caractéristiques de \( \mathfrak{g} \)-structures: Applications aux variations des classes caractéristiques de feuilletages” [Nontriviality of characteristic classes of \( \mathfrak{g} \)-structures: Applications to variations of characteristic classes of foliations], C. R. Acad. Sci., Paris, Sér. A 284 (1977), pp. 1105–1107. Another part of this (with identical title) was published in the same volume of C. R. Acad. Sci., Paris, Sér. A. MR 433470 Zbl 0348.57007 article

D. Fuchs: “Non-trivialite des classes caractéristiques de \( \mathfrak{g} \)-structures: Application aux classes caractéristiques de feuilletages” [Nontriviality of characteristic classes of \( \mathfrak{g} \)-structures: Applications to variations of characteristic classes of foliations], C. R. Acad. Sci., Paris, Sér. A 284 (1977), pp. 1017–1019. Another part of this (with identical title) was published in the same volume of C. R. Acad. Sci., Paris, Sér. A. Zbl 0348.57006 article

V. A. Rokhlin and D. B. Fuks: Nachal’nyj kurs topologii: Geometricheskie glavy [Beginner’s course in topology: Geometric chapters]. Nauka (Moscow), 1977. A French translation was published in 1981, an English translation in 1984. Zbl 0417.55002 book

D. B. Fuks: “Cohomology of infinite-dimensional Lie algebras and characteristic classes of foliations,” pp. 179–285 in Seriya sovremennye problemy matematiki [Current problems in mathematics]. Seriya sovremennye problemy matematiki 10. VINITI (Moscow), 1978. An English translation was published in J. Sov. Math. 11:6 (1978). MR 513337 incollection

I. M. Gel’fand, B. L. Feigin, and D. B. Fuks: “Cohomologies of infinite dimensional Lie algebras and Laplace operators,” Funkts. Anal. Prilozh. 12 : 4 (1978), pp. 1–5. An English translation was published in Funct. Anal. Appl. 12:4 (1979). MR 515625 Zbl 0396.17008 article

Boris L. Feigin	Related
Israïl Moiseevich Gelfand	Related

D. B. Fuks: “Cohomology of infinite-dimensional Lie algebras and characteristic classes of foliations,” J. Sov. Math. 11 : 6 (1978), pp. 922–980. English translation of Russian original published in Seriya sovremennye problemy matematiki 10 (1978). Zbl 0499.57001 article

B. L. Feĭgin and D. B. Fuks: “Invariant differential operators on the line,” Funkts. Anal. Prilozh. 13 : 4 (1979), pp. 91–92. An English translation was published in Funct. Anal. Appl. 13:4 (1980). MR 554429 Zbl 0425.58024 article

I. M. Gel’fand, B. L. Feigin, and D. B. Fuks: “Cohomology of infinite-dimensional Lie algebras and Laplace operators,” Funct. Anal. Appl. 12 : 4 (1979), pp. 243–247. English translation of Russian original published in Funkts. Anal. Prilozh. 12:4 (1978). Zbl 0404.17008 article

Although appreciable progress has been made in the last 10 years in calculating the cohomology of infinite-dimensional Lie algebras, some of the problems of this circle appear to be unapproachable to this day; this relates, in the first place, to the algebras of Hamiltonian and divergence-free vector fields and to the algebras \( L_1(n) \subset W_n \). In the solutions of these problems there has been hardly any successful progress without the introduction of new methods.

Such methods could consist of the investigation of the Laplace operators induced by any metrics in the cochain complex. The impression is added that if the metric is introduced in a reasonable way, then the eigenvalues and eigenvectors of the Laplace operator, and hence also the cohomology, will turn out to be calculable. In the following papers we propose to investigate this possibility systematically. In this paper we will analyze one example in which the program indicated can be realized successfully all the way through. It is true that the question concerns a Lie algebra whose cohomology is known: the algebra \( L_1(1) \) of formal vector fields on the line having trivial 1-jet. In what follows we will denote this algebra by \( L_1 \) and whenever the question is of cohomology, we mean continuous cohomology with trivial coefficients.

The cohomology of the algebra \( L_1 \), as well as of the algebras \( L_k(1) \) with \( k > 1 \), was found in [Goncharova 1973]. Goncharova’s calculation is awkward and does not allow one to find the cohomology of the algebra \( L_1 \) without finding the cohomology of the other algebras \( L_k \)).

Our paper significantly, as it seems to us, clarifies Goncharova’s theorem and contains some new results. In particular, we give an explicit description of cocycles representing the cohomology classes of the algebra \( L_1 \) (we also apply this method of describing cocycles to the algebras \( L_k \)).

We recall that in the cohomology theory of infinite-dimensional Lie algebras, the cohomology of the algebra \( L_1 \) has special significance; the reason is explained in Goncharova’s paper. One can add that recently Bukhshtaber and Shokurov discovered a connection between these cohomologies and the Adams–Novikov spectral sequence in complex cobordism theory [Bukhshtaber and Shokurov 1978].

Our paper owes much to Goncharova’s paper: a whole series of our arguments is implicitly contained in it. It remains to indicate also the connection or in any case the analogy between what is presented below and the theory of Kats–Muda [1974].

Boris L. Feigin	Related
Israïl Moiseevich Gelfand	Related

B. L. Feigin and D. B. Fuks: “Invariant differential operators on the line,” Funct. Anal. Appl. 13 : 4 (1979), pp. 314–315. Translation of Russian original published in Funkts. Anal. Prilozh. 13:4 (1979). Zbl 0434.58021 article

B. L. Feĭgin and D. B. Fuks: “Homology of the Lie algebra of vector fields on the line,” Funkts. Anal. Prilozh. 14 : 3 (1980), pp. 45–60. An English translation was published in Funct. Anal. Appl. 14:3 (1980). MR 583800 Zbl 0482.57010 article

B. L. Feĭgin and D. B. Fuks: “Homology of the Lie algebra of vector fields on the line,” Funct. Anal. Appl. 14 : 3 (July 1980), pp. 201–212. English translation of Russian original published in Funkts. Anal. Prilozh. 14:3 (1980). Zbl 0487.57011 article

D. B. Fuchs: “On the removal of parentheses, on Euler, Gauss, and MacDonald, and on missed opportunities,” Kvant 1981 : 8 (1981), pp. 12–20. An English translations of this appeared in Kvant selecta: Algebra and analysis, II (1999) and A mathematical omnibus (2007). article

Leonhard Euler	Related
Carl Friedrich Gauss	Related
Ian McDonald	Related

D. B. Fuks: “Foliations,” Itogi Nauki Tekh., Ser. Algebra Topologiya Geom. 18 : 2 (1981), pp. 151–213. An English translation was published in J. Sov. Math. 18:2 (1981). MR 616646 article

V. A. Rokhlin and D. B. Fuks: Premier cours de topologie: Chapitres geométriques [Beginner’s course in topology: Geometric chapters]. Mir (Moscow), 1981. French translation of 1977 Russian original. Zbl 0453.55001 book

D. B. Fuks: “Foliations,” J. Sov. Math. 18 : 2 (January 1981), pp. 255–291. English translation of Russian original published in Itogi Nauki Tekh., Ser. Algebra Topologiya Geom. 18:2 (1981). Zbl 0479.57014 article

B. L. Feĭgin and D. B. Fuks: “Invariant skew-symmetric differential operators on the line and Verma modules over the Virasoro algebra,” Funct. Anal. Appl. 16 : 2 (1982), pp. 114–126. English translation of Russian original published in Funkts. Anal. Prilozh. 16:2 (1982). MR 659165 Zbl 0505.58031 article

B. L. Feĭgin and D. B. Fuks: “Casimir operators in modules over the Virasoro algebra,” Dokl. Akad. Nauk SSSR 269 : 5 (1983), pp. 1057–1060. An English translation was published in Sov. Math., Dokl. 27 (1983). MR 701159 article

B. L. Feigin and D. B. Fuks: “Verma modules over the Virasoro algebra,” Funkts. Anal. Prilozh. 17 : 3 (July 1983), pp. 91–92. An English translation was published in Funct. Anal. Appl. 17:3 (1983). MR 714236 Zbl 0526.17010 article

D. B. Fuks: “Stable cohomology of Lie algebra of formal vector fields with tensor coefficients,” Funkts. Anal. Prilozh. 17 : 4 (1983), pp. 62–69. An English translation was published in Funct. Anal. Appl. 17:4 (1983). MR 725416 article

