Starting from a fixed elliptic curve with complex multiplication we compose lifted quotients of elliptic Jacobi theta functions to abelian functions in higher dimension. In some cases where complete Picard-Einstein metrics have been discovered on the underlying abelian surface (outside of cusp points), we are able to transform them to Picard modular forms. Basic algebraic relations of basic forms come from different multiplicative decompositions of these abelian functions in simple ones of the same lifted type. In the case of Gauß numbers the constructed basic modular forms define a Baily-Borel embedding in $\mathbb{P}^{22}$. The relations yield explicit homogeneous equations for the Picard modular image surface.

Since the early days of quantum mechanics many techniques have been developed in order to deal with manifolds with non-trivial topology. Among them two techniques have received a great attention in the literature and are shortly reviewed here as they are most geometrical in nature. These are the Kostant–Souriau geometric quantization scheme and the so called constrained quantum mechanics. A notable difference between them are the geometrical structures used in these theories. The first is based on the symplectic structure of the phase space and the second one relies on the Riemannian metric of the configurational manifold. Both approaches are illustrated in full details. Presented examples include the $n$-dimensional variants of the harmonic oscillator and the Kepler problem which are treated within geometric quantization scheme by making use of the Marsden–Weinstein reduction theorem and even a combination of both methods is applied in the study of quantum-mechanical aspects of the geodesic flows on axisymmetric ellipsoids.

The profound and beautiful interaction between smooth four-manifold topology and the quantum theory of fields often seems as impenetrable as it is impressive. The objective of this series of lectures is to provide a very modest introduction to this interaction by describing, in terms as elementary as possible, how Atiyah and Jeffrey [1] came to view the partition function of Witten’s first topological quantum field theory [21], which coincides with the zero-dimensional Donaldson invariant, as an “Euler characteristic” for an infinite-dimensional vector bundle.

In this paper we give a summary of the geometrical background of the idea of spontaneous symmetry breaking. For this purpose, we set out to discuss Yang–Mills–Higgs gauge theories from the perspective of reducible bundles. From this viewpoint, “elementary particles” are identified with vector bundles, and sections are considered to geometrically represent the states of the corresponding particle. Some physical background on the notion of “mass” is given in the introduction. Since the geometrical interpretation of a gauge boson is that of a connection, we proceed to discuss how, from a geometrical point of view, the Higgs boson can also be considered a connection. We start out with Connes’ algebraic approach, where the “shifted Higgs boson” is considered a gauge potential on a non-commutative space. We summarize how a specific generalization of the notion of a Dirac operator can be used in order to define a generalization of de Rham’s algebra. This generalization is used to define the non-commutative equivalent of the Yang–Mills action where its minima spontaneously break the gauge symmetry. In the last section we summarize how the Higgs boson can be considered a connection on a Clifford module bundle.

An exposition of the new results concerning the nonexistence of local isometric immersions of 3-dimensional Lobachevsky space $L^3$ into 5-dimensional Euclidean space $E^5$ with constant curvature of the Grassmannian image metric, on connections between curvatures of asymptotic lines on a domain of $L^3 \subset E^5$, on regularity theorems for surfaces obtained by Backlund transformation of a domain of $L^2 \subset S^3$ and $L^2 \subset E^3$.

We consider the classical null $p$-brane dynamics in $D$-dimensional curved backgrounds and apply the Batalin–Fradkin–Vilkovisky approach for BRST quantization of general gauge theories. Then we develop a method for solving the tensionless $p$-brane equations of motion and constraints. This is possible whenever there exists at least one Killing vector for the background metric. It is shown that the same method can be also applied for the tensile $1$-branes. Finally, we give two examples of explicit exact solutions in four dimensions.

In this paper we are interested in the multiplicity of the Yamabe metrics for compact Riemannain manifolds. This problem may be approached only when the scalar curvature is positive. This is the case notably of certain locally conformally flat manifolds. Furthermore, there exists a link with the singular Yamabe problem. We give various applications of this study.

By considering the set of coupled KdV differential equations as a zero curvature representation of some fourth order linear differential equation and factorizing the linear differential equation, the hierarchy of solutions of the coupled KdV differential equations have been obtained from the eigen spectrum of constant potentials.

We study degenerate curves in pseudo-Euclidean spaces of index two by introducing the Cartan reference along a degenerate curve. We obtain several different types of degenerate curves and present existence, uniqueness and congruence theorems. We also give some examples of such a curves in low dimensions.

Membranous structures that are composed of particles with in-plane $C_{2}$ group symmetry are studied. Starting from a single-constituent energy it is derived that the relevant invariants for description of such systems are the mean curvature and the curvature deviator; the energy of the system can be expressed in a simple and transparent form by these two invariants while their average values span the phase diagram of shapes that can be attained by the membrane enclosed structures.

