Related papers: Some Applications of a Multiplicative Structure on…

We show a possibility to apply certain philosophical concepts to the analysis of concrete mathematical structures. Such application gives a clear justification of topological and geometric properties of considered mathematical objects.

We study endomorphism rings of principally polarized abelian surfaces over finite fields from a computational viewpoint with a focus on exhaustiveness. In particular, we address the cases of non-ordinary and non-simple varieties. For each…

We study some graphs associated to a surface, called k-multicurve graphs, which interpolate between the curve complex and the pants graph. Our main result is that, under certain conditions, simplicial embeddings between multicurve graphs…

We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…

We study manifolds with split-complex structure and apply some general results to the study of Lorentz surfaces. In particular, we apply our results to timelike minimal immersions. The conformal realization of these surfaces is obtained…

We develop the theory of locally small spaces in a new simple language and apply this simplification to re-build the theory of locally definable spaces over structures with topologies.

We use the methods of empirical mathematics to show that iterative logarithmic operations will result in an attractor point on the complex plane. Moreover, we demonstrate that different bases converge onto different attractors. Finally, we…

In this note, we present a new look at translationally equivariant minimal Lagrangian surfaces in the complex projective plane via the loop group method.

We study harmonic surfaces in $\mathbb{R}^3$ through the framework of harmonic Enneper immersions and prove a superposition principle for such surfaces. We prove that minimal and maximal surfaces admit a decomposition into harmonic…

We define a symplectic structure on the space of non parametrized loops in $G_2$ manifold. We also develop some basics of intersection theory of Lagrangian submanifolds.

Embedding diagrams have been used extensively to visualize the properties of curved space in Relativity. We introduce a new kind of embedding diagram based on the {\it extrinsic} curvature (instead of the intrinsic curvature). Such an…

We construct three sequences of regular surfaces of general type with unbounded numerical invariants whose canonical map is 2-to-1 onto a canonically embedded surface. Only sporadic examples of surfaces with these properties were previously…

This is a latest survey article on embeddings of multibranched surfaces into 3-manifolds.

In this paper, we study the intrinsic mean curvature flow on certain closed spacelike manifolds, and prove the existence of hyperbolic structures on them.

We develop a theory of simple pentagonal subdivision of quadrilateral tilings, on orientable as well as non-orientable surfaces. Then we apply the theory to answer questions related to pentagonal tilings of surfaces, especially those…

The simple loop conjecture for 3-manifolds states that every 2-sided immersion of a closed surface into a 3-manifold is either injective on fundamental groups or admits a compression. This can be viewed as a generalization of the Loop…

The simplest version of the Spin-polynomial invariants of the underlying differentiable structures of algebraic surfaces were considered and the simplest arguments were used in order to distinguish the underlying smooth structures of…

A correspondence between different $Pin$-type structures on a compact surface and quadratic (linear) forms on its homology is constructed. Addition of structures is defined and expressed in terms of these quadratic forms.

A spatial surface is a compact surface embedded in the 3-sphere. In this paper, we provide several typical examples of spatial surfaces and construct a coloring invariant to distinguish them. The coloring is defined by using a multiple…

A classification of 2-dimensional surfaces imbedded in spacetime is presented, according to the algebraic properties of their shape tensor. The classification has five levels, and provides among other things a refinement of the concepts of…