The surfaces in three dimensional Euclidean space ${\mathbb R}^3$ obtained through the use of the soliton techniques are called integrable surfaces. Integrable equations and their Lax equations possess certain symmetries. Infinitesimal versions of these symmetries are deformations which are responsible in constructing the integrable surfaces. There are four different types of deformations. The spectral parameter, the gauge, the generalized symmetries and integration parameters deformations. We shall present here how these deformations generate surfaces in ${\mathbb R}^3$ and also in three-dimensional Minkowski ${\mathbb M}_{3}$ space. The key point here is to start with an integrable equation and its Lax representation. In this work we assume that the Lax equations of integrable equations are given in terms of a group $G$ and its algebra ${\mathfrak g}$ valued functions. The surfaces in ${\mathbb R}^3$ are also represented via respect ${\mathfrak g}$ valued functions. In constructing integrable surfaces we need the solutions of both the integrable equations and their corresponding Lax equations. In this work we use the one soliton solutions of the integrable equations. We solve the Lax equations for one soliton solutions of the integrable equations. Then choosing a deformation one can construct several types of surfaces. After obtaining these surfaces the next is to search for their properties. Most of these surfaces are Weingarten surfaces, Willmore-like surfaces and surfaces which are derivable from a variational principle. We give sketches of the interesting surfaces of Korteweg-de Vries (KdV), modified Korteweg-de Vries (mKdV) and Nonlinear Schrödinger (NLS), sine Gordon (SG) equations.

This article is a written version of author's lecture on generalized Kepler problems at the XVII-th International Conference on Geometry, Integrability and Quantization, June 5-10, 2015 Varna, Bulgaria. It begins with a review of the Kepler problem for planetary motion and its magnetized cousins, from which a surprising relationship with Lorentz transformation emerges. Next, we give a review for euclidean Jordan algebra and the associated universal Kepler problem. Finally, we demonstrate that, via the universal Kepler problem, a suitable Poisson realization of the conformal algebra for a simple euclidean Jordan algebra gives rise to a super integrable model that resembles the Kepler problem. In particular we demonstrate how the Kepler problem and its magnetized cousins are obtained this way.

These lectures notes are meant as an introduction to geometric quantization. In Section 1, I begin with presentation of the historical background of quantum mechanics. I continue with discoveries in the theory of representations of Lie groups, which lead to emergence of geometric quantization as a part of pure mathematics. This presentation is very subjective, flavored by my own understanding of the role of geometric quantization in quantum mechanics and representation theory. Section 2 is devoted to a review of geometry of Hamiltonian systems. Geometric quantization is discussed in the next two sections: prequantization in Section 3 and polarization in Section 4. In particular, I discuss geometric quantization with respect to polarizations given by Kähler structure, cotangent bundle projection and completely integrable system. More advanced topics, like metaplectic structure, pairing of polarizations, and commutation of quantization and reduction, are not included.

In this paper we study a Generalized Willmore flow for graphs and its numerical applications.

First, we derive the time dependent equation which describes the geometric evolution of a Generalized Willmore flow in the graph case. This equation is recast in divergence form as a coupled system of second order nonlinear PDEs.

Furthermore, we study finite element numerical solutions for steady-state cases obtained with the help of the FEMuS library (Finite Element Multiphysics Solver). We use automatic differentiation (AD) tools to compute the exact Jacobian of the coupled PDE system subject to Dirichlet boundary conditions.

Geometric Control Theory was initiated in the beginning of the 1970's and has now become a well established design methodology for problems of tracking prescribed reference signals while rejecting unwanted disturbance signals. In this paper we describe the error analysis for time-dependent harmonic signal tracking for general distributed parameter control systems with bounded input and output operators using an iterative numerical scheme based on the geometric design methodology.

On the total space of the cotangent bundle $T^{*}M$ of a Riemannian manifold $(M,h)$ we consider the natural Riemann extension $\bar{g}$ with respect to the Levi-Civita connection of $h$. In this setting, we construct on $T^{*}M$ a new para-complex structure $P$, whose harmonicity with respect to $\bar{g}$ is characterized here by using the reduction of $\bar{g}$ to the (classical) Riemann extension.

The gauge invariant phase difference between superconducting electrodes is a dominating dynamical degree of freedom in the Josephson junction. This rapport concerns the influence of the curvature of the junction on the dynamic of this field variable. The effects of curvature are discussed in the long and large area junctions. In particular the dynamics of the fluxion and the kink front are studied.

The Cayley maps for the Lie algebras $\mathfrak{su}(1,1)$ and $\mathfrak{so}(2,1)$ converting them into the corresponding Lie groups $\mathrm{SU}(1,1)$ and $\mathrm{SO}(2,1)$ along their natural vector-parameterizations are examined. Using the isomorphism between $\mathrm{SU}(1,1)$ and $\mathrm{SL}(2,\mathbb{R})$, the vector-parameterization of the latter is also established. The explicit form of the covering map $\mathrm{SU}(1,1) \rightarrow \mathrm{SO}(2,1)$ and its sections are presented. Using the so developed vector-parameter formalism, the composition law of $\mathrm{SO}(2,1)$ in vector-parameter form is extended so that it covers compositions of all kinds of elements including also those that can not be parameterized properly by regular $\mathrm{SO}(2,1)$ vector-parameters. The latter are characterized and it is shown that they can be represented by $\mathrm{SU}(1,1)$ vector parameters with pseudo length equal to minus four. In all cases of compositions inside $\mathrm{SO}(2,1)$, criteria for determination of their type (elliptic, parabolic, hyperbolic) have been presented. On the base of the vector-parameter formalism the problem of taking a square root in $\mathrm{SO}(2,1)$ is solved explicitly. Also, an analogue of Cartan's theorem about the decomposition of orthogonal matrix of order $n$ into product of at most $n$ reflections is formulated and proved for the subset of hyperbolic elements of the group of pseudo-orthogonal matrices from $\mathrm{SO}(2,1)$.

