### Outline of results

The text presented below was extracted from my research statement, the document that was requested in job application processes. By a research statement one often means an account of published research papers enhanced either with an account of research which has been done, but is yet not documented in papers or preprints, or a statement of research intentions. It provides a sort of research self-portrait of the author, which is especially valuable at the beginning of his/her career, right after the Ph.D, when a statement on completed research looks short.

After 48 years of research in Mathematics, an outline of the main results says more than whatever plans can say, and I removed any plans from my research statement. I do have plans, as always, but, in fact, my research plans were never on a par with results. I was never able to foresee really interesting discoveries and new twists of the subject, which ended up drastically changing the direction of my research.

The presentation below is not chronological. The research papers and results are grouped according to the subjects.

## 1 Patchworking: constructing real algebraic varieties with controlled topological properties

Although I started in low-dimensional topology, I seem to be best known for my contributions to real algebraic geometry. As far as I can judge, my most appreciated invention is the patchwork technique or “Viro method” which allows to construct real algebraic varieties by a sort of “cup and paste” technique. It was introduced in order to construct real algebraic varieties with remarkable topological properties. Using it I completed the classification up to isotopy of non-singular plane projective curves of degree 7 and disproved the classical Ragsdale conjecture formulated in 1906 [ 13 ] . See also [ 16 ] , [ 17 ] , [ 18 ] , [ 20 ] , [ 23 ] , [ 34 ] , [ 42 ] and [ 55 ] .

## 2 The dequantizing of algebraic geometry, tropical geometry and hyperfields

In my talk [ 47 ] at the third European Congress of Mathematicians in 2000, I observed that real algebraic geometry can be presented as a quantized (i.e., deformed) piecewise linear geometry. A simple version of the patchwork construction (which builds a real algebraic hypersurface in a toric variety out of a special piecewise-linear hypersurface of Euclidean space) could be understood in terms of this quantization. The generalization of these ideas gave rise to the development (by Kapranov, Kontsevich, Mikhalkin and Sturmfiels) of so called tropical geometry and its applications to problems of classical algebraic geometry.

In the tropical geometry, the role of ground field was performed by the tropical semifield. This makes algebraic aspects of tropical geometry exotic and unnatural. Tropical varieties appear also as limits of amoebas of complex algebraic varieties under a tropical degeneration (dequantization). The dequantization can be applied to the varieties, complex or real, themselves.

In [ 59 ] and [ 60 ] I observed that the dequantization of algebraic varieties can be obtained via dequantization of the ground fields $\mathbb{C}$ and $\mathbb{R}$ . The dequantizations of fields produce not fields, but hyperfields, i.e., fields in which the addition is multivalued. The dequantized $\mathbb{C}$ and $\mathbb{R}$ are comparatively simple. They have the same set of elements and the same multiplication. Only addition changes. To the best of my knowledge, the dequantized $\mathbb{C}$ and $\mathbb{R}$ had not appeared in the literature before my work.

The turn to hyperfields extends the algebraic geometry. It makes dequantized varieties legitimate algebraic varieties. Also, it improves understanding of tropical varieties (non-archimedian amoebas). They were interpreted as varieties over the tropical semifield, but at the cost of a substantial change in the definition of variety: the varieties were not defined by polynomial equations, but as loci where tropical polynomials are not differentiable. Replacing of the tropical semifield by the corresponding hyperfield returned equations to the subject.

## 3 Other results on real algebraic varieties

I found several restrictions on the topology of real algebraic curves (see [ 10 ] , [ 17 ] , [ 18 ] , [ 27 ] , [ 38 ] ), explicit elementary constructions of real algebraic surfaces with maximal total Betti numbers [ 11 ] , and non-singular real projective quartic surfaces of all but one of the isotopy types [ 12 ] . I generalized complex orientations from the case of a real algebraic curve dividing its complexification to real algebraic varieties of high dimensions [ 15 ] , [ 39 ] .

