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卷 76, 编号 4 (2021)
- 年: 2021
- 文章: 8
- URL: https://bakhtiniada.ru/0042-1316/issue/view/7523
Chaos and integrability in $\operatorname{SL}(2,\mathbb R)$-geometry
摘要
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.
Uspekhi Matematicheskikh Nauk. 2021;76(4):3-36
3-36
Integrable polynomial Hamiltonian systems and symmetric powers of plane algebraic curves
摘要
This survey is devoted to integrable polynomial Hamiltonian systems associated with symmetric powers of plane algebraic curves.We focus our attention on the relations (discovered by the authors) between the Stäckel systems, Novikov's equations for the $g$th stationary Korteweg–de Vries hierarchy, the Dubrovin–Novikov coordinates on the universal bundle of Jacobians of hyperelliptic curves, and new systems obtained by considering the symmetric powers of curves when the power is not equal to the genus of the curve.Bibliography: 52 titles.
Uspekhi Matematicheskikh Nauk. 2021;76(4):37-104
37-104
Dynamical $\mathfrak{sl}_2$ Bethe algebra and functions on pairs of quasi-polynomials
摘要
We consider the space $\operatorname{Fun}_{\mathfrak{sl}_2}V[0]$ of functions on the Cartan subalgebra of $\mathfrak{sl}_2$ with values in the zero weight subspace $V[0]$ of a tensor product of irreducible finite-dimensional $\mathfrak{sl}_2$-modules. We consider the algebra $\mathcal B$ of commuting differential operators on $\operatorname{Fun}_{\mathfrak{sl}_2}V[0]$, constructed by Rubtsov, Silantyev, and Talalaev in 2009. We describe the relations between the action of $\mathcal B$ on $\operatorname{Fun}_{\mathfrak{sl}_2}V[0]$ and spaces of pairs of quasi-polynomials.Bibliography: 25 titles.
Uspekhi Matematicheskikh Nauk. 2021;76(4):105-138
105-138
Tetrahedron equation: algebra, topology, and integrability
摘要
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.
Uspekhi Matematicheskikh Nauk. 2021;76(4):139-176
139-176
Multipoint formulae for inverse scattering at high energies
Uspekhi Matematicheskikh Nauk. 2021;76(4):177-178
177-178
Multilevel interpolation for Nikishin systems and boundedness of Jacobi matrices on binary trees
Uspekhi Matematicheskikh Nauk. 2021;76(4):179-180
179-180
Groups generated by involutions, numberings of posets, and central measures
Uspekhi Matematicheskikh Nauk. 2021;76(4):181-182
181-182
Igor' Moiseevich Krichever (on his 70th birthday)
Uspekhi Matematicheskikh Nauk. 2021;76(4):183-193
183-193