Planarizations and maps taking lines to linear webs of conics.
[Arxiv]
Preprint.
Aiming at a generalization of a classical theorem of Moebius, we study maps that take line intervals to plane curves, and also maps that take line intervals to conics from certain linear systems. 

Topological polynomials with a simple core
(joint with A. Blokh, L. Oversteegen and R. Ptacek).
[Arxiv]
Preprint.
We define the (dynamical) core of a topological polynomial (and the associated lamination). This notion extends that of the core of a unimodal interval map. Two explicit descriptions of the core are given: one related to periodic objects and one related to critical objects. We describe all laminations associated with quadratic and cubic topological polynomials with a simple core (in the quadratic case, these correspond precisely to points on the Main Cardioid of the Mandelbrot set). 

Schubert calculus and GelfandZetlin polytopes
(joint with V. Kiritchenko and E. Smirnov).
[Arxiv]
Preprint.
We describe a new approach to the Schubert calculus on complete flag varieties using the volume polynomial associated with GelfandZetlin polytopes. This approach allows us to compute the intersection products of Schubert cycles by intersecting faces of a polytope. 

Partial holomorphic semiconjugacies between rational functions.
[Arxiv]
Preprint.
We establish a general result on the existence of partially defined semiconjugacies between rational functions acting on the Riemann sphere. The semiconjugacies are defined on the complements to at most onedimensional sets. They are holomorphic in a certain sense. 

Polytopes and Equations
(joint with A. Khovanskii). (in Russian)
[PDF]
Mat. Prosv. Ser. 3, Volume 14, 2010 pp. 30—57


Sylvester's problem
(joint with Sergei Tabachnikov). (in Russian)
[PDF]
Kvant, Volumes 5 and 6, pp. 2—6 and 6—9, 2009


Moore's theorem.
[PDF]
DRAFT version 06.25.09. In this expository paper, we review a proof of the following old theorem of R.L. Moore: for a closed equivalence relation on the 2sphere such that all equivalence classes are connected and nonseparating, and not all points are equivalent, the quotient space is homeomorphic to the 2sphere. The proof uses a general topological theory close to but simpler than an original theory of Moore. The exposition is organized so that to make applications of Moore's theory (not only Moore's theorem) in complex dynamics easier, although no dynamical applications are mentioned here. 

On partial semiconjugacies of quadratic polynomials.
[PDF]
Preprint.
The notion of holomorphic function can be generalized to the case of a function, whose domain is not open. We will construct a function defined and "holomorphic" on the complement to countably many disjoint simple curves and "semiconjugating" a given quadratic polynomial with connected Julia set and accessible critical value to a hyperbolic quadratic polynomial with totally disconnected Julia set. 

Topological regluing of rational functions.
[PDF],
Inventiones Mathematicae, 2009. Volume 179. No 3. pp. 461—506
Regluing is a topological operation that helps to construct topological models for rational functions on the boundaries of certain hyperbolic components. It also has a holomorphic interpretation, with the flavor of infinite dimensional ThurstonTeichmueller theory. We will discuss a topological theory of regluing, and just trace a direction, in which a holomorphic theory can develop. 

External boundary of M_{2}.
Fields Institute Communications Volume 53: Holomorphic
Dynamics and Renormalization A Volume in Honour of John Milnor's 75th
Birthday
[PDF]
The dynamics of a rational function on the Riemann sphere is determined to a large extend by the behavior of the critical points. Quadratic rational functions have 2 critical points, therefore, people consider parameter slices of quadratic rational functions with a specified simple behavior of one critical point (each slice is then parameterized by the position of the second, ``free'', critical point). The slice corresponding to a fixed critical point consists of quadratic polynomials, and this is the most studied situation. The corresponding bifurcation locus M_{1}  the Mandelbrot set  is of fundamental importance in all holomorphic dynamics. The next slice consists of quadratic rational functions with a critical point of period 2. The bifurcation locus M_{2} in this slice shows many features that distinguish rational functions from quadratic polynomials. The external boundary of M_{2} (the boundary of a distinguished complementary component) is described, in particular, it is shown that this boundary is a topological circle, and simple topological models are given for all maps on this boundary. 

Rectifiable pencils of conics.
Moscow Mathematical Journal 7 (2007), no. 3, 561570
[PDF]
We describe analytic pencils of conics passing through the origin in C^{2} that can be mapped to straight lines locally near the origin by an analytic diffeomorphism. Under a minor nondegeneracy assumption, we prove that in a pencil with this property, almost all conics have 3 points of tangency with the same algebraic curve of class 3 (i.e. a curve projectively dual to a cubic). 