Together with Kharlamov I generalized the main topological restrictions on nonsingular plane projective real algebraic curves to singular curves [ 27 ] . Recently, in a joint paper [ 48 ] with Orevkov, by applying one of these results on singular curves, we proved Orevkov’s conjecture on the topology of nonsingular curves of degree 9.

I studied the Radon transformations based on integrals against the Euler characteristic on real and complex projective spaces, and established its relation to the projective duality for algebraic varieties [ 26 ] . This gives new relations between the numerical characteristics of projectively dual varieties.

## 4 TQFT

Jointly with V. Turaev, we found a (2+1)-dimensional topological quantum field theory based on state sums over triangulations or Heegaard diagrams and involving quantum 6j-symbols, see [ 35 ] . This paper had unexpected (to the authors) relations to Physics: it gave the first rigorous realization of the approach by G. Ponzano and T. Redge to 2+1 quantum gravity. In the context of Quantum Topology, it was widely generalized and related to other quantum invariants.

From the algebraic point of view, the invariants introduced in [ 35 ] are based on the representation theory of quantum group $U_{q}sl(2)$ where the parameter $q$ is a root of unity. In [ 51 ] I used similar constructions applied to the quantum super-group $U_{q}gl(1,1)$ and $U_{\sqrt{-1}}sl(2)$ to study quantum relatives of the Alexander polynomial.

## 5 Exotic knottings

Jointly with S.Finashin and M.Kreck, we constructed the first infinite series of surfaces smoothly embedded in the 4-sphere, which are pairwise ambiently homeomorphic, but not diffeomorphic, see [ 24 ] , [ 25 ] . The examples come from real algebraic geometry. Namely, the series of Dolgachev surfaces has real structures (that is complex conjugation involutions) the orbit spaces spaces of which are diffeomorphic to the 4-sphere. The real point sets (i.e., the fixed point sets of the involutions) are diffeomorphic to the connected sum of 5 copies of the Klein bottle. They are embedded in the 4-sphere differently from the differential viewpoint, because the Dolgachev surfaces are not diffeomorphic, but in the same way topologically. (In the paper only finiteness of the set of their topological types was proven, but later Kreck proved that there is only one topological type.)

## 6 Diagrammatic formulas for finite type invariants

In a joint work [ 40 ] with M.Polyak, we introduced diagrammatic formulas for Vassiliev knot invariants. Joint work [ 44 ] with M. Goussarov and M. Polyak extended these formulas and the notion of Vassiliev invariants to virtual knots.

For the Arnold invariants of generic immersed plane curves, I found counter-parts for real plane algebraic curves. This, together with Rokhlin’s complex orientations formula for real algebraic curves suggested combinatorial formulas for the Arnold invariants $J_{-}$ and $J_{+}$ . The formulas allowed me to prove Arnold’s conjecture about the range of these invariants. See [ 41 ] .

I initiated the topological study of generic configurations of lines in 3-space. See [ 22 ] , [ 31 ] and [ 32 ] . For non-singular real algebraic curves in 3-dimensional projective space, I defined a numerical characteristic (the “encomplexed” writhe number) invariant under rigid isotopy, which allows the proof that some real algebraic curves, which are topologically isotopic are not rigid isotopic (i.e., cannot be deformed to each other in the class of non-singular real algebraic curves). See [ 46 ] .

Recently Johan Bjorklund proved that non-singular rational real algebraic curves in the real three-dimensional projective space are defined up to rigit isotopy by the degree and encomplexed writhe number.

Curves generically immersed in the plane can be considered as the counter-part of links in 3-space, since their natural liftings to the unit tangent bundle and to the projectivized tangent bundle are knots in these 3-manifolds. Arnold’s invariants $J_{+}$ and $J_{-}$ are isotopy invariants of the corresponding knots. An even more profound invariant is the Whitney number classifying immersions of the circle into the plane up to regular homotopy. In [ 53 ] I found an expression for the Whitney number of a closed real algebraic plane affine curve dividing its complexification and equipped with a complex orientation, in terms of the behavior of its complexification at infinity.