A variational problem for axisymmetric shapes is stated where the shapes with extreme average mean curvature and extreme average curvature deviator at relevant constraints are sought for. It is shown that, at fixed membrane area and at fixed enclosed volume, the solutions of the variational problem correspond to spherical, cylindrical and toroidal shapes. These solutions form lines in the phase diagram of possible shapes that separate classes of shapes with different symmetry properties. Physically, these lines represent limits of trajectories formed by processes that cause change of the average mean curvature and/or average curvature deviator, such as intercalation of particles into the membrane. Correspondence with some experiments involving toroidal structures of erythrocyte membrane, induced by exogenously added amphiphilic molecules, is considered.

Any real hypersurface of a Kähler manifold carries a natural almost contact metric structure. There are four basic classes of real hypersurfaces of a Kähler manifold with respect to the induced almost contact metric structure. In this paper we study the basic classes of real hypersurfaces of a complex space form in terms of their Riemannian curvatures.

The reductions of the integrable $N$-wave type equations solvable by the inverse scattering method with the generalized Zakharov–Shabat system $L$ and related to some simple Lie algebra $\mathfrak{g}$ are analyzed. Special attention is paid to the $\mathbb{Z}_2$-reductions including ones that can be embedded also in the Weyl group of $\mathfrak{g}$. The consequences of these restrictions on the structure of the dresing factors are outlined. An example of 4-wave equations (with application to nonlinear optics) and its gauge equivalent are given.

The reductions for the first order linear systems of the type: \[ L \psi (x,\lambda) \equiv \left(\mathbf{i} \frac{\rm d}{{\rm d}x} + q(x) - \lambda J \right) \psi (x,\lambda) = 0,\ J \in \mathfrak{h},\ q(x) \in \mathfrak{g}_J \] are studied. This system generalizes the Zakharov–Shabat system and the systems studied by Caudrey, Beals and Coifman (CBC systems). Here $J$ is a regular complex constant element of the Cartan subalgebra $\mathfrak{h} \subset \mathfrak{g}$ of the simple Lie algebra $\mathfrak{g}$ and the potential $q(x)$ takes values in the image $\mathfrak{g}_J$ of ad $_J$. Special attention is paid to the scattering data of CBC systems and their behaviour under the Weyl group reductions. The analytical properties of the generating functional of the integrals of motion and their reduced analogs are studied. These results are demonstrated on an example of $N$-wave type equations.

This paper is a survey of some recent joint work of Hans Boden, Paul Kirk and the author, as well as work by Cappell, Lee, and Miller, on generalizing the Casson invariant to the group $SU(3)$. The main challenge here is that in this setting there are nontrivial reducible representations. Because of this, the irreducible stratum is not compact, and as a consequence an algebraic count of points does not provide a topological invariant (independent of perturbation).

After a short introduction to the program of deformation quantization we indicate why this program is of current interest. One of the reasons is Kontsevich’s generalization of the Moyal product of phase-space functions to the case of general Poisson manifolds. We discuss this generalization, including the graphical calculus for presenting the result. We then illustrate the techniques of deformation quantization for quantum mechanical problems by considering the case of the simple harmonic oscillator. We indicate the relations to more conventional approaches, including the formalisms involving operators in Hilbert space and path integrals. Finally, we sketch some new results for relativistic quantum field theories.

By considering the Boussinesq equation as a zero curvature representation of some third order linear differential equation and factorizing this linear differential equation, the hierarchy of solutions of Boussinesq equation has been obtained from the eigen spectrum of constant potentials.

The present note announces some inequalities on the number of the generators and relations of a Kähler group $\pi_1(X)$, involving the irregularity $q(X)$, the Albanese dimension $a(X)$ and the Albanese genera $g_k(X), 1 \leq k \leq a(X)$, of the corresponding compact Kähler manifold $X$. The principal ideas for their derivation are outlined and the proofs are postponed to be published elsewhere.

Complexified Hamiltonian dynamical systems are considered with subsequent construction of real forms of Hamiltonian dynamics by using compatible complex structures or involution operators mimicking the properties of complex conjugation. This provides a method of associating a class of real dynamical systems to a given initial (also real) one. Examples are given and the problem of integrability of the derived dynamical systems is also disscused.

In the present paper we consider a class of hypersurfaces of conullity two in Euclidean space, which are one-parameter systems of developable surfaces of codimension two and we prove a characterization theorem for them in terms of their second fundamental tensor.

The Björling problem for maximal surfaces in Lorentz–Minkowski space $\mathbb{L}^3$ has been recently studied by the author together with Alías and Chaves. The present paper is a natural extension of that work, and provides several variations of Björling problem. The main scheme is the following. One starts with a spacelike analytic curve in $\mathbb{L}^3$, and asks for the construction of a maximal surface which contains that curve, and satisfies additionally some other geometric condition. The solution of these Björling-type problems are then applied with a twofold purpose: to construct examples of maximal surfaces in $\mathbb{L}^3$ with prescribed properties, and to classify certain families of maximal surfaces.