B. L. Feigin and D. B. Fuks: “Verma modules over the Virasoro algebra,” Funct. Anal. Appl. 17 : 3 (1983), pp. 241–242. English translation of Russian original published in Funkts. Anal. Prilozh. 17:3 (1983). Zbl 0529.17010 article

B. L. Feigin and D. B. Fuks: “Casimir operators in modules over the Virasoro algebra,” Sov. Math., Dokl. 27 (1983), pp. 465–469. English translation of Russian original published in Dokl. Akad. Nauk SSSR 269:5 (1983). Zbl 0538.17007 article

D. B. Fuks: “Stable cohomologies of Lie algebras of formal vector fields with tensor coefficients,” Funct. Anal. Appl. 17 : 4 (1983), pp. 295–301. English translation of Russian original published in Funkts. Anal. Prilozh. 17:4 (1983). Zbl 0552.17009 article

B. L. Feigin and D. B. Fuks: “Stable cohomology of the algebra \( W_n \) and relations of the algebra \( L_1 \),” Funktsional. Anal. i Prilozhen. 18 : 3 (1984), pp. 94–95. An English translation was published in Funct. Anal. Appl. 18:3 (1984). MR 757265 article

D. B. Fuks and V. A. Rokhlin: Beginner’s course in topology: Geometric chapters. Universitext. Springer (Berlin), 1984. English translation of 1977 Russian original. MR 759162 Zbl 0562.54003 book

B. L. Feĭgin and D. B. Fuchs: “Verma modules over the Virasoro algebra,” pp. 230–245 in Topology: General and algebraic topology, and applications (Leningrad, 23–27 August 1982). Edited by L. D. Faddeev and A. A. Mal’tsev. Lecture Notes in Mathematics 1060. Springer (Berlin), 1984. MR 770243 Zbl 0549.17010 incollection

Boris L. Feigin	Related
Anatoliĭ Ivanovic Maltsev	Related
Ludvig Dmitrievich Faddeev	Related

B. L. Feigin and D. B. Fuks: “Stable cohomology of the algebra \( W_n \) and relations of the algebra \( L_1 \),” Funct. Anal. Appl. 18 : 3 (July 1984), pp. 264–266. English translation of Russian original published in Funktsional. Anal. i Prilozhen. 18:3 (1984). Zbl 0559.17008 article

Let \( W_n \) be the Lie algebra of formal vector fields in \( C^n \) and \( L_1 = L_1(n) \) be the subalgebra of this algebra composed of vector fields with trivial 1-jet. These algebras are graded, due to which their cohomology with coefficients in graded modules is also graded. We denote these gradings in cohomology by subscripts in parentheses. When we speak of cohomology we always mean continuous cohomology.

In [Fuks 1983] the stable cohomology of the Lie algebra \( L_1(n) \) with trivial coefficients is calculated, i.e., \[ H^r_{(m)}(L_1(n);C) ,\] where \( n \) is sufficiently large compared with \( r \) and \( m \); some consequences of this calculation are given there. The origin of the restrictions on \( n \) in [1983] is the fact that the basis tensor invariants in \( C^n \) are linearly independent only if \( n \) is sufficiently large compared with the rank of the tensors considered; precisely what \( n \) should be is indicated by the “second fundamental theorem of invariant theory” (cf. [Weyl 1939]). The main content of this note is the remark that an entirely simple addition to the theorem indicated from the theory of invariants (possibly not new, although we have not come across it in the literature) allows us to extend essentially the range of those \( n \) for which the results of [1983] are valid. This improvement of the results of [1983] leads to a rather unexpected corollary on a defining system of generators and relations for the algebra \( L_1(n) \).

D. B. Fuks and D. A. Leites: “Cohomology of Lie superalgebras,” C. R. Acad. Bulg. Sci. 37 : 12 (1984), pp. 1595–1596. MR 826386 Zbl 0586.17005 article

B. L. Feigin and D. B. Fuks: “Representations of the Virasoro algebra,” pp. 78–94 in Metody topologii i rimanovoj geometrii v matematicheskoj fizike [Methods of topology and Riemannian geometry in mathematical physics] (Druskininkai, Lithuania, 23–27 May 1983). Edited by A. Matuzyavichyus. Ministerstvo Vysshego i Srednego Spetsial’nogo Obrazovaniya Litovskoj SSR (Vilnius), 1984. This was later developed in a 1986 Stockholm University research report and then further in Representation of Lie groups and related topics (1990). Zbl 0642.17007 incollection

Boris L. Feigin	Related
Yuri Vladimirovich Matiyasevich	Related

D. B. Fuks: “Finite-dimensionality of the homology of the Lie algebra of Hamiltonian vector fields on the plane,” Funktsional. Anal. i Prilozhen. 19 : 4 (1985), pp. 68–73. An English translation was published in Funct. Anal. Appl. 19:4 (1985). MR 820086 article

D. B. Fuks: “Finite dimensionality of homologies of the Lie algebra of Hamiltonian vector fields on the plane,” Funct. Anal. Appl. 19 : 4 (October 1985), pp. 305–310. English translation of Russian original published in Funktsional. Anal. i Prilozhen. 19:4 (1985). Zbl 0599.17008 article

B. L. Feigin and D. B. Fuchs: Representations of the Virasoro algebra. Research report 25, Department of Mathematics, Stockholm University, 1986. A version of this was published in Representation of Lie groups and related topics (1990). techreport

D. B. Fuks: Kogomologii beskonechnomernykh algebr Li [Cohomology of infinite-dimensional Lie algebras]. Nauka (Moscow), 1986. An English translation was published in 1986. MR 772201 Zbl 0592.17011 book

F. G. Malikov, B. L. Feigin, and D. B. Fuks: “Singular vectors in Verma modules over Kac–Moody algebras,” Funktsional. Anal. i Prilozhen. 20 : 2 (1986), pp. 25–37. English translation of Russian original published in Funct. Anal. Appl. 20:2 (1986). MR 847136 article

Boris L. Feigin	Related
Fyodor G. Malikov	Related

V. I. Arnol’d, A. M. Vershik, O. Ya. Viro, A. N. Kolmogorov, S. P. Novikov, Ya. G. Sinaj, and D. B. Fuks: “Vladimir Abramovich Rokhlin (obituary),” Usp. Mat. Nauk 41 : 3(249) (1986), pp. 159–163. An English translation was published in Russ. Math. Surv. 41:3(1986). MR 854242 Zbl 0604.01010 article

Vladimir Igorevich Arnold	Related
Sergey Petrovich Novikov	Related
Oleg Yanovich Viro	Related
Ya. G. Sinai	Related
Anatoly Moiseevich Vershik	Related
Andrey Nikolaevich Kolmogorov	Related
Vladimir Abramovich Rokhlin	Related

A. T. Fomenko, D. B. Fuks, and V. L. Gutenmacher: Homotopic topology. Akadémiai Kiadó (Budapest), 1986. English translation of 1969 Russian original. A second, expanded edition was published in 2016. MR 873943 Zbl 0615.55001 book

Anatoly Timofeevich Fomenko	Related
Victor L. Gutenmacher	Related

D. B. Fuks: Cohomology of infinite-dimensional Lie algebras. Contemporary Soviet mathematics. Consultants Bureau (New York), 1986. English translation of 1984 Russian original. MR 874337 Zbl 0667.17005 book

D. B. Fuks: “Classical manifolds,” pp. 253–314. Edited by S. P. Novikov, D. B. Fuks, and R. V. Gamkrelidze. Seriya sovremennye problemy matematiki: Fundamental’nye napravleniya 12. VINITI (Moscow), 1986. An English translation appeared in Topology II (2004). MR 895593 Zbl 0671.57001 incollection

Revaz Valerianovich Gamkrelidze	Related
Sergey Petrovich Novikov	Related

V. I. Arnol’d, A. M. Vershik, O. Ya. Viro, A. N. Kolmogorov, S. P. Novikov, Ya. G. Sinaj, and D. B. Fuks: “Vladimir Abramovich Rokhlin (obituary),” Russ. Math. Surv. 41 : 3 (June 1986), pp. 189–195. English translation of Russian original published in Usp. Mat. Nauk 41:3(249) (1986). Zbl 0608.01032 article

