In this paper we review certain Lusternik-Schnirelmann categorical notions that pertain especially to Eilenberg-Mac Lane spaces. Further, we introduce a new categorical invariant that allows a refined estimate for category. Finally, we mention some results about topological complexity and relate these to category.

Reviewed are ideas underlying our concept of affinely-rigid body. We do this from the perspective of the Leonhard Euler two main achievements, known under his name: the mechanics of rigid body and the dynamics of incompressible ideal fluid. But we formulate the theory which is somehow placed between those two models. Our scheme is a finite-dimensional dynamical system, but admitting deformative degrees of freedom. We also stress the connection with the general idea of affine invariance in physics.

In the previous lecture we have introduce and discussed the concept of affinely-rigid, i.e., homogeneously deformable body. Some symmetry problems and possible applications were discussed. We referred also to our motivation by Euler ideas. Below we describe the general principles of the quantization of this theory in the Schrödinger language. The special stress is laid on highly-symmetric, in particular affinely-invariant, models and the Peter-Weyl analysis of wave functions.

Discussed are some classical and quantization problems of the affinely-rigid body in two dimensions. Strictly speaking, we consider the model of the harmonic oscillator potential and then discuss some natural anharmonic modifications. It is interesting that the considered doubly-isotropic models admit coordinate systems in which the classical and Schrödinger equations are separable and in principle solvable in terms of special functions on groups.

Studied is the problem of degeneracy of mechanical systems the configuration space of which is the three-dimensional sphere, the elliptic space, i.e., the quotient of that sphere modulo the antipodal identification, and finally, the three-dimensional pseudo-sphere, namely, the Lobatchevski space. In other words, discussed are systems on groups ${\rm{SU}}(2)$, ${{\rm{SO}}}(3,\mathbb{R})$, and ${\rm{SL}}(2,\mathbb{R})$ or its quotient ${{\rm{SO}}}(1,2)$. The main subject are completely degenerate Bertrand-like systems. We present the action-angle classical description, the corresponding quasi-classical analysis and the rigorous quantum formulas. It is interesting that both the classical action-angle formulas and the rigorous quantum mechanical energy levels are superpositions of the flat-space expression, with those describing free geodetic motion on groups.

Discussed are classical and quantized models of affinely rigid motion with degenerate dimension, i.e., such ones that the geometric dimensions of the material and physical spaces need not be equal to each other. More precisely, the material space may have dimension lower than the physical space. Physically interesting are special cases $m=2$ or $m=1$ and $n=3$, first of all $m=2$, $n=3$, i.e., roughly speaking, the affinely deformable coin in three--dimensional Euclidean space. We introduce some special coordinate systems generalizing the polar and two--polar decompositions in the regular case. This enables us to reduce the dynamics to two degrees of freedom. In quantum case this is the reduction of the Schrödinger equation to multicomponent wave functions of two deformation invariants.

Prolongation algebras which are determined by applying a version of the Wahlquist-Estabrook method to three different nonlinear partial differential equations can be employed to obtain not only Lax pairs but Bäcklund transformations as well. By solving Maurer-Cartan equations for the related group specified by the prolongation algebra, a set of differential forms is obtained which can lead directly to these kinds of results. Although specific equations are studied, the approach should be applicable to large classes of partial differential equations.

We examine the self-gravitating spherical shell in the fully nonlinear and nonsymmetric theory of the gravity. We argue that the hyperdense static finally collapsed object could not be made of any known form of matter. Also we observed that if the radius of the shell is sufficiently small then the antisymmetric part of the energy-momentum tensor exceeds its symmetric part. It seems to violate the natural physical conditions.

Let $X$ be a four-manifold with boundary three-manifold $M$. We shall describe (i) a pre-symplectic structure on the sapce $\mathcal{A}(X)$ of connections on the bundle $X\times \mathrm{SU}(n)$ that comes from the canonical symplectic structure on the cotangent space $T^{\ast}\mathcal{A}(X)$. By the boundary restriction of this pre-symplectic structure we obtain a pre-symplectic structure on the space $\mathcal{A}^{\flat}_0(M)$ of flat connections on $M\times \mathrm{SU}(n)$ that have null charge.

