Main results include: 1) A system of invariants of knots, links, and plane curves, stronger than all known polynomial invariants; 2) A universal method of the computation of homology groups of spaces of non-degenerate geometrical objects, producing in particular the knot invariants and numerous comparison theorems of the Smale–Hirsch–Gromov type for spaces of real and complex functions and maps without complicated singularities; 3) Best known (and asymptotically sharp) estimates of the numbers of branchings of algorithms of approximate computation of roots of polynomials 4) "Stratified" version of the Picard–Lefschetz theory for the homology groups of singular manifolds; 5) A proof of the Atiyah–Bott Garding conjecture on the equivalence of the sharpness of the wave front of a hyperbolic operator to the topological Petrovskii condition; an interpretation of these conditions in the terms of the geometry of the front; 6) Multidimensional analogues of the Newton's theorem on the non-integrability of ovals; 7) A construction of multi-dimensional analogues of the Maslov index by the methods of the singularity theory (the "universal complex of singularities"); 8) A calculation of the stable homotopy types of complements of plane arrangements in $\mathbf R^n$, generalizing the Goresky–MacPherson formula for the homology groups of such complements. Other results concern complexity theory of smooth maps, real algebraic geometry, generalized hypergeometric functions, topology of Lie groups, dynamical systems, geometrical combinatorics, potential theory, etc.
Biography
Graduated from the Department of Mathematics and Mechanics of M. V. Lomonosov Moscow State University (MSU) in 1965 Ph.D. since 1982 (advisor V. I. Arnold); D.Sci. since 1992. More than 100 scientific publications. Since 1997 I run a research seminar at the Independent University of Moscow. Board member of journals "Functional Analysis and its Applications" (Editor in Chief), Izvestiya RAS, Doklady DAS, Moscow Mathematical Journal, Selecta Mathematica (New Series), Topology and its Applications, Journal of Knot Theory and its Ramifications. Member of the Executive Committee of International Mathematical Union (2004–2010). President of the Moscow Mathematical Society (2010–present).
Moscow Math. Society award of 1985 for the cycle of works "Characteristic classes of wave fronts". Plenary invited address at the Zurich International Congress of Mathematicians (1994). Miller Research Professor, UC Berkeley, 1999. Visiting Fellow Commoner, Trinity College, Cambridge University, 2000. Russia State Award in the field of education.
Main publications:
V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications, 2-d extended edition, Translations of Math. Monographs, 98, AMS, Providence, RI, 1994, 268 pp. ; rasshirennyi russkii perevod: V. A. Vasilev, Topologiya dopolnenii k diskriminantam, Fazis, Moskva, 1997, xiv+536 s.
V. A. Vassiliev, Ramified integrals, singularities and lacunas, Math. Appl., 315, Kluwer Academic Publishers, Dorderecht (Netherlands), 1995, xviii+289 pp. ; rasshirennyi russkii perevod: V. A. Vasilev, Vetvyaschiesya integraly, MTsNMO, Moskva, 2000, 432 s.
V. A. Vassiliev, Lagrange and Legendre characteristic classes, 2-d edition, Gordon and Breach Publishers, New York a.o., 1993, 273 pp. ; rasshirennyi russkii perevod: V. A. Vasilev, Lagranzhevy i lezhandrovy kharakteristicheskie klassy, MTsNMO, Moskva, 2000, 312 s.
V. I. Arnold, V. A. Vasilev, V. V. Goryunov, O. V. Lyashko, Osobennosti. I. Lokalnaya i globalnaya teoriya, Itogi nauki i tekhniki. Sovr. probl. matem. Fundam. napr., 6, VINITI, Moskva, 1988, 256 s. ; Osobennosti. II. Klassifikatsiya i prilozheniya, Itogi nauki i tekhniki. Sovr. probl. matem. Fundam. napr., 39, 1989, 256 s. ; English translation: V. I. Arnold, V. A. Vasil'ev, V. V. Goryunov, O. V. Lyashko, Singularities, Encycl. Math. Sci., 6, 39, Springer-Verlag, Berlin–New York, 1993, 245+233 pp.
