Harmonic two-spheres in the unitary group may be constructed and described in terms of unitons. We present an analogue of this theory for those solutions of the noncommutative ${\rm U}(1)$ sigma-model that may be represented as finite-dimensional perturbations of zero-energy solutions. In particular, we establish that the energy of every such solution is an integer multiple of $8\pi$, describe all solutions of small energy and give many explicit examples of non-Grassmannian solutions.

A rather new physically well-motivated notion of locally noncommutative spacetimes–a refinement of the notion noncommutative spacetimes–is reviewed and discussed. It is shown that the latter model can be realized in the framework of formal deformation quantization using star products as well as in the framework of Rieffel's strict deformation quantization. For the convergent setting Rieffel's former results for $C^*$-algebras are generalized to pro-$C^*$-algebras and applied to actions well-suited to the idea of locally noncommutative spacetimes.

We consider an example of tubes of hypersurfaces in Euclidean space and generalise the tube formula to supercase. By this we assign to a point of the hypersurface in superspace a rational characteristic function. Does this rational function appear when we calculate the $\zeta$-function of an arithmetic variety?

We consider the Grassmannian version of the noncommutative $U(1)$ sigma-model, which is given by the energy functional $E(P) = \|[a, P]\|_{HS}^2$, where $P$ is an orthogonal projection on a Hilbert space $H$ and the operator $a: H \to H$ is the standard annihilation operator. Using realization of $H$ as the Bargmann-Fock space, we describe all solutions with one-dimensional image and prove that the operator $[a, P]$ is densely defined on $H$ for some class of projections $P$ with infinite-dimensional image and kernel.

We give an exposition of some recent crucial achievements in the theory of multidimensional Poisson brackets of hydrodynamic type. In particular, we solve the well-known Dubrovin–Novikov problem posed as long ago as 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensional Poisson brackets of hydrodynamic type. In contrast to the one-dimensional case, in the general case, a nondegenerate multidimensional Poisson bracket of hydrodynamic type cannot be reduced to a constant form by a local change of coordinates. We obtain the classification of all nonsingular nondegenerate multidimensional Poisson brackets of hydrodynamic type for any number $N$ of components and for any dimension $n$ by differential-geometric methods. This problem is equivalent to the classification of a special class of flat pencils of metrics. A key role in the solution of this problem was played by the theory of compatible metrics that had been earlier constructed by the present author.

We study the scattering of vortices in the Abelian (2+1)-dimensional Higgs model. We show that in the case of the symmetric head-on collision of $N$ vortices their trajectories are rotated by the angle $\pi/N$ after the collision.

We present a systematic procedure to obtain singular solutions of the constant quantum Yang-Baxter equation in arbitrary dimension. This approach, inspired by the Lie (super)algebra structure, is explicitly applied to the particular case of (graded) contractions of the orthogonal real algebra ${\mathfrak{so}}(N+1)$. In this way we show that “classical” contraction parameters which appear in the commutation relations of the contracted Lie algebras, become quantum deformation parameters, arising as entries of the resulting quantum $R$-matrices.

General properties of an Abelian connection in Fedosov deformation quantization are investigated. The definition and the criterion of being a finite formal series for an Abelian connection are presented. Examples of finite and infinite Abelian connections are given.