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In this article we study two "strong" topologies for spaces of smooth functions from a finite-dimensional manifold to a (possibly infinite-dimensional) manifold modeled on a locally convex space. Namely, we construct Whitney type topologies…

This paper reproduces the text of a part of the Author's DPhil thesis. It gives a proof of the classification of non-trivial, finite homogeneous geometries of sufficiently high dimension which does not depend on the classification of the…

We define a class of finite groups based on the properties of the closed twins of their power graphs and study the structure of those groups. As a byproduct, we obtain results about finite groups admitting a partition by cyclic subgroups.

The 2+1D topological order can be characterized by the mapping-class-group representations for Riemann surfaces of genus-1, genus-2, etc. In this paper, we use those representations to determine the possible gapped boundaries of a 2+1D…

The thesis presents the subject of synthetic topology, especially with relation to metric spaces. A model of synthetic topology is a categorical model in which objects possess an intrinsic topology in a suitable sense, and all morphisms are…

We show that the closure of the compactly supported mapping class group of an infinite-type surface is not generated by the collection of multitwists (i.e. products of powers of twists about disjoint non-accumulating curves).

We address the question of determining which mapping class groups of infinite-type surfaces admit nonelementary continuous actions on hyperbolic spaces. More precisely, let $\Sigma$ be a connected, orientable surface of infinite type with…

For any finite type connected surface $S$, we give an infinite presentation of the fundamental group $\pi_1(S,\ast)$ of $S$ based at an interior point $\ast\in{S}$ whose generators are represented by simple loops. When $S$ is…

The present paper are the notes of a mini-course addressed mainly to non-experts. It purpose it to provide a first approach to the theory of mapping class groups of non-orientable surfaces.

This paper gives a systematic construction of certain covers of finite semigroups. These covers will be used in future work on the complexity of finite semigroups.

In this paper, using some properties of fundamental groups and covering spaces of connected polyhedra and CW-complexes, we present topological proof for some famous theorems about finitely presented groups.

In a recent paper Cameron, Lakshmanan and Ajith began an exploration of hypergraphs defined on algebraic structures, especially groups, to investigate whether this can add a new perspective. Following their suggestions, we consider suitable…

In this paper we study arithmetical and structural features of a finite group that possesses exactly two conjugacy class sizes that are composite numbers.

We show how to use topological ideas, such as compactness, to establish orderability properties of infinite groups. A new application is to provide a left-ordering for the group of PL homeomorphisms of a connected surface with boundary…

We study stable commutator length on mapping class groups of certain infinite-type surfaces. In particular, we show that stable commutator length defines a continuous function on the commutator subgroups of such infinite-type mapping class…

Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…

In this note we construct an unlimited family of irregular algebraic surfaces of general type with canonical map of degree $ 8 $, irregularity $ 1 $ and arbitrarily large geometric genus such that the image of the canonical map is not a…

Recent evidence indicates that the abundance of recurring elementary interaction patterns in complex networks, often called subgraphs or motifs, carry significant information about their function and overall organization. Yet, the…

The union of a directed family of topological groups can be equipped with two noteworthy topologies: the finest topology making each injection continuous, and the finest group topology making each injection continuous. This begs the…

In this paper we study the possibility to define irreducible representations of the symmetric groups with the help of finitely many relations. The existence of finite bases is established for the classes of representations corresponding to…