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Vol 74, No 1 (2019)
- Year: 2019
- Articles: 7
- URL: https://journals.rcsi.science/0042-1316/issue/view/7508
Introduction to Heegaard Floer homology
Abstract
Heegaard Floer homology is an invariant of knots, links, and 3-manifolds introduced by Ozsvath and Szabo about 15 years ago. This survey defines Heegaard Floer homology and describes its basic properties. Also discussed is the relation between Heegaard Floer homology and invariants of singularities of curves and surfaces.Bibliography: 72 titles.
Uspekhi Matematicheskikh Nauk. 2019;74(1):3-40
3-40
Classification of Morse–Smale systems and topological structure of the underlying manifolds
Abstract
Morse–Smale systems arise naturally in applications for mathematical modelling of processes with regular dynamics (for example, in chains of coupled maps describing diffusion reactions, or in the study of the topology of magnetic fields in a conducting medium, in particular, in the study of the question of existence of separators in magnetic fields of highly conducting media). Since mathematical models in the form of Morse–Smale systems appear in the description of processes of various nature, the first step in the study of such models is to distinguish properties independent of the physical context but determining a partition of the phase space into trajectories. The relation preserving the partition into trajectories up to a homeomorphism is called topological equivalence, and the relation preserving also the time of motion along trajectories (continuous in the case of flows, and discrete in the case of cascades) is called topological conjugacy. The problem of topological classification of dynamical systems consists in finding invariants that uniquely determine the equivalence class or the conjugacy class for a given system. The present survey is devoted to a description of results on topological classification of Morse–Smale systems on closed manifolds, including results recently obtained by the authors. Also presented are recent results of the authors concerning the interconnections between the global dynamics of such systems and the topological structure of the underlying manifolds.Bibliography: 112 titles.
Uspekhi Matematicheskikh Nauk. 2019;74(1):41-116
41-116
Tensor invariants and integration of differential equations
Abstract
The connection between tensor invariants of systems of differential equations and explicit integration of them is discussed. A general result on the integrability of dynamical systems admitting a complete set of integral invariants in the sense of Cartan is proved. The existence of an invariant 1-form is related to the representability of the dynamical system in Hamiltonian form (with a symplectic structure which may be degenerate). This general idea is illustrated using an example of linear systems of differential equations. A general concept of flags of tensor invariants is introduced. General relations between the Kovalevskaya exponents of quasi-homogeneous systems of differential equations and flags of quasi-homogeneous tensor invariants having a certain structure are established. Results of a general nature are applied, in particular, to show that the general solution of the equations of rotation for a rigid body is branching in the Goryachev–Chaplygin case.Bibliography: 50 titles.
Uspekhi Matematicheskikh Nauk. 2019;74(1):117-148
117-148
Topological integrability, classical and quantum chaos, and the theory of dynamical systems in the physics of condensed matter
Abstract
This survey is devoted to questions connected with the Novikov problem of describing the geometry of level curves of quasi-periodic functions on the plane with different numbers of quasi-periods. Considered here are the history of the question, the current state of research in this field, and a number of applications of this problem to various physical problems. The main focus is on applications of results obtained in this area to the theory of transport phenomena in electron systems.Bibliography: 56 titles.
Uspekhi Matematicheskikh Nauk. 2019;74(1):149-184
149-184
On a question of Gromov concerning the generalized Liouville theorem
Uspekhi Matematicheskikh Nauk. 2019;74(1):185-186
185-186
Free algebras of Hilbert automorphic forms
Uspekhi Matematicheskikh Nauk. 2019;74(1):187-188
187-188
Boris Rufimovich Vainberg (on his 80th birthday)
Uspekhi Matematicheskikh Nauk. 2019;74(1):189-194
189-194