Vladimir Igorevich Arnold	Related
Sergey Petrovich Novikov	Related
Oleg Yanovich Viro	Related
Ya. G. Sinai	Related
Anatoly Moiseevich Vershik	Related
Andrey Nikolaevich Kolmogorov	Related
Vladimir Abramovich Rokhlin	Related

F. G. Malikov, B. L. Feigin, and D. B. Fuks: “Singular vectors in Verma modules over Kac–Moody algebras,” Funct. Anal. Appl. 20 : 2 (April 1986), pp. 103–113. English translation of Russian original published in Funktsional. Anal. i Prilozhen. 20:2 (1986). Zbl 0616.17010 article

The basic results of this paper are explicit formulas for singular vectors in reducible Verma modules over Kac–Moody algebras. As is known, Verma modules \( M(\lambda) \) over a Kac–Moody algebra \[ \mathfrak{g} = N_{-} \oplus H \oplus N_{+} \] are parametrized by linear functionals \( \lambda \) on its Cartan subalgebra \( H \). Reducible modules \( M(\lambda) \) correspond to functionals \( \lambda \) lying in the union of a countable number of hyperplanes in \( H^* \); these hyperplanes are enumerated by pairs \( (\alpha,n) \), where \( \alpha \) is a positive root of the algebra \( \mathfrak{g} \) and \( n \) is a positive integer (Kac–Kazhdan theorem [1979]). If \( \lambda \) lies in exactly one such hyperplane and the root to which, this hyperplane corresponds is real, then the module \( M(\lambda) \) contains a unique (up to proportionality) singular vector (in different terminology a zero-vector or vector of highest weight), not proportional to a vacuum vector. We give an explicit formula for this vector which however has an unusual form: to the vacuum vector oneapplies monomials in the elements of the algebra \( N_{-} \) containing these elements to nonintegral and even nonreal powers. We start by explaining in Sec. 2 how to reduce such monomials to traditional form. Then we begin to derive the formulas for singular vectors. In Sec. 3 the case of the algebra \( \mathfrak{sl}(2)\hat{} \), the simplest of the infinite-dimensional Kac–Moody algebras, is analyzed in detail; this section also contains important information about singular vectors in reducible Verma modules over \( \mathfrak{sl}(2)\hat{} \) corresponding to imaginary roots. In Sec. 4 the results of Sec. 3 are extended to the case of an arbitrary Kac–Moody algebra, where in the case of real roots one gets exhaustive info mation. In the concluding Sec. 5 we turn to the consideration of a special case, this time the case of the finite-dimensional Lie algebra \( \mathfrak{sl}(n) \); in this case we are able to reduce the answer to especially convenient form. We note that Verma modules over finite-dimensional simple Lie algebras are studied among other things in the recent paper of Zhelobenko [1986]; this paper also contains some expressions for singular vectors in Verma modules, which differ in form from those we find. The connection between our results and the results of [1986] is still unclear.

Boris L. Feigin	Related
Fyodor G. Malikov	Related

Topologiya–I [Topology–I]. Edited by S. P. Novikov, D. B. Fuks, and R. V. Gamkrelidze. Seriya Sovremennye Problemy Matematiki. Fundamental’nye Napravleniya 12. VINITI (Moscow), 1986. An English translation was published in 1996. MR 895591 Zbl 0639.00031 book

Revaz Valerianovich Gamkrelidze	Related
Sergey Petrovich Novikov	Related

B. L. Feĭgin and D. B. Fuks: “Cohomology of Lie groups and Lie algebras,” pp. 121–209 in Lie groups and Lie algebras–2. Seriya sovremennye problemy matematiki: Fundamental’nye napravleniya 21. VINITI (Moscow), 1988. An English translation appeared in Lie groups and {Lie algebras, {II}} (2000). MR 968446 Zbl 0653.17008 incollection

B. L. Feigin, D. B. Fuks, and V. S. Retakh: “Massey operations in the cohomology of the infinite dimensional Lie algebra \( L_1 \),” pp. 13–31 in Topology and geometry: Rohlin semininar. Edited by O. Ya. Viro. Lecture Notes in Mathematics 1346. Springer (Berlin), 1988. MR 970070 Zbl 0653.17010 incollection

Boris L. Feigin	Related
Vladimir Solomonovich Retakh	Related
Oleg Yanovich Viro	Related

O. Ya. Viro and D. B. Fuks: “Introduction to homotopy theory,” pp. 6–121 in Topologiya–2 [Topology–2]. Seriya sovremennye problemy matematiki: Fundamental’nye napravleniya 24. VINITI (Moscow), 1988. An English translation appeared in Topology II (2004). MR 987942 Zbl 0674.55001 incollection

O. Ya. Viro and D. B. Fuks: “Homology and cohomology,” pp. 123–240 in Topologiya–2 [Topology–2]. Edited by S. P. Novikov, V. A. Rokhlin, and R. V. Gamkrelidze. Seriya sovremennye problemy matematiki: Fundamental’nye napravleniya 24. VINITI (Moscow), 1988. An English translation appeared in Topology II (2004). MR 987943 Zbl 0674.55002 incollection

Revaz Valerianovich Gamkrelidze	Related
Sergey Petrovich Novikov	Related
Oleg Yanovich Viro	Related
Vladimir Abramovich Rokhlin	Related

B. L. Feigin and D. B. Fuks: “Cohomology of some nilpotent subalgebras of the Virasoro and Kac–Moody Lie algebras,” J. Geom. Phys. 5 : 2 (1988), pp. 209–235. Dedicated to I. M. Gelfand on his 75th birthday. MR 1029428 Zbl 0692.17008 article

Boris L. Feigin	Related
Israïl Moiseevich Gelfand	Related

D. Fuchs: “On poetic feet,” Kvant 2 (1988), pp. 17–24. In Russian. article

D. B. Fuks: “Two projections of singular vectors of Verma modules over the affine Lie algebra \( A^1_1 \),” Funkts. Anal. Prilozh. 23 : 2 (1989), pp. 81–83. An English translation was published in Funct. Anal. Appl. 23:2 (1989). MR 1011370 article

A. T. Fomenko and D. B. Fuks: Kurs gomotopicheskoj topologii [A course in homotopic topology]. Nauka (Moscow), 1989. Zbl 0675.55001 book

D. B. Fuks: “Two projections of singular vectors of Verma modules over the affine Lie algebra \( A^1_1 \),” Funct. Anal. Appl. 23 : 2 (April 1989), pp. 154–156. English translation of Russian original published in Funkts. Anal. Prilozh. 23:2 (1989). Zbl 0707.17014 article

The singular vectors of Verma modules are important in the representation theory of Lie algebras in the Bernshtein–Gel’fand–Gel’fand spirit [1971]. In particular, knowledge of these vectors is useful for the computation of homologies. In the article [1986] of Malikov, Feigin, and the author, formulas for the singular vectors in the case of the Kac–Moody algebras are written out, but in these formulas the nonintegral (even nonreal) powers of elements of the algebra are present. The formulas are reduced in [1986] to the normal form only for the finite-dimensional Lie algebra \( \mathfrak{sl}(n) \).

The results of the present note relate to the case of an elementary infinite-dimensional Kac–Moody algebra: the Lie algebra \( A^1_1 = \mathfrak{sl}(2)\hat{} \). Using a formula from [1986], we explicitly indicate the projections of the singular vectors of Verma modules over \( A^1_1 \) into \( U(\mathfrak{sl}(2)) \) and into \( U(H) \), where \( H \) is the Heisenberg algebra and \( U \) denotes the universal enveloping algebra. The results are applied to the computation of homologies.