## 8 Branched coverings

During my study at Leningrad State University, I proved that any closed orientable 3-manifold of genus two is a two-fold branched covering of the 3-sphere branched over a link with 3-bridges [ 1 ] , [ 3 ] (this was proven independently by Joan S.Birman and H.Hilden).

I also found interpretations of the signature invariants of a link of codimension 2 as signature invariants of cyclic branched covering spaces of a ball, and proved estimates for the slice genus of links and the genus of non locally flat surfaces in 4-manifolds [ 4 ] , [ 6 ] .

For most of the results in the latter group, cyclic branched coverings can be replaced by local coefficient systems, see [ 58 ] .

## 9 Relations among reflections

In [ 62 ] I proved that the isometry group of the Euclidean plane is defined by a simple collection of generators and relations. Since the group has the cardinality of continuum, the set of generators also must be of this cardinality. This system generators was well-known: it is the set of reflections about all the lines. A complete system of relations among them consist of relations of length two and four. The relations of length two state that each of the reflections is of order two. The composition of reflections in two different lines is either translation or rotation. The relations of length four account when two such compositions coincide.

Similar presentations by generators and relations were found for other isometry groups. In [ 63 ] I considered relations among isometries of order two with fixed point sets of any dimension. Any isometry of a classical homogeneous space is a composition of two such involutions. (For reflections about hyperplanes, the minimal number of reflections in a presentation of isometry depends on the dimension, in a space of dimension $n$ , one may need $n+1$ reflections.) In low dimensional classical homogeneous spaces, I proposed a new graphical calculus for operating with isometries. It generalizes a well-known graphical representation for vectors and translations in an affine space. Instead of arrows, we use biflippers, which are arrows framed at the end points with subspaces. The head to tail addition of vectors and composition of translations is generalized to head to tail composition rules for isometries.