V. A. Vassiliev, “Combinatorial formulas for cohomology of spaces of knots”, Moscow Math. J., 1:1 (2001), 91–123
V.A. Vassiliev, “Real Function Singularities and Their Bifurcation Sets”, Handbook of Geometry and Topology of Singularities VII, eds. José Luis Cisneros-Molina, Lê Dũng Tráng, José Seade, Springer, 2025, 71–119https://link.springer.com/chapter/10.1007/978-3-031-68711-2_2
V. A. Vassiliev, “Average intersection numbers of low-harmonic plane curves in L_2 and W_2^r statistics”, Journal of Knot Theory and Its Ramifications, 34:08 (2025), 2550042 , 10 pp., arXiv: 2303.12645
V.A. Vassiliev, “Cohomology of spaces of Hopf equivariant maps of spheres”, Homology, Homotopy and Applications, 26:2 (2024), 219–227 , arXiv: 2102.07157
13.
V.A. Vassiliev, On the cohomology of varieties of chord diagrams, 2024 , 8 pp., arXiv: 2412.18508
2023
14.
V.A. Vassiliev, "Complements of discriminants of real singularities of type $X_{10}$", 2023 , 17 pp., arXiv: 2301.06994
15.
Mark Agranovksy, Jan Boman, Alexander Koldobsky, Victor Vassiliev, Vladyslav Yaskin, “Algebraically integrable bodies and related properties of the Radon transform”, Harmonic Analysis and Convexity, eds. Alexander Koldobsky and Alexander Volberg, De Gruyter, Berlin, Boston, 2023, 1–36; arXiv: 2212.07510
16.
V.A. Vassiliev, “Complements of discriminants of real parabolic function singularities”, Moscow Mathematical Journal, 23:3 (2023), 401–432 , arXiv: 2208.10929
Victor A. Vassiliev, “Varieties of chord diagrams, braid group cohomology and degeneration of equality conditions”, Pacific Journal of Mathematics, 326:1 (2023), 135–160 , arXiv: 2108.00463
2020
18.
Victor A. Vassiliev, “New Examples of Irreducible Local Diffusion of Hyperbolic PDE's”, SIGMA, 16 (2020), 9 , 21 pp., arXiv: 1909.13211
V. A. Vassiliev, “New examples of locally algebraically integrable bodies”, Mathematical Notes, 106:6 (2019), 894–898 , arXiv: 1902.07235
2018
23.
Victor A.Vassiliev, “Twisted homology of configuration spaces, homology of spaces of equivariant maps, and stable homology of spaces of non-resultant systems of real homogeneous polynomials”, 2018 , 32 pp., arXiv: 1809.05632
24.
V. A. Vassiliev, “Lacunas and local algebraicity of volume functions”, J. Singul., 18 (2018), 350–357 , arXiv: 1803.07829
V. A. Vassiliev, “Local Petrovskii lacunas close to parabolic singular points of the wavefronts of strictly hyperbolic partial differential equations”, Sb. Math., 207:10 (2016), 1363–1383
2015
34.
V. A. Vassiliev, “Newton's lemma XXVIII on integrable ovals in higher dimensions and reflection groups”, Bull. Lond. Math. Soc., 47:2 (2015), 290–300 , arXiv: 1407.7221
V. A. Vassiliev, “Homology of spaces of non-resultant polynomial systems in $\mathbb R^2$ and $\mathbb C^2$”, Arnold Math. J., 1:3 (2015), 233–242 , arXiv: 1409.6005
V. A. Vassiliev, “Rational Homology of the Order Complex of Zero Sets of Homogeneous Quadratic Polynomial Systems in $\mathbb R^3$”, Proc. Steklov Inst. Math., 290 (2015), 197–209 , arXiv: 1501.06063
2014
38.
V. Vassiliev, “Topology in Arnold's Work”, Arnold: Swimming Against the Tide, eds. B. A. Khesin and S. L. Tabachnikov, Amer. Math. Soc., Providence, RI, 2014, 165–171http://www.ams.org/bookstore-getitem/item=mbk-86
S. A. Grishanov, V. A. Vassiliev, “Invariants of links in 3-manifolds and splitting problem of textile structures”, J. Knot Theory Ramifications, 20:3 (2011), 345–370http://ag.hse.ru/data/2012/03/12/1211315126/4gv.pdf
S. A. Grishanov, V. A. Vassiliev, “Gauss diagram invariants of links in $M^2\times\mathbf R^1$”, J. Knot Theory Ramifications, 20:3 (2011), 371–387http://www.pdmi.ras.ru/ãrnsem/papers/5gv.pdf
42.
V. A. Vassiliev, “Topological complexity and Schwarz genus of general real polynomial equation”, Mosc. Math. J., 11:3 (2011), 617–625
V. A. Vasilev, “Integriruemye ovaly i monodromiya. Annotatsiya doklada 10.12.2009”, Trudy seminara po vektornomu i tenzornomu analizu, XXVII, Izd-vo Mosk. un-ta, 2011, 18
2009
45.
