Russian Mathematical Surveys

This is a survey of the theory of attractors of nonlinear Hamiltonian partial differential equations since its appearance in 1990. Included are results on global attraction to stationary states, to solitons, and to stationary orbits, together with results on adiabatic effective dynamics of solitons and their asymptotic stability, and also results on numerical simulation. The results obtained are generalized in the formulation of a new general conjecture on attractors of $G$-invariant nonlinear Hamiltonian partial differential equations. This conjecture suggests a novel dynamical interpretation of basic quantum phenomena: Bohr transitions between quantum stationary states, de Broglie's wave-particle duality, and Born's probabilistic interpretation.

Bibliography: 212 titles.

The purpose of the survey is to systematize a vast amount of information about the minimal model program for varieties with group actions. We discuss the basic methods of the theory and give sketches of the proofs of some principal results.

Bibliography: 243 titles.

Contents §0. Introduction §1. Abelian problem on the stabilizer §2. Classical models of computations 2.1. Boolean schemes and sequences of operations 2.2. Reversible computations §3. Quantum formalism 3.1. Basic notions and notation 3.2. Transformations of mixed states 3.3. Accuracy §4. Quantum models of computations 4.1. Definitions and basic properties 4.2. Construction of various operators from the elements of a basis 4.3. Generalized quantum control and universal schemes §5. Measurement operators §6. Polynomial quantum algorithm for the stabilizer problem §7. Computations with perturbations: the choice of a model §8. Quantum codes (definitions and general properties) 8.1. Basic notions and ideas 8.2. One-to-one codes 8.3. Many-to-one codes §9. Symplectic (additive) codes 9.1. Algebraic preparation 9.2. The basic construction 9.3. Error correction procedure 9.4. Torus codes §10. Error correction in the computation process: general principles 10.1. Definitions and results 10.2. Proofs §11. Error correction: concrete procedures 11.1. The symplecto-classical case 11.2. The case of a complete basis

Bibliography

The theory of limits of discrete combinatorial objects has been thriving for the last decade or so. The syntactic, algebraic approach to the subject is popularly known as 'flag algebras', while the semantic, geometric approach is often associated with the name 'graph limits'. The language of graph limits is generally more intuitive and expressible, but a price that one has to pay for it is that it is better suited for the case of ordinary graphs than for more general combinatorial objects. Accordingly, there have been several attempts in the literature, of varying degree of generality, to define limit objects for more complicated combinatorial structures. This paper is another attempt at a workable general theory of dense limit objects. Unlike previous efforts in this direction (with the notable exception of [5] by Aroskar and Cummings), our account is based on the same concepts from first-order logic and model theory as in the theory of flag algebras. It is shown how our definitions naturally encompass a host of previously considered cases (graphons, hypergraphons, digraphons, permutons, posetons, coloured graphs, and so on), and the fundamental properties of existence and uniqueness are extended to this more general case. Also given is an intuitive general proof of the continuous version of the Induced Removal Lemma based on the compactness theorem for propositional calculus. Use is made of the notion of open interpretation that often allows one to transfer methods and results from one situation to another. Again, it is shown that some previous arguments can be quite naturally framed using this language.

Bibliography: 68 titles.

We review the integrability of the geodesic flow on a threefold $\mathcal M^3$ admitting one of the three group geometries in Thurston's sense. We focus on the $\operatorname{SL}(2,\mathbb R)$ case. The main examples are the quotients $\mathcal M^3_\Gamma=\Gamma\backslash \operatorname{PSL}(2,\mathbb R)$, where $\Gamma \subset \operatorname{PSL}(2,\mathbb R)$ is a cofinite Fuchsian group. We show that the corresponding phase space $T^*\mathcal M_\Gamma^3$ contains two open regions with integrable and chaotic behaviour, with zero and positive topological entropy, respectively.

As a concrete example we consider the case of the modular threefold with the modular group $\Gamma=\operatorname{PSL}(2,\mathbb Z)$. In this case $\mathcal M^3_\Gamma$ is known to be homeomorphic to the complement of a trefoil knot $\mathcal K$ in a 3-sphere. Ghys proved the remarkable fact that the lift of a periodic geodesic on the modular surface to $\mathcal M^3_\Gamma$ produces the same isotopy class of knots as that which appears in the chaotic version of the celebrated Lorenz system and was studied in detail by Birman and Williams. We show that these knots are replaced by trefoil knot cables in the integrable limit of the geodesic system on $\mathcal M^3_\Gamma$.

Bibliography: 60 titles.

The practice of bringing together the concepts of 'Newton polytopes', 'toric varieties', 'tropical geometry', and 'Gröbner bases' has led to the formation of stable and mutually beneficial connections between algebraic geometry and convex geometry. This survey is devoted to the current state of the area of mathematics that describes the interaction and applications of these concepts.

Bibliography: 68 titles.

The Zamolodchikov tetrahedron equation inherits almost all the richness of structures and topics in which the Yang–Baxter equation is involved. At the same time, this transition symbolizes the growth of the order of the problem, the step from the Yang–Baxter equation to the local Yang–Baxter equation, from the Lie algebra to the 2-Lie algebra, from ordinary knots in $\mathbb{R}^3$ to 2-knots in $\mathbb{R}^4$. These transitions are followed in several examples, and there are also discussions of the manifestation of the tetrahedron equation in the long-standing question of integrability of the three-dimensional Ising model and a related model of neural network theory: the Hopfield model on a two-dimensional lattice.

Bibliography: 82 titles.

