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We introduce the concept of a soft ditopological space as the "soft generalization" of the concept of a ditopological space as it is defined in the papers by L.M. Brown and co-authors, see e.g. L. M. Brown, R. Erturk, S. Dost,…
A method to define the complex structure and separate the conformal mode is proposed for a surface constructed by two-dimensional dynamical triangulation. Applications are made for surfaces coupled to matter fields such as $n$ scalar fields…
In [1] we introduced the concept of structured space, which is a topological space that locally resembles some algebraic structures. In [2] we proceeded the study of these spaces, developing two cohomology theories. The aim of this paper is…
We study automorphisms of smooth hypersurfaces in projective space $\mathbb{P}^{n+1}$ whose fixed loci have codimension at most two for $n\geq2$. While classifications of possible orders of automorphisms are known, our aim is to explore the…
We give a systematic approach to constructing non-reduced, locally Cohen-Macaulay schemes with reduced support a smooth projective variety. The hierarchy of such structures includes a lot of information about the underlying variety, its…
We explain in some detail the geometric structure of spheres in any dimension. Our approach may be helpful for other homogeneous spaces (with other signatures) such as the de Sitter and anti-de Sitter spaces. We apply the procedure to the…
Closed (and simply-connected) manifolds whose dimensions are greater than 4 are classified via sophisticated algebraic and abstract theory such as surgery theory and homotopy theory. It is difficult to handle 3 or 4-dimensional closed…
Syzygies capture intricate geometric properties of a subvariety in projective space. However, when the ambient space is a product of projective spaces or a more general smooth projective toric variety, minimal free resolutions over the Cox…
This is an introduction to the subject of the differential topology of the space of smooth loops in a finite dimensional manifold. It began as the background notes to a series of seminars given at NTNU and subsequently at Sheffield. I am…
This is the third in a series of papers on the geometry and analysis of singular area minimizing hypersurfaces. We show how to derive obstruction and structure theories for scalar curvature constraints without imposing dimensional or…
Deformational structures, in many aspects generalizing standard elasticity theory, are investigated in abstract form. Within free deformational structures we define algebra of deformations, classify them by its special properties, define…
We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and…
In this paper which is the first of a series of papers on smooth structures, the concepts of C-structures and smooth structures are introduced and studied. The notion of smooth structure on semi-integral domains is given. It is shown that…
Concepts and techniques from the theory of G-structures of higher order are applied to the study of certain structures (volume forms, conformal structures, linear connections and projective structures) defined on a pseudo-Riemanniann…
We survey known results about the complexity of surjective homomorphism problems, studied in the context of related problems in the literature such as list homomorphism, retraction and compaction. In comparison with these problems,…
This survey focuses on the geometric problem of log-surfaces, which are pairs consisting of a smooth projective surface and a reduced non-empty boundary divisor. In the first part, we focus on the geography problem for complex log-surfaces…
This article is a continuation of a previous work that dealt with the topological obstructions to the reductions of the bundle of linear frames on a spacetime manifold for a particular chain of subgroups of GL(4). In this article, the…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
The $s$-th higher topological complexity of a space $X$, $TC_s(X)$, can be estimated from above by homotopical methods, and from below by homological methods. We give a thorough analysis of the gap between such estimates when $X=RP^m$, the…
Higher dimensional supersymmetric quantum mechanics is studied. General properties of the two dimensional case are presented. For three spatial dimesions or higher, a spin structure is shown to arise naturally from the nonrelativistic…