S. A. Grishanov, V. A. Vassiliev, “Fiedler type combinatorial formulas for generalized Fiedler type invariants of knots in $M^2\times\mathbf R^1$”, Topology Appl., 156:14 (2009), 2307–2316http://www.pdmi.ras.ru/ãrnsem/papers/3gv.pdf
S. A. Grishanov, V. R. Meshkov, V. A. Vassiliev, “Recognizing textile structures by finite type knot invariants”, J. Knot Theory Ramifications, 18:2 (2009), 209–235http://www.pdmi.ras.ru/ãrnsem/papers/1gmv.pdf
S. A. Grishanov, V. A. Vassiliev, “Two constructions of weight systems for invariants of knots in non-trivial 3-manifolds”, Topology Appl., 155:16 (2008), 1757–1765http://www.pdmi.ras.ru/ãrnsem/papers/2gv.pdf
V. A. Vassiliev, “First-order invariants and cohomology of spaces of embeddings of self-intersecting curves in $\mathbb R^n$”, Izv. Math., 69:5 (2005), 865–912http://www.pdmi.ras.ru/ãrnsem/papers/
50.
V. I. Arnold, A. A. Davydov, V. A. Vassiliev, V. M. Zakalyukin, “Mathematical models and control of catastrophic processes”, UNESCO Encyclopaedia of Life Support Systems, v. II, eds. V. I. Agoshkov, J.-P. Puel, EOLSS Publishers, Oxford, UK, 2005, 3–46http://www.eolss.net/Sample-Chapters/C02/E6-03A-07-05.pdf
2004
51.
V. A. Vassiliev, “Combinatorial formulas for cohomology of spaces of knots”, Advances in topological quantum field theory, NATO Sci. Ser. II Math. Phys. Chem., 179, Kluwer Acad. Publ., Dorderecht, 2004, 1–21http://www.pdmi.ras.ru/ãrnsem/papers/cofor.ps.gz
52.
“Ramified integals and Picard-Lefschetz theories”, Globus. Seminar of the Moscow Independent University, 2004, 22–58
2003
53.
V. A. Vasilev, “Algoritmy dlya kombinatornoi realizatsii kogomologii prostranstv uzlov”, Fundamentalnaya matematika segodnya, NMU, M., 2003, 10–31http://booksmania.biz/other/textbook4117.html
V. Vassiliev, “Resolutions of discriminants and topology of their complements”, New developments in singularity theory (Cambridge, 2000), NATO Sci. Ser. II Math. Phys. Chem., 21, Kluwer Acad. Publ., Dorderecht, 2001, 87–115http://www.pdmi.ras.ru/~arnsem/papers/SIERSHOOL.pdf
V. A. Vasilev, Lagranzhevy i lezhandrovy kharakteristicheskie klassy, MTsNMO, M., 2000 , 312 pp.
61.
V. A. Vasilev, Vetvyaschiesya integraly, MTsNMO, M., 2000 , 432 pp.
2004
62.
V. A. Vassiliev, Comments to some 32 problems: V. I. Arnold and his seminar, Arnold's problems, Springer, Berlin, 2004, 238–241, 263, 265–267, 284, 299–300, 310, 313, 316–317, 347, 371–372, 388–389, 394, 407, 416, …
1999
63.
V. A. Vassiliev, “Topological order complexes and resolutions of discriminant sets”, Publ. de l'institut mathematique (Beograd), Nouvelle serie,, Publ. Inst. Math. (Beograd) (N.S.), 66(80), 1999, 165–185http://elib.mi.sanu.ac.rs/files/journals/publ/86/n080p165.pdf
64.
V. A. Vassiliev, “Homology of $i$-connected graphs and invariants of knots, plane arrangements, etc.”, The Arnoldfest (Toronto, ON, 1997), Fields Inst. Commun., 24, Amer. Math. Soc., Providence, RI, 1999, 451–469http://www.pdmi.ras.ru/~arnsem/papers/TORON.pdf
65.
V. A. Vassiliev, “Topology of two-connected graphs and homology of spaces of knots”, Differential and symplectic topology of knots and curves, Amer. Math. Soc. Transl. Ser. 2, 190, Amer. Math. Soc., Providence, RI, 1999, 253–286https://arxiv.org/abs/math/9707212
V. A. Vassiliev, “On a Problem by M. Kazarian”, Funct. Anal. Appl., 33:3 (1999), 220–221
68.
