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This paper discusses an algorithmic way of constructing integrable evolution equations based on Lie algebraic structure. We derive, in a pedagogical style, a large class of two-component peakon type dual systems from their two-component soliton equations counter part. We study the essential aspects of Hamiltonian flows on coadjoint orbits of the centrally extended semidirect product group to give a systematic derivation of the dual counter parts of various two-component of integrable systems, viz., the dispersive water wave equation, the Kaup–Boussinesq system and the Broer–Kaup system, using moment of inertia operators method and the (frozen) Lie–Poisson structure. This paper essentially gives Lie algebraic explanation of Olver–Rosenau's paper [31].

We show that an n-dimensional real ellipsoid in ℝⁿ⁺¹ with the induced Riemannian metric does not admit an unbounded adapted complexification in the sense of Lempert/Szőke and Guillemin/Stenzel, unless it is a round sphere. In other words, an ellipsoid whose (maximal) Grauert tube has infinite radius must be a round sphere. For the proof we take advantage of the integrability of the geodesic flow and use a classical theorem on umbilic geodesics. We carry out an extension of this result to Liouville metrics elsewhere.

Suppose that $\Sigma$ is a closed and oriented surface equipped with a Riemannian metric. In the literature, there are three seemingly distinct constructions of open books on the unit (co)tangent bundle of $\Sigma$, having complex, contact and dynamical flavors, respectively. Each one of these constructions is based on either an admissible divide or an ordered Morse function on $\Sigma$. We show that the resulting open books are pairwise isotopic provided that the ordered Morse function is adapted to the admissible divide on $\Sigma$. Moreover, we observe that if $\Sigma$ has positive genus, then none of these open books are planar and furthermore, we determine the only cases when they have genus one pages.

Every oriented closed geodesic on the modular surface has a canonically associated knot in its unit tangent bundle coming from the periodic orbit of the geodesic flow. We study the volume of the associated knot complement with respect to its unique complete hyperbolic metric. We show that there exist sequences of closed geodesics for which this volume is bounded linearly in terms of the period of the geodesic’s continued fraction expansion. Consequently, we give a volume’s upper bound for some sequences of Lorenz knots complements, linearly in terms of the corresponding braid index.

Also, for any punctured hyperbolic surface we give volume’s bounds for the canonical lift complement relative to some sequences of sets of closed geodesics in terms of the geodesics length.

We prove that several evolution equations arising as mathematical models for fluid motion cannot be realized as metric Euler equations on the Lie group DIFF^∞(𝕊¹) of all smooth and orientation-preserving diffeomorphisms on the circle. These include the quasi-geostrophic model equation, cf. [A. Córdoba, D. Córdoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. 162 (2005) 1377–1389], the axisymmetric Euler flow in ℝ^d (see [H. Okamoto and J. Zhu, Some similarity solutions of the Navier–Stokes equations and related topics, Taiwanese J. Math. 4 (2000) 65–103]), and De Gregorio's vorticity model equation as introduced in [S. De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation, J. Stat. Phys. 59 (1990) 1251–1263].

This paper studies the application of the Jacobi–Eisenhart lift, Jacobi metric and Maupertuis transformation to the Kepler system. We start by reviewing fundamentals and the Jacobi metric. Then we study various ways to apply the lift to Kepler-related systems: first as conformal description and Bohlin transformation of Hooke’s oscillator, second in contact geometry and third in Houri’s transformation [T. Houri, Liouville integrability of Hamiltonian systems and spacetime symmetry (2016), www.geocities.jp/football_physician/publication.html], coupled with Milnor’s construction [J. Milnor, On the geometry of the Kepler problem, Am. Math. Mon.90 (1983) 353–365] with eccentric anomaly.

We show that the geodesic flow on the infinite-dimensional group of diffeomorphisms of the circle, endowed with a fractional Sobolev metric at the identity, is described by the generalized Constantin–Lax–Majda equation with parameter a = -½.

We extend the Series [The modular surface and continued fractions, J. London Math. Soc. (2) 31(1) (1985) 69–80] connection between the modular surface ${ℳ}=PSL(2,\mathbb{Z})\backslash ℍ$, cutting sequences, and regular continued fractions to the slow converging Lehner and Farey continued fractions with digits $(1,+1)$ and $(2,-1)$ in the notation used for the Lehner continued fractions. We also introduce an alternative insertion and singularization algorithm for Farey expansions and other non-semiregular continued fractions, and an alternative dual expansion to the Farey expansions so that $\frac{dxdy}{{(1+xy)}^2}$ is invariant under the natural extension map.

We prove that on a surface of negative Euler characteristic, two real-analytic Finsler metrics have the same unparametrized oriented geodesics, if and only if they differ by a scaling constant and addition of a closed 1-form.

Dynamical systems on Riemannian manifolds are examined which are either generated by a semi-flow or by a C¹-smooth map. The exponential growth rate of the k-volumina provides a lower bound of the topological entropy of the system if its phase space is compact and has box-counting dimension k. If the system possesses an equivariant sub-bundle instead of the tangent map its restriction onto the fibers of this sub-bundle can be considered. Inequalities for systems with not necessarily compact phase space are also given. Examples address linear maps in ℝ^m, the time-reversed Lorenz system, and the geodesic flow on a (not necessarily compact) Riemannian manifold without conjugate points.

Riemann surfaces are of fundamental importance in many areas of mathematics and theoretical physics. The study of the moduli space of Riemann surfaces of a fixed topological type is intimately related to the study of the Teichmüller space of that surface, together with the action of the mapping class group. Classical Teichmüller theory has many facets and involves the interplay of various methods from geometry, analysis, dynamics and algebraic geometry. In recent years, higher Te-ichmüller theory emerged as a new field in mathematics. It builds as well on a combination of methods from different areas of mathematics. The goal of my talk is to invite the reader to get to know and to get involved into higher Teichmüller theory by describing some of its many facets.

We describe the topological structure of cocompact singular Riemannian foliations on Riemannian manifolds without conjugate points. We prove that such foliations are regular and developable and have regular closures. We deduce that in some cases such foliations do not exist.

Understanding the limiting distributions of translates of measures on subman-ifolds of homogeneous spaces of Lie groups lead to very interesting number theoretic and geometric applications. We explore this theme in various generalities, and in specific cases. Our main tools are Ratner's theorems on unipotent flows, nondivergence theorems of Dani and Margulis, and dynamics of linear actions of semisimple groups.

This is a survey on generalized Liouville manifolds which were studied in the joint work with K. Sugahara ([3]).Without assuming that the metric is positive definite, we define a generalized Liouville structure and generalized Liouville manifolds whose geodesic flows are completely integrable. We give many generalized Liouville manifolds and give a classification of them.