Our main idea is to suggest a new model of non-perturbative and geometrically motivated nonlinearity in quantum mechanics. The Schrödinger equation and corresponding relativistic linear wave equations derivable from variational principles are analyzed as usual self-adjoint equations of mathematical physics. It turns out that introducing the second-order time derivatives to dynamical equations, even as small corrections, can help to obtain the regular Legendre transformation. Following the conceptual transition from the special to general theory of relativity, where the metric tensor loses its status of the absolute geometric object and becomes included into degrees of freedom (gravitational field), in our treatment the Hilbert-space scalar product becomes a dynamical quantity which satisfies together with the state vector the system of differential equations. The structure of obtained Lagrangian and equations of motion is very beautiful, as usually in high-symmetry problems.

We discuss some classical and quantization problems of infinitesimal affinely-rigid bodies moving in two-dimensional manifolds. Considered are highly symmetric models for which the variables can be separated. We follow the standard procedure of quantization in Riemannian manifolds, i.e., we use the ${\rm L}^{2}$-Hilbert space of wave functions in the sense of the usual Riemannian measure (volume element).

The classical $\mathrm{U}(1)$-Kepler problems at level $n\ge 2$ were formulated, and their trajectories are determined via an idea similar to the one used by Kustaanheimo and Stiefel in the study of Kepler problem. It is found that a non-colliding trajectory is an ellipse, a parabola or a branch of hyperbola according as the total energy is negative, zero or positive, and the complex orientation-preserving linear automorphism group of $\mathbb C^n$ acts transitively on both the set of elliptic trajectories and the set of parabolic trajectories.

Let $(M,{\rm g})$ be an arbitrary pseudo-Riemannian manifold of dimension at least three. We determine the form of all the conformal symmetries of the conformal Laplacian on $(M,{\rm g})$, which are given by differential operators of second order. They are constructed from conformal Killing two-tensors satisfying a natural and conformally invariant condition. As a consequence, we get also the classification of the second order symmetries of the conformal Laplacian. We illustrate our results on two families of examples in dimension three. Besides, we explain how the (conformal) symmetries can be used to characterize the $R$-separation of some PDEs.

TWe give a master equation expressing the zero curvature representation of the equations of motion of a two-dimensional non-linear sigma models. The geometrical properties of this equation are highlighted.

Here we present a new parametrization of the Mylar balloon via the Weierstrassian functions which is used for the derivation of the basic geometrical characteristics of the balloon.

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the theory of visible actions on complex manifolds.

Our main results give a classification of triples $(G,H,L)$ for a compact Lie group $G$ and its Levi subgroups $H,L$, which satisfy $G=HBL$. Here, $B$ is a subset of a Chevalley--Weyl involution $\sigma$-fixed points subgroup $G^{\sigma}$ of $G$. The point here is that one decomposition $G=LBH$ produces three strongly visible actions on generalized flag varieties, and thus three finite-dimensional multiplicity-free representations (Kobayashi's triunity principle).

Furthermore, we can also prove that the visibility of actions of compact Lie groups, the existence of a decomposition $G=LBH$ and the multiplicity-freeness property of finite-dimensional tensor product representations are all equivalent.

This research is motivated by a study of special types of surfaces of revolution, using methods from differential geometry, elasticity theory and variational calculus. In particular, we present an elastic membrane model for the beta barrels in protein biology, via a certain Generalized Willmore type energy functional. We study the corresponding Euler--Lagrange equation, as well as a specific boundary value problem whose solutions are Generalized Willmore surfaces of revolution. We study the corresponding solutions both theoretically and numerically.

We consider the recursion operator related to a system introduced recently that could be considered as a generalization to a pole gauge generalized Zakharov-Shabat system on $\mathfrak{sl}(3,\mathbb{C})$ but involving rational dependence on the spectral parameter and subject to $\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$ reduction of Mikhailov type. We calculate the hierarchies of nonlinear evolution equations related to this system through the recursion operators we introduce.

We give a review on non formal star product and its star exponentials with concrete examples.