Showing papers in "Russian Mathematical Surveys in 2016"

Operator Lipschitz functions

Abstract: The purpose of this survey article is a comprehensive study of operator Lipschitz functions. A continuous function $f$ on the real line ${\Bbb R}$ is called operator Lipschitz if $\|f(A)-f(B)\|\le{\rm const}\|A-B\|$ for arbitrary self-adjoint operators $A$ and $B$. We give sufficient conditions and necessary conditions for operator Lipschitzness. We also study the class of operator differentiable functions on ${\Bbb R}$. Then we consider operator Lipschitz functions on closed subsets of the plane as well as commutator Lipschitz functions on such subsets. Am important role is played by double operator integrals and Schur multipliers.

Perturbations of self-adjoint and normal operators with discrete spectrum

Abstract: The spectral properties of operators of the form A = T +B are analyzed, where B is a non-symmetric operator subordinate to a self-adjoint or normal operator T . The different definitions of perturbations with respect to T are considered: completely subordinated, subordinate with order p < 1, locally subordinate. Analogues of these types of perturbations are considered also for operators defined in terms of quadratic forms. For perturbations of different types, series of statements on the completeness property of the root vectors of the operator and on the basis or unconditional basis property are proved. The spectra of the operators T and T +B are compared as well. A survey of research in this area is presented. Bibliography: 89 titles.

Operator estimates in homogenization theory

Abstract: This paper gives a systematic treatment of two methods for obtaining operator estimates: the shift method and the spectral method. Though substantially different in mathematical technique and physical motivation, these methods produce basically the same results. Besides the classical formulation of the homogenization problem, other formulations of the problem are also considered: homogenization in perforated domains, the case of an unbounded diffusion matrix, non-self-adjoint evolution equations, and higher-order elliptic operators. Bibliography: 62 titles.

Dispersion estimates for one-dimensional Schrödinger and Klein-Gordon equations revisited

Abstract: It is shown that for a one-dimensional Schrodinger operator with a potential whose first moment is integrable the elements of the scattering matrix are in the unital Wiener algebra of functions with integrable Fourier transforms. This is then used to derive dispersion estimates for solutions of the associated Schrodinger and Klein-Gordon equations. In particular, the additional decay conditions are removed in the case where a resonance is present at the edge of the continuous spectrum. Bibliography: 29 titles.

Proximity of probability distributions in terms of Fourier-Stieltjes transforms

Abstract: A survey is given of some results on smoothing inequalities for various probability metrics (in particular, for the Kolmogorov distance), and some analogues of these results in the class of functions of bounded variation are presented. Bibliography: 61 titles.

Homotopy theory in toric topology

Abstract: In toric topology, to each simplicial complex K on m vertices one associates two key spaces, the Davis-Januskiewicz space DJK and the moment-angle complex ZK , which are related by a homotopy fibration ZK w̃ −→ DJK −→ ∏m i=1 CP∞. A great deal of work has been done to study properties of DJK and ZK , their generalisations to polyhedral products, and applications to algebra, combinatorics and geometry. In the first part of this paper we survey some of the main results in the study of the homotopy theory of these spaces. In the second part we break new ground by initiating a study of the map w̃. We show that, for a certain family of simplicial complexes K, the map w̃ is a sum of higher and iterated Whitehead products.

Lax operator algebras and integrable systems

Abstract: A new class of infinite-dimensional Lie algebras, called Lax operator algebras, is presented, along with a related unifying approach to finite-dimensional integrable systems with a spectral parameter on a Riemann surface such as the Calogero-Moser and Hitchin systems. In particular, the approach includes (non-twisted) Kac-Moody algebras and integrable systems with a rational spectral parameter. The presentation is based on quite simple ideas about the use of gradings of semisimple Lie algebras and their interaction with the Riemann-Roch theorem. The basic properties of Lax operator algebras and the basic facts about the theory of the integrable systems in question are treated (and proved) from this general point of view. In particular, the existence of commutative hierarchies and their Hamiltonian properties are considered. The paper concludes with an application of Lax operator algebras to prequantization of finite-dimensional integrable systems. Bibliography: 51 titles.

Self-adjoint commuting differential operators of rank two

Abstract: This is a survey of results on self-adjoint commuting ordinary differential operators of rank two. In particular, the action of automorphisms of the first Weyl algebra on the set of commuting differential operators with polynomial coefficients is discussed, as well as the problem of constructing algebro-geometric solutions of rank l > 1 of soliton equations. Bibliography: 59 titles.

On manifolds defined by 4-colourings of simple 3-polytopes

Abstract: Let $\mathcal{P}$ be the class of combinatorial 3-dimensional simple polytopes $P$, different from a tetrahedron, without 3- and 4-belts of facets. By the results of Pogorelov and Andreev, a polytope $P$ admits a realisation in Lobachevsky space $\mathbb{L}^3$ with right dihedral angles if and only if $P \in \mathcal{P}$. We consider two families of smooth manifolds defined by regular 4-colourings of Pogorelov polytopes P: six-dimensional quasitoric manifolds over $P$ and three-dimensional small covers of $P$; the latter are also known as three-dimensional hyperbolic manifolds of Loebell type. We prove that two manifolds from either of the families are diffeomorphic if and only if the corresponding 4-colourings are equivalent.

