Given any possibly unbounded, locally finite link, we show that there exists a smooth diffeomorphism transforming this link into a set of stream (or vortex) lines of a vector field that solves the steady incompressible Euler equation in R 3 . Furthermore, the diffeomorphism can be chosen arbitrarily close to the identity in any C r norm.

We prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in R 3 . More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in R 3 , we show that they can be transformed with a C m -small diffeomorphism into a set of vortex tubes of a Beltrami field that tends to zero at infinity. The structure of the vortex lines in the tubes is extremely rich, presenting a positive-measure set of invariant tori and infinitely many periodic vortex lines. The problem of the existence of steady knotted thin vortex tubes can be traced back to Lord Kelvin.

We introduce four types of SU(2M + 1) spin chains which can be regarded as the BC N versions of the celebrated Haldane-Shastry chain. These chains depend on two free parameters and, unlike the original Haldane-Shastry chain, their sites need not be equally spaced. We prove that all four chains are solvable by deriving an exact expression for their partition function using Polychronakos's "freezing trick". From this expression we deduce several properties of the spectrum, and advance a number of conjectures that hold for a wide range of values of the spin M and the number of particles. In particular, we conjecture that the level density is Gaussian, and provide a heuristic derivation of general formulas for the mean and the standard deviation of the energy.

The full spectrum and eigenfunctions of the quantum version of a nonlinear oscillator defined on an N -dimensional space with nonconstant curvature are rigorously found. Since the underlying curved space generates a position-dependent kinetic energy, three different quantization prescriptions are worked out by imposing that the maximal superintegrability of the system has to be preserved after quantization. The relationships among these three Schrödinger problems are described in detail through appropriate similarity transformations. These three approaches are used to illustrate different features of the quantization problem on N -dimensional curved spaces or, alternatively, of position-dependent mass quantum Hamiltonians. This quantum oscillator is, to the best of our knowledge, the first example of a maximally superintegrable quantum system on an N -dimensional space with nonconstant curvature.