B. L. Feigin and D. B. Fuchs: “Representations of the Virasoro algebra,” pp. 465–554 in Representation of Lie groups and related topics. Edited by A. M. Vershik and D. P. Zhelobenko. Advanced Studies in Contemporary Mathematics 7. Gordon and Breach (New York), 1990. An earlier version of this appeared as a Stockholm University research report (1986). MR 1104280 Zbl 0722.17020 incollection

Boris L. Feigin	Related
Anatoly Moiseevich Vershik	Related
Dmitry Petrovich Zhelobenko	Related

D. B. Fuchs: “On Soviet mathematics of the 1950s and 1960s,” pp. 213–222 in Golden years of Moscow mathematics. Edited by S. Zdravkovska and P. L. Duren. History of Mathematics 6. American Mathematical Society (Providence, RI), 1993. MR 1246573 incollection

Peter L. Duren	Related
Smilka Zdravkovska	Related

Unconventional Lie algebras. Edited by D. Fuchs. Advances in Soviet Mathematics 17. American Mathematical Society (Providence, RI), 1993. MR 1254723 Zbl 0782.00026 book

A. B. Astashkevich and D. B. Fuchs: “On the cohomology of the Lie superalgebra \( W(m|n) \),” pp. 1–13 in Unconventional Lie algebras. Edited by D. Fuchs. Advances in Soviet Mathematics 17. American Mathematical Society (Providence, RI), 1993. MR 1254724 Zbl 0801.17021 incollection

D. Fuchs: “Singular vectors over the Virasoro algebra and extended Verma modules,” pp. 65–74 in Unconventional Lie algebras. Edited by D. Fuchs. Advances in Soviet Mathematics 17. American Mathematical Society (Providence, RI), 1993. MR 1254726 Zbl 0827.17028 incollection

D. Fuchs and A. Schwarz: “Matrix Vieta theorem,” pp. 15–22 in Lie groups and Lie algebras: E. B. Dynkin’s seminar. Edited by S. G. Gindikin and E. B. Vinberg. American Mathematical Society Translations. Series 2 169. American Mathematical Society (Providence, RI), 1995. This book is also no. 26 in the Advances in Mathematics series. MR 1364450 Zbl 0837.15011 ArXiv math/9410207 incollection

We consider generalizations of the Vieta formula (relating the coefficients of an algebraic equation to the roots) to the case of equations whose coefficients are order-\( k \) matrices.

Specifically, we prove that if \( X_1,\dots \), \( X_n \) are solutions of an algebraic matrix equation \[ X^n + A_1 X^{n-1} + \cdots + A_n = 0 ,\] independent in the sense that they determine the coefficients \( A_1,\dots \), \( A_n \), then the trace of \( A_1 \) is the sum of the traces of the \( X_i \), and the determinant of \( A_n \) is, up to a sign, the product of the determinants of the \( X_i \). We generalize this to arbitrary rings with appropriate structures.

This result is related to and motivated by some constructions in non-commutative geometry.

Evgeniĭ Borisovich Dynkin	Related
Simon Grigorevich Gindikin	Related
Albert S. Schwarz	Related	Volume
Ernest Borisovich Vinberg	Related

D. Fuchs and L. Lang: Massey products and deformations. Preprint, February 1996. ArXiv q-alg/9602024 techreport

The classical deformation theory of Lie algebras involves different kinds of Massey products of cohomology classes. Even the condition of extendibility of an infinitesimal deformation to a formal one-parameter deformation of a Lie algebra involves Massey powers of two dimensional cohomology classes which are not powers in the usual definition of Massey products in the cohomology of a differential graded Lie algebra. In the case of deformations with other local bases, one deals with other, more specific Massey products. In the present work a construction of generalized Massey products is given, depending on an arbitrary graded commutative, associative algebra. In terms of these products, the above condition of extendibility is generalized to deformations with arbitrary local bases. Dually, a construction of generalized Massey products on the cohomology of a differential graded commutative associative algebra depends on a nilpotent graded Lie algebra. For example, the classical Massey products correspond to the Lie algebra of strictly upper triangular matrices, while the matric Massey products correspond to the Lie algebra of block strictly upper triangular matrices.

Topology, I. Edited by S. P. Novikov. Encyclopaedia of Mathematical Sciences 12. Springer (Berlin), 1996. English translation of 1986 Russian original. MR 1392484 Zbl 0830.00014 book

Revaz Valerianovich Gamkrelidze	Related
Sergey Petrovich Novikov	Related

A. Astashkevich and D. Fuchs: “Asymptotics for singular vectors in Verma modules over the Virasoro algebra,” Pac. J. Math. 177 : 2 (1997), pp. 201–209. MR 1444780 Zbl 0898.17012 article

D. Fuchs and S. Tabachnikov: “Invariants of Legendrian and transverse knots in the standard contact space,” Topology 36 : 5 (1997), pp. 1025–1053. MR 1445553 Zbl 0904.57006 article

A. Fialowski and D. Fuchs: “Singular deformations of Lie algebras: Example: Deformations of the Lie algebra \( L_1 \),” pp. 77–92 in Topics in singularity theory: V. I. Arnold’s 60th anniversary collection. Edited by A. Khovanskiĭ, A. Varchenko, and V. Vassiliev. AMS Translations. Series 2 180. American Mathematical Society (Providence, RI), 1997. MR 1767113 Zbl 0883.17023 ArXiv q-alg/9706027 incollection

Vladimir Igorevich Arnold	Related
Victor Anatolyevich Vassiliev	Related
Alice Fialowski	Related
Askold Georgievich Khovanskii	Related
Alexander Nicholaevich Varchenko	Related

A. Fialowski and D. Fuchs: “Construction of miniversal deformations of Lie algebras,” J. Funct. Anal. 161 : 1 (1999), pp. 76–110. MR 1670210 Zbl 0944.17015 ArXiv math/0006117 article

We consider deformations of finite or infinite dimensional Lie algebras over a field of characteristic 0. There is substantial confusion in the literature if one tries to describe all the non-equivalent deformations of a given Lie algebra. It is known that there is in general no “universal” deformation of a Lie algebra \( L \) with a commutative algebra base \( A \) with the property that for any other deformation of \( L \) with base \( B \) there exists a unique homomorphism \( f:A\to B \) that induces an equivalent deformation. Thus one is led to seek a miniversal deformation. For a miniversal deformation such a homomorphism exists, but is unique only at the first level. If we consider deformations with base \( \operatorname{spec} A \), where \( A \) is a local algebra, then under some minor restrictions there exists a miniversal element. In this paper we give a construction of a miniversal deformation.

D. Fuchs and S. Tabachnikov: “More on paperfolding,” Am. Math. Monthly 106 : 1 (January 1999), pp. 27–35. MR 1674137 Zbl 1037.53501 article

D. B. Fuchs and M. B. Fuchs: “The arithmetic of binomial coefficients,” pp. 1–12 in Kvant selecta: Algebra and analysis, I. Edited by S. Tabachnikov. Mathematical World 14. American Mathematical Society (Providence, RI), 1999. English translation of Russian original published in Kvant 1970:6 (1970). MR 1727594 incollection

Every student knows the formulas \begin{align*} (1+x)^2 &= 1 + 2x + x^2, \\ (1+x)^3 &= 1 + 3x + 3x^2 + x^3. \end{align*} The numbers \( (1,\,2,\,1) \), \( (1,\,3,\,3,\,1) \), as well as numbers obtained in an analogous way by raising \( (1+x) \) to the fourth power, the fifth power, and so on, are called binomial coefficients. This article deals with various properties of binomial coefficients. In the first section we lay out the “general theory”: Many of the theorems we prove here used to be part of the school curriculum. In the second section we will show very easy way to find the remainder when a binomial coefficient is divided by a prime number. The third, and concluding, section deals with certain remarkable properties of binomial coefficients. The main assertions in this section are formulated as hypotheses.

Serge Lvovich Tabachnikov	Related
M. B. Fuchs	Related

D. B. Fuchs and M. B. Fuchs: “On best approximations, I,” pp. 27–35 in Kvant selecta: Algebra and analysis, I. Edited by S. Tabachnikov. Mathematical World 14. American Mathematical Society (Providence, RI), 1999. English translation of Russian original published in Kvant 1971:6 (1971). MR 1727597 incollection

How can one find the best rational approximation for an irrational number? For example, which approximation of the number \( \sqrt{2} \) is the best: \( \frac{3}{2} \), \( \frac{7}{5} \) or \( 1.41 \)? The answer seems to be fairly simple: The smaller the error, the better the approximation. But this is not the whole story, since we write \( \pi \approx 3.14 \) even though we might know five or more decimal digits. It stands to reason that when choosing an approximation we want to decrease not only the error but also the denominator: The smaller the denominator, the less awkward the fraction and the easier to manage it (to store, substitute in formulas, etc.).

In this article we examine the problem of how to meet these two contradictory demands, of finding a rational approximation for a given irrational number that is as far as possible simultaneously the most precise and least awkward.