Thick morphisms of supermanifolds and oscillatory integral operators

Abstract: We show that thick morphisms (or microformal morphisms) between smooth (super)manifolds, introduced by us before, are classical limits of 'quantum thick morphisms' defined here as particular oscillatory integral operators on func- tions. In (3, 4) we introduced nonlinear pullbacks of functions with respect to 'micro- formal' or 'thick' morphisms of (super)manifolds, which generalize ordinary smooth maps. By definition, such a morphism is a formal canonical relation between the cotangent bundles of a special kind, namely, specified by a generating function de- pending on position coordinates on the source and momentum coordinates on the target. This function is seen as a power expansion near the zero section. Thick morphisms form a formal category; that means that the composition law for the gen- erating functions is a formal power series. Likewise, the pullback of a function w.r.t. a thick morphism is given by a formal power series whose terms are nonlinear dif- ferential operators. There is a parallel construction based on anticotangent bundles yielding nonlinear pullbacks of odd functions (the former construction applies to even functions). Our main application was to L1-morphisms between homotopy Schouten or Poisson algebras of functions. Another application was the construction of an 'adjoint operator' for nonlinear maps of vector bundles. (Thick morphisms in the even version are close to symplectic micromorphisms of Cattaneo-Dherin-Weinstein, see (1) and subsequent works, defined as germs of canonical relations between germs of symplectic manifolds at Lagrangian submanifolds; analogs of our pullbacks do not arise in such a setting. See further discussion of this in (4).) We show here that thick morphisms of microformal geometry can be seen as the classical limit of certain 'quantum thick morphisms', which are given by oscillatory integral operators of a particular kind. Let us point out that oscillatory integral operators (and Fourier integral operators) are well known, as well as well known is their connection with canonical relations between cotangent bundles. Roughly, each such relation defines a class of Fourier integral operators (see, e.g., (2)). We, however, consider a very special integral operator in this class, generalizing the operator of pullback w.r.t. a smooth map. It is defined by a 'quantum' version of a generating function specifying a thick morphism. 'Quantum' here mean depending on ~. The action of such operators on oscillatory wave functions in the classical limit exactly reproduces the nonlinear pullback of (3, 4). The same holds true for the composition of our operators: in the classical limit it reduces to the composition of thick morphisms.

Topics in sub-Riemannian geometry

Abstract: Sub-Riemannian geometry is the geometry of spaces with nonholonomic constraints. This paper presents an informal survey of some topics in this area, starting with the construction of geodesic curves and ending with a recent definition of curvature. Bibliography: 28 titles.

Gluing of categories and Krull-Schmidt partners

Abstract: We discuss Krull--Schmidt partners for DG categories and produce bunches of them for smooth projective varieties.

On embedding Morse–Smale diffeomorphisms on the sphere in topological flows

Abstract: One important indicator of the adequacy of a numerical solution of an autonomous system of differential equations is the topological conjugacy of the discrete model obtained to the time-one shift map of the original flow. The most significant results in this direction have been obtained for structurally stable flows. In particular, it was shown in [1] and [2] that the Runge–Kutta discretization of a Morse–Smale flow (n > 2) without periodic trajectories on the n-disk is topologically conjugate to the time-one shift (for a sufficiently small step size). In this connection the question, going back to Palis [3], of necessary and sufficient conditions for embedding a Morse–Smale diffeomorphism in a topological flow arises naturally. Recall that a diffeomorphism f on a closed manifold M is called a Morse–Smale diffeomorphism if its non-wandering set Ωf is finite and consists of hyperbolic periodic points, and for any two points p, q ∈ Ωf the intersection of the stable manifold W s p of p and the unstable manifold W u q of q is transversal. In [3] the following necessary conditions for embedding a Morse–Smale diffeomorphism f : M → M in a topological flow were stated, and we call them the Palis conditions: 1) the non-wandering set Ωf coincides with the set of fixed points; 2) the restriction of the diffeomorphism f to each invariant manifold of each fixed point p ∈ Ωf preserves its orientation; 3) if for any distinct saddle points p, q ∈ Ωf the intersection W s p ∩W u q is non-empty, then it contains no compact connected components. According to [3], in the case when n = 2 these conditions are not only necessary but also sufficient. In [4] examples of Morse–Smale diffeomorphisms on the three-dimensional sphere were constructed that satisfy the Palis conditions but do not embed in topological flows, and also necessary and sufficient conditions were obtained for embedding a three-dimensional Morse–Smale diffeomorphism in a topological flow. An additional obstruction to embedding such diffeomorphisms in topological flows is connected with the possibility of a non-trivial embedding of the separatrices of saddle points in the ambient manifold. In the present paper we show that for the class G(S) of Morse–Smale diffeomorphisms without heteroclinic intersections defined on the sphere S of dimension n > 4 and satisfying the Palis conditions no such obstruction exists and the following theorem holds.

Moutard type transformation for matrix generalized analytic functions and gauge transformations

Abstract: A Moutard type transform for matrix generalized analytic functions is derived. Relations between Moutard type transforms and gauge transforms are demonstrated.