Variations on the TaitKneser theorem
(joint with Sergei Tabachnikov).
Preprint
[PDF]
The classical TaitKneser theorem states that the osculating circles of a smooth plane curve, free from curvature extrema, are pairwise disjoint. We prove a number of analogs of this theorem, e.g. for ovals of osculating cubics, osculating polynomials and trigonometric polynomials; in each case, we will obtain a nondifferentiable foliation with smooth leaves. 

Quadratic Mathematics (in Russian).
Preprint
[PDF]
Lecture notes of a short course for senior highschool students and 1st year undergraduate students given in Dubna, 2005 

On binary quadratic forms with semigroup property (joint work with F. Aicardi ).
To appear in Proceedings of Steklov Institute
dedicated to the 70th birthday of V. Arnold
[PDF]
A quadratic form f is said to have semigroup property if its values at points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all binary integer quadratic forms with semigroup property. If there is an integer bilinear map s such that f(s(x,y))=f(x)f(y) for all vectors x and y from the integer 2dimensional lattice, then the form f has semigroup property. We give an explicit description of all pairs (f,s) with the property stated above. We do not know any other examples of forms with semigroup property. It turns out that certain pairs (f,s are closely related with order 3 elements in class groups. 

Maps That Take Lines To Circles, in Dimension 4.
Functional Analysis and its Applications
40 (2006), no. 2, 108116
[PDF]
We list all diffeomorphisms between an open subset of the 4dimensional projective space and an open subset of the 4dimensional sphere that take all line segments to arcs of round circles. These are the following: restrictions of the quaternionic Hopf fibrations and projections from a hyperplane to a sphere from some point. 

Circles and quadratic maps between spheres. Geometriae Dedicata (2005) 115, pp. 1932
[PDF]
Consider an analytic map from a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of lines passing through 0 to germs of circles. We call such map a rounding. Two roundings are equivalent if they take the same lines to the same circles. We prove that any rounding whose differential at 0 has rank at least 2 gives rise to a quadratic map between spheres. Results of P. Yiu on quadratic maps between spheres have some interesting implications concerning roundings. 

Circles and Clifford algebras.
Funktsional. Anal. i Prilozhen. 38 (2004), No. 1, 5664.
[PDF]
Consider a smooth map from a neighborhood of 0 in a real vector space to a neighborhood of 0 in a Euclidean space. Suppose that this map takes all germs of lines passing through 0 to germs of Euclidean circles, or lines, or points. We prove that under some simple additional assumptions this map takes all vector lines to the same circles as a Hopf map coming from a representation of a Clifford algebra. We also describe a connection between our result and the HurwitzRadon theorem about sums of squares. 

Kahler metrics whose geodesics are circles.
Proceedings of the Conference "Fundamental
Mathematics Today", Ed. S.K. Lando and O.K. Sheinman,
pp. 284293
[PDF]
We classify all Kahler metrics in an open subset of C^{2} whose real geodesics are circles. All such metrics are equivalent (via complex projective transformations) to Fubini metrics (i.e. to FubiniStudy metric on CP^{2} restricted to an affine chart, to the complex hyperbolic metric in the unit ball model or to the Euclidean metric). 

Combinatorics of Convex Polytopes.
MCCME Moscow, 2002 Introductory lecture notes for 1styear undergraduate students 

Rectification of circles and quaternions.
Michigan Mathematical Journal, 51 (2003), 153  167
[PDF]
Consider a pencil of circles passing through 0 in 4dimensional space. It is said to be rectifiable if there is a germ of diffeomorphism at 0 that takes all circles from this pencil to straight lines. We will give a classification of all rectifiable pencils of circles containing sufficiently many circles in general position. This result is surprisingly different from those in dimensions 2 and 3 (Khovanskii and Izadi) due to a connection with the quaternionic algebra. 

On polytopes that are simple at the edges.
Functional Analysis and its Applications, 35, No. 3 (2001), 189  198 The volume polynomial technique is used to prove some geometric inequalities for polytopes simple at the edges. In particular, we generalize Khovanskii's inequalities and prove a partial case of Staley's conjecture. 

An analogue of the HodgeRiemann relations for simple polytopes.
Russian Mathematical Surveys, 54, No.2 (1999), 381  426 We prove that algebra constructed by the volume polynomial of a simple polytope (introduced by Pukhlikov and Khovanskii) has analogues of all Kahler package theorems. This gives a simplification of McMullen's geometric proof of the gtheorem. Moreover, analogues of Hard Lefschetz theorem and the HodgeRiemann bilinear relations are formulated as inequalities on the volume polynomial and proved in terms of the volume polynomial only. 

Mixed HodgeRiemann relations in linear context.
Functional Analysis and its Applications, 32, No. 4 (1999), 268  272 We prove mixed linear analogues of the HodgeRiemann bilinear relations. They give a generalization of the Aleksandrov inequalities for mixed discriminants. 