Serge Lvovich Tabachnikov	Related
M. B. Fuchs	Related

D. B. Fuchs and M. B. Fuchs: “On best approximations, II,” pp. 37–47 in Kvant selecta: Algebra and analysis, I. Edited by S. Tabachnikov. Mathematical World 14. American Mathematical Society (Providence, RI), 1999. English translation of Russian original published in Kvant 1971:11 (1971). MR 1727598 incollection

It is inconvenient to work with irrational numbers. Rational numbers are much easier to work with. Hence the problem of rational approximations to irrational numbers and their efficiency is most important. In Part I of our article we compared different approximations to the number \( \alpha \) and proved that for any irrational \( \alpha \) and arbitrarily large \( N \) there exist infinitely many rational approximations \( p/q \) such that \[ q\Bigl|\alpha - \frac{p}{q}\Bigr| < \frac{1}{N} .\] This, second part of the article deals with the subtler question of whether for any \( \alpha \) there exist infinitely many approximations \( p/q \) such that \[ q^2\Bigl|\alpha - \frac{p}{q}\Bigr| \] is less than a given number. It turns out that this number cannot be arbitrary, in accordance with the Hurwitz–Borel theorem, which states that for any \( \alpha \) there exist infinitely many different approximations \( p/q \) such that \[ q^2\Bigl|\alpha - \frac{p}{q}\Bigr| < \frac{1}{\sqrt{5}} .\] Note that the number \( \sqrt{5} \) cannot be replaced by a larger number. Before reading any further, the reader is advised to have another look at the previous part of the article and also to read the article by N. M. Baskin in Kvant 1970, no. 8.

Serge Lvovich Tabachnikov	Related
M. B. Fuchs	Related

D. B. Fuchs and M. B. Fuchs: “Rational approximations and transcendence,” pp. 65–69 in Kvant selecta: Algebra and analysis, I. Edited by S. Tabachnikov. Mathematical World 14. American Mathematical Society (Providence, RI), 1999. English translation of Russian original published in Kvant 1973:12 (1973). MR 1727601 incollection

Serge Lvovich Tabachnikov	Related
M. B. Fuchs	Related

D. B. Fuchs: “On the removal of parentheses, on Euler, Gauss, and MacDonald, and on missed opportunities,” pp. 39–49 in Kvant selecta: Algebra and analysis, II. Edited by S. Tabachnikov. Mathematical World 15. American Mathematical Society (Providence, RI), 1999. English translation of Russian original published in Kvant 1981:8 (1981). A version of this also appeared in A mathematical omnibus (2007). MR 1728803 incollection

Leonhard Euler	Related
Ian McDonald	Related
Carl Friedrich Gauss	Related
Serge Lvovich Tabachnikov	Related

Differential topology, infinite-dimensional Lie algebras, and applications: D. B. Fuchs’ 60th anniversary collection. Edited by A. Astashkevich and S. Tabachnikov. AMS Translations. Series 2 194. American Mathematical Society (Providence, RI), 1999. MR 1729355 Zbl 0921.00044 book

Alexander B. Astashkevich	Related
Serge Lvovich Tabachnikov	Related

Northern California symplectic geometry seminar (Stanford and Berkeley, CA, 1989–1998). Edited by Ya. Eliashberg, D. Fuchs, T. Ratiu, and A. Weinstein. AMS Translations. Series 2 196. American Mathematical Society (Providence, RI), 1999. This is also no. 45 of the series Advances in the Mathematical Sciences (no ISSN). MR 1736209 Zbl 0930.00050 book

Tudor Ștefan Rațiu	Related
Alan Weinstein	Related

B. L. Feigin and D. B. Fuchs: “Cohomologies of Lie groups and Lie algebras,” pp. 125–223 in Lie groups and Lie algebras, II. Edited by E. B. Vinberg. Encyclopaedia of Mathematical Sciences 21. Springer (Berlin), 2000. Translation of Russian original published in Seriya sovremennye problemy matematiki 21 (1988). MR 1756408 Zbl 0931.17014 incollection

Boris L. Feigin	Related
Ernest Borisovich Vinberg	Related

D. Fuchs and L. L. Weldon: “Massey brackets and deformations,” J. Pure Appl. Algebra 156 : 2–3 (2001), pp. 215–229. MR 1808824 Zbl 1054.17018 article

J. Epstein, D. Fuchs, and M. Meyer: “Chekanov–Eliashberg invariants and transverse approximations of Legendrian knots,” Pac. J. Math. 201 : 1 (2001), pp. 89–106. MR 1867893 Zbl 1049.57005 article

Judith Karen Epstein	Related
Maike Meyer	Related

D. B. Fuks and T. D. Èvans: “On the structure of the restricted Lie algebra for the Witt algebra in a finite characteristic,” Funktsional. Anal. i Prilozhen. 36 : 2 (2002), pp. 69–74. An English translation was published in Funct. Anal. Appl. 36:2 (2002). MR 1922020 article

T. J. Evans and D. B. Fuchs: “On the restricted Lie algebra structure of the Witt Lie algebra in finite characteristic,” Funct. Anal. Appl. 36 : 2 (2002), pp. 140–144. English translation of Russian original published in Funktsional. Anal. i Prilozhen. 36:2 (2002). Zbl 1029.17018 ArXiv math/0111271 article

J. Epstein and D. Fuchs: “On the invariants of Legendrian mirror torus links,” pp. 103–115 in Symplectic and contact topology: Interactions and perspectives (Toronto and Montreal, 26 March–7 April 2001). Edited by Y. Eliashberg, B. Khesin, and F. Lalonde. Fields Institute Communications 35. American Mathematical Society (Providence, RI), 2003. MR 1969270 Zbl 1044.57007 incollection

We will compute several invariants of Legendrian mirror torus links in the standard contact space \( \{\mathbb{R}^3 \), \( y\,dx - dz\} \). Our purpose is not to classify Legendrian mirror torus links. A classification of Legendrian torus links, both algebraic and mirror, is contained in a recent paper by Etnyre and Honda [2001] who prove that topologically isotopic algebraic and mirror torus links with equal Thurston–Bennequin and Maslov numbers must be Legendrian isotopic. Nevertheless, the behavior of both classical and non-classical invariants of Legendrian mirror torus links is interesting, becuase it contradicts some natural conjectures and gives rise to new ones. In particular, we note that for \( q \leq p \), the behavior of the invariants of the Legendrian mirror torus link \( K(-p,q) \) depends on the parity of \( q \), where \( q \) is the braid index of \( K(-p,q) \).

François Lalonde	Related
Judith Karen Epstein	Related
Yakov M. Eliashberg	Related
Boris A. Khesin	Related

D. Fuchs: “Chekanov–Eliashberg invariant of Legendrian knots: Existence of augmentations,” J. Geom. Phys. 47 : 1 (July 2003), pp. 43–65. MR 1985483 Zbl 1028.57005 article

O. Ya. Viro and D. B. Fuchs: “Introduction to homotopy theory,” pp. 1–93 in Topology II. Edited by S. P. Novikov and V. A. Rokhlin. Encyclopaedia of Mathematical Sciences 24. Springer (Berlin), 2004. Translated from the Russian by C. J. Shaddock. Translation of Russian original published in Seriya sovremennye problemy matematiki 24 (1988). MR 2054456 incollection

Oleg Yanovich Viro	Related
Sergey Petrovich Novikov	Related
Vladimir Abramovich Rokhlin	Related

O. Ya. Viro and D. B. Fuchs: “Homology and cohomology,” pp. 95–196 in Topology II. Edited by S. P. Novikov and V. A. Rokhlin. Encyclopaedia of Mathematical Sciences 24. Springer (Berlin), 2004. Translation of Russian original published in Seriya sovremennye problemy matematiki 24 (1988). MR 2054457 Zbl 1078.57013 incollection

A complex is a sequence of Abelian groups and homomorphisms of the form \[ \dots \stackrel{\partial_3}{\rightarrow} C_3 \stackrel{\partial_2}{\rightarrow} C_1 \stackrel{\partial_1}{\rightarrow}C_0 \stackrel{\partial_0}{\rightarrow} C_{-1} \rightarrow \dots, \] in which \( \partial_q \circ \partial_{q+1} = 0 \) for each \( q \). A complex is positive if \( C_q = 0 \) for \( q < 0 \), free if all the \( C_q \) are free Abelian groups, and a complex of finite type if the sum \( \bigoplus_q C_q \) is finitely generated. (In dealing with positive complexes, the sequence \( \{C_q \), \( \partial_q\} \) is often cut short, and we speak of the sequence \[ \dots\to C_1 \stackrel{\partial_1}{\rightarrow} C_0 \,.) \] The homomorphisms \( \partial_q \) are called the differentials or boundary operators; elements of \( C_q \) are called the \( q \)-chains of the complex; chains that are annihilated by the differential are called cycles, and cycles whose difference lies in the image of the differential are said to be homologous.