In this paper we study some non-positively curved Riemannian manifolds acted on by a Lie group of isometries with principal orbits of codimension one. Among other results it is proved that if the universal covering manifold satisfies some conditions then every non-exceptional singular orbit is a totally geodesic submanifold. When $M$ is flat and is not toruslike, it is proved that either each orbit is isometric to $\mathbb{R}^k \times \mathbb{T}^m$ or there is a singular orbit. If the singular orbit is unique and non-exceptional, then it is isometric to $\mathbb{R}^k \times \mathbb{T}^m$.

In this paper, we will report a recent study about moduli spaces of branched and complete minimal surfaces in Euclidean space of genus one with two ends and total curvature $−4 \pi$.

The two dimensional Hamiltonian with generalized shape invariance symmetry over $S^2$, has been obtained via Fourier transformation over the three coordinates of the $SU(3)$ Casimir operator defined on $SU(3)/SU(2)$ symmetric space. It is shown that the generalized shape invariance is equivalent to $SU(3)$ symmetry and that there is one to one correspondence between the representations of the generalized shape invariance and $SU(3)$ Verma modules. Also the two dimensional Hamiltonian in $\mathbb{R}^2$ space which posseses ordinary shape invariance symmetry with respect to two parameters, has been obtained via Inönü–Wigner contraction over $SU(3)$ manifold.

This paper surveys and compares some recent approaches to stochastic infinite-dimensional geometry on the space $\Gamma$ of configurations (i. e. locally finite subsets) of a Riemannian manifold $M$ under Poisson measures. In particular, different approaches to Bochner–Weitzenböck formulas are considered. A unitary transform is also introduced by mapping functions of $n$ configuration points to their multiple stochastic integral.

I investigate the Poisson-sigma model on the classical and quantum level. First I show how the interaction can be obtained by a deformation of the classical master equation of an Abelian BF theory in two dimensions. On the classical level this model includes various known two-dimensional field theories, in particular the Yang–Mills theory. On the quantum level the perturbation expansion of the path integral in the covariant gauge yields the Kontsevich deformation formula. Finally I perform the calculation of the path integral in a general gauge, and demonstrate how the derived partition function reduces in the special case of a linear Poisson structure to the familiar form of 2D Yang–Mills theory.

The Lamb shift in hydrogen spectrum is obtained here in the framework of some weak generalization of the classical Maxwell electrodynamics. It is shown that weakly generalized classical Maxwell electrodynamics can describe the intra-atomic phenomena with the same success as relativistic quantum mechanics can do. Group-theoretical grounds for the description of fermionic states by bosonic system are presented briefly. The advantages of generalized electrodynamics in intra-atomic region in comparison with standard Maxwell electrodynamics are demonstrated on testing example of hydrogen atom. We are able to obtain some results which are impossible in the framework of standard Maxwell electrodynamics. The Sommerfeld–Dirac formula for the fine structure of the hydrogen atom spectrum is obtained on the basis of such Maxwell equations without appealing to the Dirac equation. The Bohr postulates are proved to be the consequences of the equations under consideration. The relationship of the new model with the Dirac theory is investigated. Possible directions of unification of such electrodynamics with gravity are mentioned.

We study a quantum system in a Riemannian manifold $M$ on which a Lie group $G$ acts isometrically. The path integral on $M$ is decomposed into a family of path integrals on quotient space $Q = M/G$ and the reduced path integrals are completely classified by irreducible unitary representations of $G$. It is not necessary to assume that the action of $G$ on $M$ is either free or transitive. Hence the quotient space $M/G$ may have orbifold singularities. Stratification geometry, which is a generalization of the concept of principal fiber bundle, is necessarily introduced to describe the path integral on $M/G$. Using it we show that the reduced path integral is expressed as a product of three factors; the rotational energy amplitude, the vibrational energy amplitude, and the holonomy factor.

We describe a framework suggesting how to “deform” in Planck’s constant $h$ the classical concept of integrability. The key point is to use well defined counterparts of Feynman’s measure on the space of paths of the mechanical system. Then we introduce the associated deformation of quantum conservation laws. The method is tested on elementary systems and provide, indeed, more information than expected.

We solve in two-dimensional theta functions the integrable case $\ddot{r} = -ar + 2zr,\ \ddot{z} = -bz + 6z^2 + r^2$ (a and b are constant parameters) of the generalizied Hénon–Heiles system. The general solution depends on six arbitrary constants, called algebraic–geometric coordinates. Three of them are coordinates on the degree two (and dimension three) Siegel upper half-plane and define two-dimensional tori $\mathbb{T}^2.$ Each trajectory of the Hénon–Heiles system lies on certain torus $\mathbb{T}^2$. Next two arbitrary constants define the initial position on $\mathbb{T}^2$. The speed of the flow depends multiplicatively on the last arbitrary constant.