The dynamics of vortex filaments has provided for almost a century one of the most interesting connections between differ ential geometry and soliton equations, and an example in which knotted curves arise as solutions of differential equations possessing an infinite family of symmetries and a remarkably rich geometrical structure. These lectures discuss several aspects of the integrable dynamics of closed vortex filaments in an Eulerian fluid, including its Hamiltonian formulation, the construction of a large class of special solutions, and the role of the Floquet spectrum in characterizing the geometric and topological properties of the evolving curves.

Fermions and bosons are the fundamental particles of nature, and they are naturally described mathematically by using the methods of super-symmetry. These methods are illustrated here by consideration of a number of physical examples which arise in non-relativistic and relativistic quantum mechanics.

This paper surveys the new, algorithmic theory of moving frames. Applications in geometry, computer vision, classical invariant theory, and numerical analysis are indicated.

Discussed is affine model of collective degrees of freedom of multi-particle systems or continuous media. The novelty of our approach is that it is not only kinematics, i.e. geometry of degrees of freedom, that is invariant under affine group, but rather we study affinely-invariant geodetic models of such affine systems. It is shown that the dynamics of bounded elastic vibrations may be encoded in such geodetic models in the very form of the kinetic energy expression. Some special solutions like the relative equilibria are discussed. We start with the general approach to group-theoretical degrees of freedom and then discuss peculiarities of the affine group and certain other groups underlying collective dynamics.

In the Batalin-Vilkovisky field-antifield formalism a classical mechanical system is described by a solution of the classical master equation. The quantization of this general gauge theory in the Lagrangian approach can be accomplished in closed form [2]. The AKSZ-formalism is a geometrical construction of such a solution as a QP-manifold [1]. This can be extended and applied to topological quantum field theories.

The governing equations of the hydrodynamic stability theory are separable only with the parallel steady-state flow assumption, when they can be reduced to an ordinary differential equation, the Orr-Sommerfeld equation. For nonparallel flows, a basic flow and the equations for disturbance flow are dependent on the downstream coordinate so that the corresponding operator does not separate unless certain terms are ignored. If the basic flow is non-steady, this brings about great difficulties in theoretical studies of the instability since the normal modes containing an exponential time factor $\mathbf{exp}\ t$ are not applicable here. The objective of this work was to obtain new results in the problem of linear stability of non-parallel and unsteady flows by applying the recently developed symmetry-based approach to the separation of variables in PDEs with variable coefficients.

The Poisson structures and Hamiltonian formulation of three dimensional systems is considered in general. A class of degenerate structures in higher dimensions is also briefly discussed.

The center-of-mass system for many bodies in $\mathbb{R}^3$ admits a natural action of the rotation group $SO(3)$. According to the orbit types for the $SO(3)$ action, the center-of-mass system $M$ is stratified into strata. A quantum Hamiltonian system and a classical Lagrangian system are defined on $L^2(M)$ and on $T(M)$, respectively. These systems are also stratified according to the stratification of $M$, and then reduced by the rotational symmetry, respectively.

A few explicit formulas providing conformal coordinates of the axially symmetric constant mean curvature surfaces introduced by Delaunay and their duals are derived. These results give also new examples in a long line of research connected with finding isothermic immersions of surfaces and their duals.

The curvature tensor on a 3-dimensional almost contact manifold with B-metric belonging to two main classes is studied. These classes are the rest of the main classes which were not considered in the first part of this work. The dimension 3 is the lowest possible dimension for the almost contact manifolds with B-metric. The corresponding curvatures are found and the respective geometric characteristics of the considered manifolds are given.

We construct the sequence of orthogonal polynomials with respect to an inner product defined in the sense of $q$-integration over several intervals in the complex plane. For such introduced polynomials we prove that all zeros lie in the smallest convex hull over the intervals in the complex plane. The results are stated precisely in some special cases, as some symmetric cases of equal lengths, angles and weights.

The notion of a phase space in classical mechanics is of course well known. The extension of this concept to field theory however, is a challenging endeavor, and over the years numerous propositions for such a generalization have appeared in the literature. In this contribution we review a Hamiltonian formulation of Lagrangian field theory based on an extension to infinite dimensions of J.-M. Souriau’s symplectic approach to mechanics. Following G. Zuckerman, we state our results in terms of the variational bicomplex. We present a basic example, and briefly discuss some possible avenues of research.

In this paper we describe the Euclidean signature curves for two dimensional closed curves in the plane and their generalization to closed space curves. The focus will be on discrete numerical methods for approximating such curves. Further we will apply these numerical methods to plot the signature curves related to three-dimensional simulated DNA supercoils. Our primary focus will be on statistical analysis of the data generated for the signature curves of the supercoils. We will try to establish some relationships between the statistics and the shape of the signature curve. The hope is that these findings will provide a more efficient way for computers to search for and identify signature curves corresponding to similar DNA supercoils.

Position uncertainty (delocalization) measures for a particle on the sphere are proposed and illustrated on several examples of states. The new measures are constructed using suitably the standard multiplication angle operator variances. They are shown to depend solely on the state of the particle and to obey uncertainty relations of the Schrödinger-Robertson type. A set of Hermitian operators with continuous spectrum is pointed out the variances of which are complementary to the longitudinal angle uncertainty measure.

The hyperbolic trigonometry, fully analogous to the common Euclidean trigonometry, is presented and employed to calculate the hyperbolic triangle defect in the Poincaré ball model of $n$-dimensional hyperbolic geometry. It is shown that hyperbolic trigonometry allows the hyperbolic triangle defect to be expressed in terms of the triangle hyperbolic side lengths by a remarkably elegant identity.

In this work a hypothesis about the fluid flow character is proposed which consists in the splitting of a flow in two parts. The first one is treated dynamically, and the second statistically. The instability generating energy coming from the dynamical flow being distributed by the statistical flow results in an influence on the first one and forms the observed flow. The connection between the two flows is obtained by the requirement of the mechanical energy conservation. A corresponding model is analyzed in some details and applied to the case of developed turbulence in a channel Poiseuille flow. The mean velocity profiles for values of $Re =$ 13800, 23200, 32800 are computed and compared with the existing experimental data. Numerically derived velocity profiles with $Re$ from 100 to 143000 are given.

The present paper is concerned with the geometric (point) Lie symmetry groups of the Willmore equation – the Euler-Lagrange equation associated with the Willmore functional. The ten-parameter group of special conformal transformations in three-dimensional Euclidean space, which is known to be the symmetry group of the Willmore functional, is recognized as the largest group of geometric transformations admitted by the Willmore equation in Mongé representation. The conserved currents of ten linearly independent conservation laws, which correspond to the variational symmetries of the Willmore equation and hold on its smooth solutions, are derived. All types of non-equivalent group-invariant solutions of the Willmore equation are identified, an optimal system of one-dimensional subalgebras being given together with the invariants of the corresponding one-parameter groups, up to one exception. Special attention is paid to the rotationally-invariant (axially-symmetric) solutions.