Oleg Yanovich Viro	Related
Sergey Petrovich Novikov	Related
Vladimir Abramovich Rokhlin	Related

D. B. Fuchs: “Classical manifolds,” pp. 197–252 in Topology II. Edited by S. P. Novikov and V. A. Rokhlin. Encyclopaedia of Mathematical Sciences 24. Springer (Berlin), 2004. MR 2054458 incollection

Sergey Petrovich Novikov	Related
Vladimir Abramovich Rokhlin	Related

D. Fuchs and T. Ishkhanov: “Invariants of Legendrian knots and decompositions of front diagrams,” Mosc. Math. J. 4 : 3 (July–September 2004), pp. 707–717. To Borya Feigin, with love. MR 2119145 Zbl 1073.53106 article

Boris L. Feigin	Related
Tigran Ishkhanov	Related

D. Fuchs: Is it possible to cut a cube into pieces and to assemble a tetrahedron? Hilbert’s problem and Dehn’s theorem, 2004–2005. Berkeley Math Circle handout. misc

D. B. Fuchs: “Jewish University,” pp. 183–190 in You failed your math test, comrade Einstein: Aventures and misadventures of young mathematicians or test your skills in almost recreational mathematics. Edited by M. A. Shifman. World Scientific (Singapore), 2005. To the memory of Bella Subbotovskaya. incollection

Mikhail A. Shifman	Related
Bella Abramovna Subbotovskaya	Related

D. Fuchs and E. Fuchs: “Closed geodesics on regular polyhedra,” Mosc. Math. J. 7 : 2 (April–June 2007), pp. 265–279. To Askold Khovanskii, a dear friend and an admired mathematician. MR 2337883 Zbl 1129.53022 article

Ekaterina Fuchs	Related
Askold Georgievich Khovanskii	Related

Mathematical omnibus: Thirty lectures on classic mathematics. Edited by D. Fuchs and S. Tabachnikov. American Mathematical Society (Providence, RI), 2007. A German translation was published in 2011. MR 2350979 Zbl 1318.00004 book

T. J. Evans and D. Fuchs: “A complex for the cohomology of restricted Lie algebras,” J. Fixed Point Theory Appl. 3 : 1 (2008), pp. 159–179. MR 2402915 Zbl 1178.17018 article

D. Fuchs and C. Wilmarth: “Projections of singular vectors of Verma modules over rank 2 Kac–Moody Lie algebras” in Special issue on Kac–Moody algebras and applications, published as SIGMA 4. Issue edited by R. Borcherds, E. Frenkel, V. Kac, R. Moody, J. Patera, A. Pianzola, and P. Ramond. National Academy of Sciences of Ukraine (Kiev), 2008. Article no. 059, 11 pages. MR 2434939 Zbl 1218.17014 ArXiv 0806.1976 incollection

Victor G. Kac	Related
Jiří Patera	Related
Pierre Ramond	Related
Richard Ewen Borcherds	Related
Constance Wilmarth	Related
Robert V. Moody	Related
Edward Vladimir Frenkel	Related
Arturo Pianzola	Related

D. Fuchs: Geodesics on a regular dodecahedron, 2009. Max Planck Institute for Mathematics ebook. misc

D. Fuchs and S. Tabachnikov: “Self-dual polygons and self-dual curves,” Funct. Anal. Other Math. 2 : 2–4 (2009), pp. 203–220. MR 2506116 Zbl 1180.51012 ArXiv 0707.1048 article

D. Fuchs and C. Wilmarth: “Laplacian spectrum for the nilpotent Kac–Moody Lie algebras,” Pac. J. Math. 247 : 2 (2010), pp. 323–334. MR 2734151 Zbl 1255.17011 ArXiv 0808.0890 article

D. Fuchs and D. Rutherford: “Generating families and Legendrian contact homology in the standard contact space,” J. Topol. 4 : 1 (2011), pp. 190–226. MR 2783382 Zbl 1237.57026 ArXiv 0807.4277 article

D. Davis, D. Fuchs, and S. Tabachnikov: “Periodic trajectories in the regular pentagon,” Mosc. Math. J. 11 : 3 (July–September 2011), pp. 439–461. To the memory of V. I. Arnold. MR 2894424 Zbl 1276.37033 ArXiv 1102.1005 article

Vladimir Igorevich Arnold	Related
Diana Davis	Related
Serge Lvovich Tabachnikov	Related

D. B. Fuchs and S. Tabachnikov: Ein Schaubild der Mathematik: 30 Vorlesungen über klassische Mathematik [Mathematical omnibus: Thirty lectures on classic mathematics]. Springer (Berlin), 2011. German translation of 2007 English original. Zbl 1211.00003 book

D. Fuchs, Y. Ilyashenko, B. Khesin, V. Vassiliev, and H. Hofer: “Memories of Vladimir Arnold,” Notices Am. Math. Soc. 59 : 4 (April 2012), pp. 482–502. Coordinating editors were Boris Khesin and Serge Tabchnikov. MR 2951953 Zbl 1284.37002 article

Helmut Hofer	Related
Vladimir Igorevich Arnold	Related
Victor Anatolyevich Vassiliev	Related
Yulij Sergeevich Ilyashenko	Related
Boris A. Khesin	Related
Serge Lvovich Tabachnikov	Related

D. Fuchs: “Evolutes and involutes of spatial curves,” Am. Math. Monthly 120 : 3 (March 2013), pp. 217–231. MR 3030294 Zbl 1273.53004 article

D. Fuchs and S. Tabachnikov: “Periodic trajectories in the regular pentagon, II,” Mosc. Math. J. 13 : 1 (January–March 2013), pp. 19–32. MR 3112214 Zbl 1347.37066 ArXiv 1201.0026 article

D. Fuchs: Cohomology of Lie algebra of Hamiltonian vector fields: Experimental data, conjectures, and theorems, 2014. Slides from Toronto University’s Vladimir Arnold conference. misc

D. Fuchs: “Dima Arnold in my life,” pp. 133–140 in Arnold: Swimming against the tide. Edited by B. A. Khesin and S. L. Tabachnikov. American Mathematical Society (Providence, RI), 2014. incollection

Vladimir Igorevich Arnold	Related
Boris A. Khesin	Related
Serge Lvovich Tabachnikov	Related

D. Fuchs: “Periodic billiard trajectories in regular polygons and closed geodesics on regular polyhedra,” Geom. Dedicata 170 (2014), pp. 319–333. MR 3199491 Zbl 1296.53085 article

S. Mokhammadzade and D. B. Fuks: “Cohomology of the Lie algebra \( \mathfrak{h}_2 \): Experimental results and hypotheses,” Funktsional. Anal. i Prilozhen. 48 : 2 (2014), pp. 67–78. An English translation was published in Funct. Anal. Appl. 48:2 (2014). MR 3288177 article

S. Mohammadzadeh and D. B. Fuchs: “Cohomology of the Lie algebra \( \mathfrak{h}_2 \): Experimental results and conjectures,” Funct. Anal. Appl. 48 : 2 (2014), pp. 128–137. English translation of Russian original published in Funktsional. Anal. i Prilozhen. 48:2 (2014). Zbl 1354.17014 article

A. Fomenko and D. Fuchs: Homotopical topology, 2nd translated edition. Graduate Texts in Mathematics 273. Springer (Cham), 2016. First two chapters written in collaboration with Victor Gutenmacher. Second, expanded, edition of the 1986 English translation of the 1968 Russian original. MR 3497000 Zbl 1346.55001 book

Anatoly Timofeevich Fomenko	Related
Victor L. Gutenmacher	Related

D. Fuchs and S. Tabachnikov: “On Lagrangian tangent sweeps and Lagrangian outer billiards,” Geom. Dedicata 182 (2016), pp. 203–213. MR 3500383 Zbl 1351.37161 ArXiv 1502.03177 article

Given a Lagrangian submanifold in linear symplectic space, its tangent sweep is the union of its (affine) tangent spaces, and its tangent cluster is the result of parallel translating these spaces so that the foot point of each tangent space becomes the origin. This defines a multivalued map from the tangent sweep to the tangent cluster, and we show that this map is a local symplectomorphism (a well known fact, in dimension two). We define and study the outer billiard correspondence associated with a Lagrangian submanifold. Two points are in this correspondence if they belong to the same tangent space and are symmetric with respect to its foot point. We show that this outer billiard correspondence is symplectic and establish the existence of its periodic orbits. This generalizes the well studied outer billiard map in dimension two.

