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 ] .

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.

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.

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.

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.)

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.

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 ] .

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.