V. A. Vassiliev, “How to Calculate Homology Groups of Spaces of Nonsingular Algebraic Projective Hypersurfaces”, Proc. Steklov Inst. Math., 225 (1999), 121–140 , arXiv: 1407.7229
1998
69.
V. A. Vassiliev, “On invariants and homology of spaces of knots in arbitrary manifolds”, Topics in quantum groups and finite-type invariants, Amer. Math. Soc. Transl. Ser. 2, 185, Amer. Math. Soc., Providence, RI, 1998, 155–182 , arXiv: math/9703207
V. A. Vasilyev, “Ramification in integral geometry and monodromy of complex links”, Dynamical systems, 3, J. Math. Sci. (New York), 83:4 (1997), 554–558
V. A. Vassiliev, “Topological complexity of root-finding algorithms”, The mathematics of numerical analysis (Park City, UT, 1995), Lectures in Appl. Math., 32, Amer. Math. Soc., Providence, RI, 1996, 831–856
81.
V. A. Vassiliev, “Topological complexity and real roots of polynomials”, Math. Notes, 60:5 (1996), 503–509
82.
V. A. Vassiliev, “On spaces of polynomial knots”, Sb. Math., 187:2 (1996), 193–213 , arXiv: q-alg/9505008
V. Vassiliev, “The mathematical legacy of “Lacunas for hyperbolic differential equations”. Comment on: “Lacunas for hyperbolic differential operators with constant coefficients. I, II” [Acta Math. 124 (1970), 109–189; ibid. 131 (1973), 145–206] by M. F. Atiyah, R. Bott and L. Gårding”, Raoul Bott: collected papers, v. 2, Contemp. Math., Birkhäuser Boston, Boston, MA, 1994, xxiii–xxviii
1993
89.
V. A. Vassiliev, “Invariants of knots and complements of discriminants”, Developments in mathematics: the Moscow school, Chapman & Hall, London, 1993, 194–250
90.
V. A. Vassiliev, “Complexes of connected graphs”, The Gel'fand Mathematical Seminars, 1990–1992, Birkhäuser Boston, Boston, MA, 1993, 223–235 https://arxiv.org/abs/1409.5999 http://www.pdmi.ras.ru/ãrnsem/papers/congraph.pdf
V. A. Vassil’ev, “Cohomology of braid groups and complexity”, From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990), Springer, New York, NY, 1993, 359–367
92.
V. A. Vassil'ev, “The Smale-Hirsch principle in catastrophe theory”, From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990), Springer, New York, 1993, 117–128
93.
V. I. Arnol'd, V. V. Goryunov, O. V. Lyashko, V. A. Vassiliev, Singularity theory. II, Dynamical Systems, 39, Springer, Berlin–New York, 1993 , 233 pp.
1992
94.
V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications, Translated from the Russian by B. Goldfarb, Trans. Math. Monogr., 98, Amer. Math. Soc., Providence, RI, 1992 , vi+208 pp. ; Second extended edition, 1994
95.
V. A. Vassiliev, “On function spaces that are interpolating at any $k$ nodes”, Funct. Anal. Appl., 26:3 (1992), 209–210
1993
96.
V. A. Vassiliev, “Geometry of local lacunae of hyperbolic operators with constant coefficients”, Russian Acad. Sci. Sb. Math., 75:1 (1993), 111–123
1992
97.
V. A. Vassiliev, “Geometric realization of the homology of classical Lie groups, and complexes that are $S$-dual to flag manifolds”, St. Petersburg Math. J., 3:4 (1992), 809–815
1991
98.
V. A. Vassiliev, V. V. Serganova, “On the number of real and complex moduli of singularities of smooth functions and realizations of matroids”, Math. Notes, 49:1 (1991), 15–20
1990
99.
V. A. Vasilev, Kogomologii prostranstv uzlov, Preprint IPM AN SSSR, № 91, IPM AN SSSR, Moskva, 1990 , 29 pp.
100.
V. A. Vasilev, Gomologicheskie invarianty uzlov: algoritmy i vychisleniya, Preprint IPM AN SSSR, № 90, 1990 , 23 pp.
V. A. Vassiliev, “Topology of complements to discriminants and loop spaces”, Theory of singularities and its applications, Adv. Soviet Math., 1, American Mathematical Society, Providence, RI, 1990, 9–21
1992
103.
V. A. Vassiliev, “Lacunas of hyperbolic partial differential equations and singularity theory”, Amer. Math. Soc. Transl. (2), 153 (1992), 25–37
104.