## References

• 1 Links, two-fold branched coverings and braids, Matem. sbornik 87:2 (1972) 216 – 228 (Russian); English translation in Soviet Math. Sbornik.
• 2 Local knotting of submanifolds, Matem. sbornik 90:2 (1973) 172-181 (Russian); English translation in Soviet Math. Sbornik.
• 3 Two-fold branched coverings of three-dimensional sphere, Zap. Nauchn. Semin. LOMI 36 (1973) 6-39 (Russian); English transl. in J. Soviet Math 8:5 (1977) 531-553.
• 4 Branched coverings of manifolds with boundary and invariants of links. I, Izvestiya AN SSSR, ser. Matem. 37:6 (1973) 1242-1259 (Russian); English translation in Soviet Math. Izvestia.
• 5 Non-projecting isotopies and knots with homeomorphic covering spaces, Zap. Nauchn. Semin. LOMI 66 (1976) Russian; English transl. in J. Soviet Math. 12:1 (1979) 86-96.
• 6 Placements in codimension 2 and boundary, Uspekhi Mat. Nauk 30:1 (1975) 231-232 (Russian).
• 7 The Volodin-Kuznetsov-Fomenko conjecture on Heegaard diagrams of 3-dimensional sphere is not true, (joint with V. L. Kobelsky), Uspekhi Mat. Nauk 32:5 (1977) 175-176 (Russian).
• 8 Signatures of links, Tezisy VII Vsesoyuznoj topologicheskoj konferencii (1977) 41 (Russian).
• 9 Estimates for twisted homology,(joint with V. G. Turaev), Tezisy VII Vsesoyuznoj topologicheskoj konferencii (1977) 42 (Russian).
• 10 Generalizing Petrovsky and Arnold inequalities for curves with singularities, Uspekhi Mat. Nauk 33:3 (1978) 145-146 (Russian).
• 11 Constructing M-surfaces, Funkts. analiz i ego prilozh. 13:3 (1979) 71-72 (Russian); English transl. in Functional Anal. Appl. 13:3 (1979).
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• 14 Colored knots, Kvant (1981) No. 3, 8-14 (Russian); English translation: Tied into Knot Theory: unraveling the basics of mathematical knots. Quantum 8 (1998), no. 5, 16–20.
• 15 Complex orientations of real algebraic surfaces, Uspekhi Mat. Nauk 37:4 (1982) 93 (Russian).
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• 17 Plane real algebraic curves of degrees 7 and 8: new restrictions, Izvestiya AN SSSR, ser. Matem. 47:5 (1983) 1135-1150 (Russian); English transl. in Soviet Math. Izvestia.
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• 19 Intersections of loops on two-dimensional manifolds. II. Free loops (joint with V. G. Turaev), Mat. sbornik 121:3 (1983) 359-369 (Russian); English translation in Soviet Math. Sbornik.
• 20 Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7, Lecture Notes in Math. 1060 (1984) 187-200, Springer-Verlag, Berlin and New York.
• 21 The signature of a branched covering, Mat. zametki 36:4 (1984) 549-557 (Russian); English translation in Math. Notes 36:3 4772-776.
• 22 Topological problems on lines and points of the three-dimensional space, Doklady AN SSSR 284:5 (1985) 1049-1052 (Russian); English translation in Soviet Math. Doklady 32:2 (1985) 528-531 .
• 23 Progress of the last six years in topology of real algebraic varieties, Uspekhi Mat. Nauk 41:3 (1986) 45-67 (Russian) English translation in Russian Math. Surveys 41:3 (1986) 55-82.
• 24 Exotic knottings of surfaces in the 4-sphere (joint with S. M. Finashin and M. Kreck), Bull. Amer. Math. Soc. 17:2 (1987) 287-290.
• 25 Non-diffeomorphic but homeomorphic knottings of surfaces in the 4-sphere (joint with S. M. Finashin and M. Kreck), Lecture Notes in Math. 1346 (1988) 157-198, Springer-Verlag, Heidelberg and New-York.
• 26 Some integral calculus based on Euler characteristic, Lecture Notes in Math. 1346 (1988) 127-138, Springer-Verlag, Heidelberg and New-York.
• 27 Extensions of the Gudkov-Rohlin congruence (joint with V.M.Kharlamov) Lecture Notes in Math. 1346 (1988) 357-406, Springer-Verlag, Heidelberg and New-York.
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• 30 Problems in Topology (joint with O. A. Ivanov, N. Yu. Netsvetaev and V. M. Kharlamov), Leningrad, LGU, 1988 (Russian); Second, extended edition: St. Petersburg, SPbU, 2000.
• 31 Interlacing of skew lines (joint with J.V.Drobotukhina) Kvant (1988) No. 3, 12-19 (Russian).
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• 33 Compact four-dimensional exotica with small homology, Algebra i analiz 1:4 (1989) 67-77 (Russian); English translation in Leningrad Math. J. 1:4 (1990) 871-880.