D. Fuchs: “Geodesics on regular polyhedra with endpoints at the vertices,” Arnold Math. J. 2 : 2 (2016), pp. 201–211. MR 3504350 Zbl 1432.53058 article

In a recent work of Davis et al. [2017], the authors consider geodesics on regular polyhedra which begin and end at vertices (and do not touch other vertices). The cases of regular tetrahedra and cubes are considered. The authors prove that (in these cases) a geodesic as above never begins at ends at the same vertex and compute the probabilities with which a geodesic emanating from a given vertex ends at every other vertex. The main observation of the present article is that there exists a close relation between the problem considered in [Davis et al. 2017] and the problem of classification of closed geodesics on regular polyhedra considered in articles [Fuchs and Fuchs 2007; Fuchs 2014]. This approach yields different proofs of result of [Davis et al. 2017] and permits to obtain similar results for regular octahedra and icosahedra (in particular, such a geodesic never ends where it begins).

M. Arnold, D. Fuchs, I. Izmestiev, S. Tabachnikov, and E. Tsukerman: “Iterating evolutes and involutes,” Discrete Comput. Geom. 58 : 1 (2017), pp. 80–143. MR 3658331 Zbl 1381.51007 ArXiv 1510.07742 article

This paper concerns iterations of two classical geometric constructions, the evolutes and involutes of plane curves, and their discretizations: evolutes and involutes of plane polygons. In the continuous case, our main result is that the iterated involutes of closed locally convex curves with rotation number one (possibly, with cusps) converge to their curvature centers (Steiner points), and their limit shapes are hypocycloids, generically, astroids. As a consequence, among such curves only the hypocycloids are homothetic to their evolutes. The bulk of the paper concerns two kinds of discretizations of these constructions: the curves are replaced by polygons, and the evolutes are formed by the circumcenters of the triples of consecutive vertices (\( \mathcal{P} \)-evolutes), or by the incenters of the triples of consecutive sides (\( \mathcal{A} \)-evolutes). For equiangular polygons, the theory is parallel to the continuous case: we define discrete hypocycloids (equiangular polygons whose sides are tangent to hypocycloids) and a discrete Steiner point. The space of polygons is a vector bundle over the space of the side directions; our main result here is that both kinds of evolutes define vector bundle morphisms. In the case of \( \mathcal{P} \)-evolutes, the induced map of the base is 4-periodic, and the dynamics reduces to the linear maps on the fibers. We prove that the spectra of these linear maps are symmetric with respect to the origin. The asymptotic dynamics of linear maps is determined by their eigenvalues with the maximum modulus, and we show that all types of behavior can occur: in particular, hyperbolic, when this eigenvalue is real, and elliptic, when it is complex. We also study \( \mathcal{P} \)- and \( \mathcal{A} \)-involutes and prove that the side directions of iterated \( \mathcal{P} \)-involutes of polygons with odd number of sides behave ergodically; this generalizes well-known results concerning iterations of the construction of the pedal triangle. In addition to the theoretical study, we performed numerous computer experiments; some of the observations remain unexplained.

Maxim D. Arnold	Related
Emmanuel Tsukerman	Related
Serge Lvovich Tabachnikov	Related
Ivan Veniaminovich Izmestiev	Related

A. Chernavski, D. Fuchs, S. Gussein-Zade, Yu. Ilyashenko, O. Karpenkov, A. Kirillov, S. Lando, S. Matveev, M. Skopenkov, V. Tikhomirov, M. Tsfasman, and V. Vassiliev: “Alexei Bronislavovich Sossinsky turns 80,” Mosc. Math. J. 17 : 3 (July–September 2017), pp. 555–558. MR 3711006 Zbl 1427.01015 article

Mikhail B. Skopenkov	Related
Vladimir Mikhailovich Tikhomirov	Related
Alexey Viktorovich Chernavsky	Related
Mikhail Anatolyevich Tsfasman	Related
Sabir Medgidovich Gusein-Zade	Related
Victor Anatolyevich Vassiliev	Related
Alexei Bronislavovich Sossinsky	Related
Sergei Konstantinovich Lando	Related
Oleg Nikolaevich Karpenkov	Related
Yulij Sergeevich Ilyashenko	Related
Sergei Vladimirovich Matveev	Related
Alexandre Aleksandrovich Kirillov	Related

D. Fuchs and S. Tabachnikov: “Iterating evolutes of spacial polygons and of spacial curves,” Mosc. Math. J. 17 : 4 (October–December 2017), pp. 667–689. MR 3734657 Zbl 1422.53005 ArXiv 1611.08836 article

The evolute of a smooth curve in an \( m \)-dimensional Euclidean space is the locus of centers of its osculating spheres, and the evolute of a spatial polygon is the polygon whose consecutive vertices are the centers of the spheres through the consecutive \( (m{+}1) \)-tuples of vertices of the original polygon. We study the iterations of these evolute transformations. This work continues the recent study of similar problems in dimension two. Here is a sampler of our results.

The set of \( n \)-gons with fixed directions of the sides, considered up to parallel translation, is an \( (n{-}m) \)-dimensional vector space, and the second evolute transformation is a linear map of this space. If \( n=m+2 \), then the second evolute is homothetic to the original polygon, and if \( n=m+3 \), then the first and the third evolutes are homothetic. In general, each eigenvalue of the second evolute map has double multiplicity. We also study curves, with cusps, in 3-dimensional Euclidean space and their evolutes. We provide continuous analogs of the results obtained for polygons, and present a class of curves which are homothetic to their second evolutes; these curves are spatial analogs of the classical hypocycloids.

D. Fuchs, A. Kirillov, S. Morier-Genoud, and V. Ovsienko: “On tangent cones of Schubert varieties,” Arnold Math. J. 3 : 4 (2017), pp. 451–482. MR 3766071 Zbl 1427.14103 ArXiv 1606.07846 article

We consider tangent cones of Schubert varieties in the complete flag variety, and investigate the problem when the tangent cones of two different Schubert varieties coincide. We give a sufficient condition for such coincidence, and formulate a conjecture that provides a necessary condition. In particular, we show that all Schubert varieties corresponding to the Coxeter elements of the Weyl group have the same tangent cone. Our main tool is the notion of pillar entries in the rank matrix counting the dimensions of the intersections of a given flag with the standard one. This notion is a version of Fulton’s essential set. We calculate the dimension of a Schubert variety in terms of the pillar entries of the rank matrix.

Sophie Morier-Genoud	Related
Alexandre Aleksandrovich Kirillov	Related
Valentin Yurevich Ovsienko	Related

M. Arnold, D. Fuchs, I. Izmestiev, and S. Tabachnikov: Cross-ratio dynamics on ideal polygons. Preprint, December 2018. ArXiv 1812.05337 techreport

Two ideal polygons, \( (p_1,\dots,p_n) \) and \( (q_1,\dots,q_n) \), in the hyperbolic plane or in hyperbolic space are said to be \( \alpha \)-related if the cross-ratio \[ [p_i,p_{i+1},q_i,q_{i+1}] = \alpha \] for all \( i \) (the vertices lie on the projective line, real or complex, respectively). For example, if \( \alpha = -1 \), the respective sides of the two polygons are orthogonal. This relation extends to twisted ideal polygons, that is, polygons with monodromy, and it descends to the moduli space of Möbius-equivalent polygons. We prove that this relation, which is, generically, a \( 2{-}2 \) map, is completely integrable in the sense of Liouville. We describe integrals and invariant Poisson structures, and show that these relations, with different values of the constants \( \alpha \), commute, in an appropriate sense. We investigate the case of small-gons, describe the exceptional ideal polygons, that possess infinitely many \( \alpha \)-related polygons, and study the ideal polygons that are \( \alpha \)-related to themselves (with a cyclic shift of the indices).