V. A. Vassiliev, “Characteristic classes of singularities”, Amer. Math. Soc. Transl. (2), 1992, no. 153, 11–23
V. A. Vassiliev, “Topological complexity of algorithms for the approximate solution of systems of polynomial equations”, Leningrad Math. J., 1:6 (1990), 1401–1417
1989
107.
V. A. Vassiliev, “Topology of spaces of functions without compound singularities”, Funct. Anal. Appl., 23:4 (1989), 277–286
108.
V. I. Arnol'd, V. A. Vassiliev, V. V. Goryunov, O. V. Lyashko, “Singularities. II. Classification and applications”, Dynamical systems – 8, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 39, VINITI, Moscow, 1989, 5–249 , 256 pp.
109.
V. A. Vassiliev, “On the topology of spaces of functions without complex singularities”, Russian Math. Surveys, 44:3 (1989), 218–219
1988
110.
V. A. Vassilyev, Lagrange and Legendre characteristic classes, Adv. Stud. Contemp. Math., 3, Frst edition, Gordon and Breach Sci. Publ., New York, 1988 , x+268 pp. ; Second revised edition, 1993 , x+267 pp.
111.
V. A. Vassiliev, “Braid group cohomologies and algorithm complexity”, Funct. Anal. Appl., 22:3 (1988), 182–190
1990
112.
V. A. Vassiliev, “Stable cohomology of complements to the discriminants of deformations of singularities of smooth functions”, J. Soviet Math., 52:4 (1990), 3217–3230
1988
113.
V. I. Arnol'd, V. A. Vassiliev, V. V. Goryunov, O. V. Lyashko, “Singularities. I. Local and global theory”, Dynamical systems – 6, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 6, VINITI, Moscow, 1988, 5–250
1987
114.
V. A. Vassiliev, I. M. Gel'fand, A. V. Zelevinskii, “General hypergeometric functions on complex Grassmannians”, Funct. Anal. Appl., 21:1 (1987), 19–31
115.
V. A. Vassiliev, “Stable cohomologies of complements to discriminant manifolds of singularities of holomorphic functions”, Russian Math. Surveys, 42:2 (1987), 307–308
116.
V. A. Vassiliev, “Sharpness and the local Petrovskii condition for strictly hyperbolic operators with constant coefficients”, Math. USSR-Izv., 28:2 (1987), 233–273
1986
117.
V. A. Vasilev, “Rezkie fronty giperbolicheskikh operatorov s postoyannymi koeffitsientami”, Sovmestnye zasedaniya seminara imeni I.G. Petrovskogo po differentsialnym uravneniyam i matematicheskim problemam fiziki i Moskovskogo matematicheskogo obschestva (Moskva, 20–23 yanvarya 1986 g.), UMN, 41:4 (1986), 162
V. A. Vassiliev, I. M. Gel'fand, A. V. Zelevinskii, “The behavior of general hypergeometric functions in a complex domain”, Dokl. Akad. Nauk SSSR, 290:2 (1986), 277–281
1982
119.
V. A. Vassiliev, “Self-intersections of wave fronts and Legendre (Lagrangian) characteristic numbers”, Funct. Anal. Appl., 16:2 (1982), 131–133
1981
120.
V. A. Vassiliev, “Characteristic classes of Lagrangian and Legendre manifolds dual to singularities of caustics and wave fronts”, Funct. Anal. Appl., 15:3 (1981), 164–173
1979
121.
V. A. Vassiliev, “Asymptotic behavior of exponential integrals in the complex domain”, Funct. Anal. Appl., 13:4 (1979), 239–247
1978
122.
V. A. Vassiliev, “Affineness of normal forms of strata $\mu=\mathrm{const}$ of germs of smooth functions”, Funct. Anal. Appl., 12:3 (1978), 218–220
1977
123.
V. A. Vassiliev, “Asymptotic exponential integrals, Newton's diagram, and the classification of minimal points”, Funct. Anal. Appl., 11:3 (1977), 163–172
Presentations in Math-Net.Ru
1.
Algebraically integrable bodies V. A. Vassiliev Seminar "Complex analysis in several variables" (Vitushkin Seminar) February 26, 2020 16:45
Вступительное слово V. A. Vassiliev Conference "Moscow Mathematical Society and Lomonosov Moscow State University" dedicated to the 150th anniversary of Moscow Mathematical Society December 23, 2014 18:00
On $k$-convex submanifolds in Eucledean space V. A. Vassiliev General Mathematics Seminar of the St. Petersburg Division of Steklov Institute of Mathematics, Russian Academy of Sciences October 15, 1997
2025