• 34 Plane real algebraic curves: constructions with controlled topology Algebra i analiz 1:5 (1989) 1-73 (Russian); English translation in Leningrad Math. J. 1:5 (1990) 1059-1134.
• 35 State sum invariants of 3-manifolds and quantum 6j-symbols (joint with V. G. Turaev), Topology 31:4 (1992) 865-902.
• 36 Lectures on combinatorial presentations of manifolds, In book Differential geometry and topology (Alghero, 1992) World Sci. Publ., River Edge, NJ (1993) 244-264.
• 37 Moves of triangulations of a PL-manifold. Quantum groups (Leningrad, 1990), 367–372, Lecture Notes in Math., 1510, Springer, Berlin, 1992.
• 38 An inequality for the number of nonempty ovals of a curve of odd degree (joint with V. I. Zvonilov), Algebra i analiz 4:3 (1992) 159-170 (Russian); English translation in St. Petersburg Math. J. 4:3 (1993).
• 39 Complex orientations of real algebraic surfaces, Topology of manifolds and varieties, Advances of Soviet Math. 18 (1994), 261-284; see also arXive: math.AG/0611396.
• 40 Gauss diagram formulas for Vassiliev invariants,(joint with Michael Polyak) International Mathematics Research Notes 1994:11.
• 41 Generic immersions of circle to surfaces and complex topology of real algebraic curves, Topology of real algebraic varieties and relate d topics, ( a volume dedicated to memory of D. A. Gudkov ), AMS Translations, Series 2, 173, (1995) 231-252 .
• 42 Patchworking algebraic curves disproves the Ragsdale conjecture, (joint with Ilia Itenberg), The Mathematical Intelligencer 18:1 (1996), 19-28.
• 43 Mutual position of hypersurfaces in projective space. Geometry of differential equations, 161–176, Amer. Math. Soc. Transl. Ser. 2, 186, Amer. Math. Soc., Providence, RI, 1998.
• 44 Finite type invariants of classical and virtual knots, (joint with M. Goussarov and M. Polyak), Topology 39:5, (2000) 1045-1068; see also arXiv: math. GT/9810073.
• 45 On the Casson knot invariant, (joint with Michael Polyak), J. Knot Theory and Its Ramifications 10:5 (2001) 711–738; see also arXiv:math.GT/9903158.
• 46 Encomplexing the writhe. Topology, ergodic theory, real algebraic geometry , 241–256, Amer. Math. Soc. Transl. Ser. 2, 202, Amer. Math. Soc., Providence, RI, 2001; see also arXiv: math.AG/0005162.
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• 49 What is an amoeba, Notices AMS, 49:8 (2002), 916-917.
• 50 Remarks on definition of Khovanov homology, arXiv:math.GT/0202199 (2002).
• 51 Quantum relatives of Alexander polynomial, arXiv:math.GT/0204290 (2002), St.Petersburg Mathematical Journal , 18:3 (2006) 63–157 (Russian), to be published in English in St.Petersburg Math. J.
• 52 Khovanov homology, its definitions and ramifications, Fund. Math. 184 (2004), 317-342. math.GT/0204290
• 53 Whitney Number of Closed Real Algebraic Affine Curve of Type I, Moscow Mathematical Journal 6:1 (2006); see also arXiv: math.AG/0602256.
• 54 Virtual Links, Orientations of Chord Diagrams and Khovanov Homology, Proceedings of 12th Gökova Geometry-Topology Conference 2005, International Press, 2006, 187–212; see also arXiv: math.GT/0611406.
• 55 Asymptotically Maximal Real Algebraic Hypersurfaces of Projective Space, (joint with Ilia Itenberg) Proceedings of 13th Gökova Geometry-Topology Conference 2006, International Press, 2007, accepted for publication.
• 56 From the sixteenth Hilbert problem to Tropical Geometry, Japan. J. Math , vol. 3 (2008), pp. 1-30
• 57 Elementary Topology Problem Textbook. (joint with O.A.Ivanov, N.Yu.Netsvetaev, V.M.Kharlamov) American Mathematical Society, 2008, 400 pages. ISBN 978-0-8218-4506-6.
• 58 Twisted acyclicity of a circle and signatures of a link, Journal of Knot Theory and Its Ramifications , 18:6 (2009), 729-755.
• 59 Hyperfields for Tropical Geometry I. Hyperfields and dequantization, arXiv:1006.3034.
• 60 On basic concepts of tropical geometry (Russian) Trudy MIAN , 2011, vol. 273, 271-303; English translation in Trudy Matematicheskogo Instituta imeni V.A. Steklova , 2011, Vol. 273, pp. 271-303
• 61 1-manifolds, Bulletin of the Manifold Atlas (2013), Accepted:28th May 2013 http://www.boma.mpim-bonn.mpg.de/.
• 62 Defining relations for reflections, arXiv:1405.1460 [math.MG].
• 63 Biflippers and head to tail composition rules, arXiv:1412.2397. View
• 64 Pictorial calculus for isometries, American Mathematical Monthly, Vol. 123, No. 1, January 2016, 82-87.