Maxim D. Arnold	Related
Serge Lvovich Tabachnikov	Related
Ivan Veniaminovich Izmestiev	Related

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@article {key3859406m,
	AUTHOR = {Aptekarev, A. I. and Buchstaber, V.
		 M. and Vassiliev, V. A. and Gromov,
		 M. L. and Ilyashenko, Yu. S. and Kashin,
		 B. S. and Keselman, V. M. and Kozlov,
		 V. V. and Kontsevich, M. L. and Krichever,
		 I. M. and Kruzhilin, N. G. and Lando,
		 S. K. and Manin, Yu. I. and Margulis,
		 G. A. and Nemirovski, S. Yu. and Novikov,
		 S. P. and Reshetnyak, Yu. G. and Sinai,
		 Ya. G. and Suetin, S. P. and Treschev,
		 D. V. and Fuchs, D. B. and Khovanskii,
		 A. G. and Chirka, E. M. and Schwarz,
		 A. S. and Shiryaev, A. N.},
	TITLE = {Vladimir {A}ntonovich {Z}orich (on his
		 80th birthday)},
	JOURNAL = {Usp. Mat. Nauk},
	FJOURNAL = {Uspekhi Matematicheskikh Nauk},
	VOLUME = {73},
	NUMBER = {5(443)},
	YEAR = {2018},
	PAGES = {193--196},
	DOI = {10.4213/rm9828},
	NOTE = {An English translation was published
		 in \textit{Russ. Math. Surv.} \textbf{73}:5
		 (2018). MR:3859406.},
	ISSN = {0042-1316},
}

A. I. Aptekarev, V. M. Buchstaber, V. A. Vassiliev, M. L. Gromov, Yu. S. Ilyashenko, B. S. Kashin, V. M. Keselman, V. V. Kozlov, M. L. Kontsevich, I. M. Krichever, N. G. Kruzhilin, S. K. Lando, Yu. I. Manin, G. A. Margulis, S. Yu. Nemirovski, S. P. Novikov, Yu. G. Reshetnyak, Ya. G. Sinai, S. P. Suetin, D. V. Treschev, D. B. Fuchs, A. G. Khovanskii, E. M. Chirka, A. S. Schwarz, and A. N. Shiryaev: “Vladimir Antonovich Zorich (on his 80th birthday),” Russ. Math. Surv. 73 : 5 (2018), pp. 935–939. English translation of Russian original published in Usp. Mat. Nauk 73:5(443) (2018). Zbl 1417.01018 article

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@article {key1417.01018z,
	AUTHOR = {Aptekarev, A. I. and Buchstaber, V.
		 M. and Vassiliev, V. A. and Gromov,
		 M. L. and Ilyashenko, Yu. S. and Kashin,
		 B. S. and Keselman, V. M. and Kozlov,
		 V. V. and Kontsevich, M. L. and Krichever,
		 I. M. and Kruzhilin, N. G. and Lando,
		 S. K. and Manin, Yu. I. and Margulis,
		 G. A. and Nemirovski, S. Yu. and Novikov,
		 S. P. and Reshetnyak, Yu. G. and Sinai,
		 Ya. G. and Suetin, S. P. and Treschev,
		 D. V. and Fuchs, D. B. and Khovanskii,
		 A. G. and Chirka, E. M. and Schwarz,
		 A. S. and Shiryaev, A. N.},
	TITLE = {Vladimir {A}ntonovich {Z}orich (on his
		 80th birthday)},
	JOURNAL = {Russ. Math. Surv.},
	FJOURNAL = {Russian Mathematical Surveys},
	VOLUME = {73},
	NUMBER = {5},
	YEAR = {2018},
	PAGES = {935--939},
	DOI = {10.1070/RM9828},
	NOTE = {English translation of Russian original
		 published in \textit{Usp. Mat. Nauk}
		 \textbf{73}:5(443) (2018). Zbl:1417.01018.},
	ISSN = {0036-0279},
}

D. Fuchs: “Cusps,” pp. 185–200 in Mathematical adventures for students and amateurs. Edited by D. F. Hayes and T. Shubin. American Mathematical Society (Providence, RI), 2020. incollection

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@article {key4081971m,
	AUTHOR = {Astashov, A. M. and Astashova, I. V.
		 and Bocharov, A. V. and Buchstaber,
		 V. M. and Vassiliev, V. A. and Verbovetsky,
		 A. M. and Vershik, A. M. and Veselov,
		 A. P. and Vinogradov, M. M. and Vitagliano,
		 L. and Vitolo, R. F. and Voronov, Th.
		 Th. and Kac, V. G. and Kosmann-Schwarzbach,
		 Y. and Krasil\cprime shchik, I. S. and
		 Krichever, I. M. and Krishchenko, A.
		 P. and Lando, S. K. and Lychagin, V.
		 V. and Marvan, M. and Maslov, V. P.
		 and Mishchenko, A. S. and Novikov, S.
		 P. and Rubtsov, V. N. and Samokhin,
		 A. V. and Sossinsky, A. B. and Stasheff,
		 J. and Fuchs, D. B. and Khelemsky, A.
		 Ya. and Khor\cprime kova, N. G. and
		 Chetverikov, V. N. and Schwarz, A. S.},
	TITLE = {Alexandre {M}ikha\u{\i}lovich {V}inogradov},
	JOURNAL = {Usp. Mat. Nauk},
	FJOURNAL = {Uspekhi Matematicheskikh Nauk},
	VOLUME = {75},
	NUMBER = {2(452)},
	YEAR = {2020},
	PAGES = {185--190},
	DOI = {10.4213/rm9931},
	NOTE = {An English translation was published
		 in \textit{Russ. Math. Surv.} \textbf{75}:2
		 (2020). MR:4081971.},
	ISSN = {0042-1316},
}

A. M. Astashov, I. V. Astashova, A. V. Bocharov, V. M. Buchstaber, V. A. Vassiliev, A. M. Verbovetsky, A. M. Vershik, A. P. Veselov, M. M. Vinogradov, L. Vitagliano, R. F. Vitolo, T. T. Voronov, V. G. Kac, Y. Kosmann-Schwarzbach, I. S. Krasil’shchik, I. M. Krichever, A. P. Krishchenko, S. K. Lando, V. V. Lychagin, M. Marvan, V. P. Maslov, A. S. Mishchenko, S. P. Novikov, V. N. Rubtsov, A. V. Samokhin, A. B. Sossinsky, J. Stasheff, D. B. Fuchs, A. Ya. Khelemsky, N. G. Khor’kova, V. N. Chetverikov, and A. S. Schwarz: “Alexandre Mikhaĭlovich Vinogradov,” Russ. Math. Surv. 75 : 2 (April 2020), pp. 369–375. English translation of Russian original published in Usp. Mat. Nauk 75:2 (2020). Zbl 07229698 article

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@article {key07229698z,
	AUTHOR = {Astashov, A. M. and Astashova, I. V.
		 and Bocharov, A. V. and Buchstaber,
		 V. M. and Vassiliev, V. A. and Verbovetsky,
		 A. M. and Vershik, A. M. and Veselov,
		 A. P. and Vinogradov, M. M. and Vitagliano,
		 L. and Vitolo, R. F. and Voronov, Th.
		 Th. and Kac, V. G. and Kosmann-Schwarzbach,
		 Y. and Krasil\cprime shchik, I. S. and
		 Krichever, I. M. and Krishchenko, A.
		 P. and Lando, S. K. and Lychagin, V.
		 V. and Marvan, M. and Maslov, V. P.
		 and Mishchenko, A. S. and Novikov, S.
		 P. and Rubtsov, V. N. and Samokhin,
		 A. V. and Sossinsky, A. B. and Stasheff,
		 J. and Fuchs, D. B. and Khelemsky, A.
		 Ya. and Khor\cprime kova, N. G. and
		 Chetverikov, V. N. and Schwarz, A. S.},
	TITLE = {Alexandre {M}ikha\u{\i}lovich {V}inogradov},
	JOURNAL = {Russ. Math. Surv.},
	FJOURNAL = {Russian Mathematical Surveys},
	VOLUME = {75},
	NUMBER = {2},
	MONTH = {April},
	YEAR = {2020},
	PAGES = {369--375},
	DOI = {10.1070/RM9931},
	NOTE = {English translation of Russian original
		 published in \textit{Usp. Mat. Nauk}
		 \textbf{75}:2 (2020). Zbl:07229698.},
	ISSN = {0036-0279},
}

Dmitry Fuchs

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