Izvestiya: Mathematics

The structure of proper holomorphic maps with multiplicity higher than one from bounded Reinhardt domains in $\mathbb C^2$ onto two-dimensional complex manifolds is described.

We consider functions $f(T,R)$ of pairs of noncommuting contractions on Hilbert space and study the problem as to which functions $f$ we have Lipschitz type estimates in Schatten–von Neumann norms. We prove that if $f$ belongs to the Besov class $(B_{\infty,1}^1)_+(\mathbb{T}^2)$ of analytic functions in the bidisc, then we have a Lipschitz type estimate for functions $f(T,R)$ of pairs of not necessarily commuting contractions $(T,R)$ in the Schatten–von Neumann norms $\mathbf{S}_p$ for $p\in[1,2]$. On the other hand, we show that for functions in $(B_{\infty,1}^1)_+(\mathbb{T}^2)$, there are no such Lipschitz type estimates for $p>2$, nor in the operator norm.

In the holomorphic version of the inverse scattering method, we prove that the determinant of a Toeplitz-type Fredholm operator arising in the solution of the inverse problem is an entire function of the spatial variable for all potentials whose scattering data belong to a Gevrey class strictly less than 1. As a corollary, we establish that, up to a constant factor, every local holomorphic solution of the Korteweg–de Vries equation is the second logarithmic derivative of an entire function of the spatial variable. We discuss the possible order of growth of this entire function. Analogous results are given for all soliton equations of parabolic type.

The aim of this paper is to describe the structure of finitely generated subgroups of a family of branch groups containing the first Grigorchuk group and the Gupta–Sidki $3$-group. We then use this to show that all the groups in this family are subgroup separable (LERF).

These results are obtained as a corollary of a more general structural statement on subdirect products of just infinite groups.

This paper is the first in a series of three devoted to constructing a finitely presented infinite nilsemigroup satisfying the identity $x^9=0$. This solves a problem of Lev Shevrin and Mark Sapir.

In this first part we obtain a sequence of complexes formed of squares ($4$-cycles) having the following geometric properties.

1) Complexes are uniformly elliptic. A space is said to be uniformly elliptic if there is a constant $\lambda>0$ such that in the set of shortest paths of length $D$ connecting points $A$ and $B$ there are two paths such that the distance between them is at most $\lambda D$. In this case, the distance between paths with the same beginning and end is defined as the maximal distance between the corresponding points.

2) Complexes are nested. A complex of level $n+1$ is obtained from a complex of level $n$ by adding several vertices and edges according to certain rules.

3) Paths admit local transformations. Assume that we can transform paths by replacing a path along two sides of a minimal square by the path along the other two sides. Two shortest paths with the same ends can be transformed into each other locally if these ends are vertices of a square in the embedded complex.

The geometric properties of the sequence of complexes will be further used to define finitely presented semigroups.

We study real Hurwitz numbers enumerating real meromorphic functions of a special kind, referred to as framed purely real functions. We deduce partial differential equations of cut-and-join type for the generating functions for these numbers. We also construct a topological field theory for them.

We classify uniruled compact Kähler threefolds whose groups of bimeromorphic selfmaps do not have the Jordan property.

We prove new results on the derivative of the Minkowski question mark function.

The Torelli group of a closed oriented surface $S_g$ of genus $g$ is the subgroup $\mathcal{I}_g$ of the mapping class group $\operatorname{Mod}(S_g)$ consisting of all mapping classes that act trivially on the homology of $S_g$. One of the most intriguing open problems concerning Torelli groups is the question of whether the group $\mathcal{I}_3$ is finitely presented. A possible approach to this problem relies on the study of the second homology group of $\mathcal{I}_3$ using the spectral sequence $E^r_{p,q}$ for the action of $\mathcal{I}_3$ on the complex of cycles. In this paper we obtain evidence for the conjecture that $H_2(\mathcal{I}_3;\mathbb{Z})$ is not finitely generated and hence $\mathcal{I}_3$ is not finitely presented. Namely, we prove that the term $E^3_{0,2}$ of the spectral sequence is not finitely generated, that is, the group $E^1_{0,2}$ remains infinitely generated after taking quotients by the images of the differentials $d^1$ and $d^2$. Proving that it remains infinitely generated after taking the quotient by the image of $d^3$ would complete the proof that $\mathcal{I}_3$ is not finitely presented.

We prove some properties of real Segre cubics. In particular, we find the topological types of the real parts of Segre cubics as well as the topological types of the real parts of the complements of the Segre planes. We prove some differential-geometric properties of the real parts of real Segre cubics and Kummer quartics. We study the automorphism groups of real Segre cubics and, in particular, their action on the real parts of these cubics.

We prove a multiple coloured Tverberg theorem and a balanced coloured Tverberg theorem, applying different methods, tools and ideas. The proof of the first theorem uses a multiple chessboard complex (as configuration space) and the Eilenberg–Krasnoselskii theory of degrees of equivariant maps for non-free group actions. The proof of the second result relies on the high connectivity of the configuration space, established by using discrete Morse theory.

The intersection of two quadrics is called a biquadric. If we mark a non-singular quadric in the pencil of quadrics through a given biquadric, then the given biquadric is called a marked biquadric. In the classical papers of Plücker and Klein, a Kummer surface was canonically associated with every three-dimensional marked biquadric (that is, with a quadratic line complex provided that the Plücker–Klein quadric is marked). In Reid’s thesis, this correspondence was generalized to odd-dimensional marked biquadrics of arbitrary dimension $\ge 3$. In this case, a Kummer variety of dimension $g$ corresponds to every biquadric of dimension $2g-1$. Reid only constructed the generalized Plücker–Klein correspondence. This map was not studied later. The present paper is devoted to a partial solution of the problem of creating the corresponding theory.

We consider the problem of the optimal start control for two-dimensional Boussinesq equations describing non-isothermal flows of a viscous fluid in a bounded domain. Using the study of the properties of admissible tuples and of the evolution operator, we prove the solubility of the optimization problem under natural assumptions about the model data. We derive a variational inequality which is satisfied by the optimal control provided that the objective functional is determined by the final observation. We also obtain sufficient conditions for the uniqueness of an optimal control.

We construct an analogue of classical Lie theory in the case of Lie groups and Lie algebras defined over the algebra of dual numbers. As an application, we study approximate symmetries of differential equations and construct analogues of Hjelmslev’s natural geometry.

It is well known that the number of homomorphisms from a group $F$ to a group $G$ is divisible by the greatest common divisor of the order of $G$ and the exponent of $F/[F,F]$. We study the question of what can be said about the number of homomorphisms satisfying certain natural conditions like injectivity or surjectivity. A simple non-trivial consequence of our results is the fact that in any finite group the number of generating pairs $(x,y)$ such that $x^3=1=y^5$ is divisible by the greatest common divisor of fifteen and the order of the group $[G,G]\cdot\{g^{15}\mid g\in G\}$.