In this paper we introduce the concept of a Gibbs measure, which generalizes the concept of an equilibrium Gibbs distribution in statistical physics. The new concept is important in the study of Anosov dynamical systems. By means of this concept we construct a wide class of invariant measures for dynamical systems of this kind and investigate the problem of the existence of an invariant measure consistent with a smooth structure on the manifold; we also study the behaviour under small random excitations as . The cases of discrete time and continuous time are treated separately.

CONTENTS Introduction § 1. Classical statistical physics on a two-dimensional lattice and quantum mechanics on a chain § 2. Connection with the inverse problem method § 3. The six-vertex model § 4. Generating vectors and permutation relations § 5. The general Bethe Ansatz § 6. Integral equations Conclusion Appendix 1 Appendix 2 References

CONTENTS Introduction Chapter I. General properties of walks on an inhomogeneous lattice § 1. The invariant measure and the law of large numbers § 2. The Lindeberg-Brown theorem and its corollaries § 3. Time dependence. The weak central limit theorem § 4. Ergodic theorems for fields with stationary increments. L_p(Ω)-estimates Chapter II. The central limit theorem for specific types of walks § 1. Reflective symmetry § 2. Symmetric walks § 3. Two-fold stochastic walks § 4. Symmetrizable walks § 5. Walks on manifolds § 6. Further results. Unsolved problems References

This paper is a survey of recent effective results in the theory of birational rigidity of higher-dimensional Fano varieties and Fano–Mori fibre spaces.

Bibliography: 59 titles.

This paper focuses on Richard Thompson's group $F$, which was discovered in the 1960s. Many papers have been devoted to this group. We are interested primarily in the famous problem of amenability of this group, which was posed by Geoghegan in 1979. Numerous attempts have been made to solve this problem in one way or the other, but it remains open.

In this survey we describe the most important known properties of this group related to the word problem and representations of elements of the group by piecewise linear functions as well as by diagrams and other geometric objects. We describe the classical results of Brin and Squier concerning free subgroups and laws. We include a description of more modern important results relating to the properties of the Cayley graphs (the Belk–Brown construction) as well as Bartholdi's theorem about the properties of equations in group rings. We consider separately the criteria for (non-)amenability of groups that are useful in the work on the main problem. At the end we describe a number of our own results about the structure of the Cayley graphs and a new algorithm for solving the word problem.

Bibliography: 69 titles.

In this survey we describe the history and current state of one of the key areas in the qualitative theory of elliptic partial differential equations related to the strong maximum principle and the boundary point principle (normal derivative lemma).

Bibliography: 234 titles.

We provide local formulae for the pressure of incompressible fluids. The pressure can be expressed in terms of its average and averages of squares of velocity increments in arbitrarily small neighbourhoods. As an application, we give a brief proof of the fact that $C^{\alpha}$ velocities have $C^{2\alpha}$ (or Lipschitz) pressures. We also give some regularity criteria for 3D incompressible Navier–Stokes equations.

Bibliography: 9 titles.

An explicit Lyapunov function is constructed for scalar parabolic reaction-advection-diffusion equations under periodic boundary conditions. The non-linearity is assumed to be even with respect to the advection term. The method followed was originally suggested by H. Matano for, and limited to, separated boundary conditions.

Bibliography: 20 titles.

This paper is devoted to boundary layer theory for singularly perturbed convection-diffusion equations in the unit circle. Two characteristic points appear, $(\pm 1,0)$, in the context of the equations considered here, and singularities may occur at these points depending on the behaviour there of a given function $f$, namely, the flatness or compatibility of $f$ at these points as explained below. Two previous articles addressed two particular cases: [24] dealt with the case where the function $f$ is sufficiently flat at the characteristic points, the so-called compatible case; [25] dealt with a generic non-compatible case ($f$ polynomial). This survey article recalls the essential results from those papers, and continues with the general case ($f$ non-flat and non-polynomial) for which new specific boundary layer functions of parabolic type are introduced in addition.

Bibliography: 49 titles.

This is a survey of basic facts presently known about non-linear Poisson structures in the analysis of integrable systems in non-holonomic mechanics. It is shown that by using the theory of Poisson deformations it is possible to reduce various non-holonomic systems to dynamical systems on well-understood phase spaces equipped with linear Lie–Poisson brackets. As a result, not only can different non-holonomic systems be compared, but also fairly advanced methods of Poisson geometry and topology can be used for investigating them.

Bibliography: 95 titles.

Consider a quantum three-particle system consisting of two fermions of unit mass and another particle of mass $m\gt 0$ interacting in a point-like manner with the fermions. Such systems are studied here using the theory of self-adjoint extensions of symmetric operators: the Hamiltonian of the system is constructed as an extension of the symmetric energy operator

$$\begin{equation*} H_0=-\frac{1}{2}\biggl(\frac{1}{m}\Delta_y+\Delta_{x_1}+\Delta_{x_2}\biggr), \end{equation*}$$

$L_2(\mathbb{R}^3)\otimes L_2^{\operatorname{asym}}(\mathbb{R}^3\times\mathbb{R}^3)$

$H_0$

$\{T_l,\ l=0,1,2,\dots\}$

$L_2(\mathbb{R}^3)$

$T_l$

$l$

$l$

$0\lt m_l^{(1)} \lt m_l^{(2)} \lt \infty$

$T_l$

$m \gt m_l^{(2)}$

$m\leqslant m_l^{(2)}$

$m\in[m_l^{(1)}, m_l^{(2)}]$

$T_l$

$\mathbb{R}^3$

$0 \lt m \lt m_l^{(1)}$

$\{\lambda_n\}$

$2l+1$

$\lambda_n\to-\infty$

$n\to\infty$

$H_0$

$P^2/(2(m+2))+z_n$

$z_n<0$

Bibliography: 19 titles.