The energy spectrum of two 0-branes for fixed angular momentum in 2+1 dimensions is calculated by the Rayleigh-Ritz method. The basis function used for each angular momentum consists of 80 eigenstates of the harmonic oscillator problem on the corresponding space. It is seen that the spectrum exhibits a definite linear Regge trajectory behavior. It is argued how this behavior supports the picture by which the bound-states of quarks and QCD-strings are governed by the quantum mechanics of matrix coordinates.

We construct a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with lightlike axis and call these surfaces meridian surfaces of parabolic type. They are analogous to the meridian surfaces of elliptic or hyperbolic type. Using the invariants of these surfaces we give the complete classification of the meridian surfaces of parabolic type with constant Gauss curvature or constant mean curvature. We also classify the Chen meridian surfaces of parabolic type and the meridian surfaces of parabolic type with parallel normal bundle.

In this paper, we give explicit parametrizations for Bour type surfaces in various three-dimensional space forms, using Weierstrass-type representations. We also determine classes and degrees of some Bour type zero mean curvature surfaces in three-dimensional Minkowski space.

We show that the torus in ${\mathbb R}^3$ is a critical point of a sequence of functionals ${\mathcal F}_{n}$ ($n=1,2,3, \ldots$) defined over compact surfaces (closed membranes) in ${\mathbb R}^3$. When the Lagrange function ${\mathcal E}$ is a polynomial of degree $n$ of the mean curvature $H$ of the torus, the radii ($a,r$) of the torus are constrained to satisfy $\frac{a^2}{r^2}=\frac{n^2-n}{n^2-n-1},~~ n \ge 2$. A simple generalization of torus in ${\mathbb R}^3$ is a tube of radius $r$ along a curve ${\bf \alpha}$ which we call it toroidal surface (TS). We show that toroidal surfaces with non-circular curve ${\bf \alpha}$ do not provide minimal energy surfaces of the functionals ${\mathcal F}_{n}$ ($n=2,3$) on closed surfaces. We discuss possible applications of the functionals discussed in this work on cell membranes.

Via the Lagrangian formalism, an example of extended harmonic CMC immersion and conservation laws are obtained.

In this paper we consider eight special Frenet ruled surfaces along to the involute-evolute curves, $\alpha^{\ast}$ and $\alpha $ respectively$,$ with curvature $k_{1}\neq 0$. First we find the excplicit equation of Frenet ruled surfaces along the involute curves in terms of the Frenet apparatus of evolute curve $\alpha$. Also normal vector fields of these Frenet ruled surfaces have been calculated too.

Further we give all results for sixteen positions of Normal vector fields of four Frenet ruled surfaces in terms of Frenet apparatus of evolute curve $\alpha$. These results also give us the positions of eight special Frenet ruled surfaces along to the involute-evolute curves, based on their normal vectors, in terms of curvatures of evolute curve $\alpha$. We can give the answer of the questions that in which condition we can produce orthogonal surfaces or surfaces with constant angle. For example Darboux ruled surface and involutive tangent ruled surface of an evolute $\alpha$ have the perpendicular normal vector fields.

Here we present a new characterization of Euler elastica via the Weierstrassian functions.

We construct surfaces from modified Korteweg-de Vries (mKdV) and sine-Gordon (SG) soliton solutions by the use of parametric deformations. For each case there are two types of deformations. The first one gives surfaces on spheres and the second one gives highly complicated surfaces in three dimensional Euclidean space (${\mathbb R}^3$). The SG surfaces that we obtained are not the critical points of functional where the Lagrange function is a polynomial function of the Gaussian ($K$) and mean ($H$) curvatures of the surfaces. We also give the graph of interesting mKdV and SG surfaces arise from parametric deformations.

In this work first, we survey the most recent classification results for submanifolds with pointwise 1-type Gauss map. Then, we study a class of hypersurfaces with vanishing Gauss-Kronecker curvature in terms of type of their Gauss map.

The Lorentz transformations of order $(m,n)$ in pseudo-Euclidean spaces with indefinite inner product of signature $(m,n)$ are extended in this work from $m=1$ and $n\ge1$ to all $m,n\ge1$. A parametric realization of the Lorentz transformation group of any order $(m,n)$ is presented, giving rise to generalized gyrogroups and gyrovector spaces called bi-gyrogroups and bi-gyrovector spaces. The latter, in turn, form the setting for generalized analytic hyperbolic geometry that underlies generalized balls called eigenballs.

The work is concerned with a special family of axially symmetric surfaces providing local extrema to the so-called Willmore functional, which assigns to each surface its total squared mean curvature. The components of the position vector of the profile curves of the regarded Willmore surfaces satisfy a system of first-order ordinary differential equations. The solutions of this system are expressed by quadratures in terms of the tangent angle and, in this way, the corresponding Willmore surfaces are determined.

We consider some aspects of the spectral theory of a system that is a generalization to a pole gauge Zakharov-Shabat type system on the Lie algebra $\frak{sl}(3,\mathbb{C})$ but involving rational dependence on the spectral parameter and subjected to $\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$ reduction of Mikhailov type. The question of the existence of analytic fundamental solutions under some special type of boundary conditions has been considered, recently we consider boundary conditions in general position.

We give a brief review on Weyl manifold as a quantization of symplectic manifold, equipped with a system of quantized canonical charts and quantized canonical transformations among them called Weyl diffeomorphism. Weyl manifold is deeply related to deformation quantization on symplectic manifolds. We explain a relation between Weyl manifolds and deformation quantization.