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We study rack and quandle coverings from a universal algebraic viewpoint and we show how they can be understood using the notion of strongly abelian congruences. We provide an abstract characterization of several particular types of…
We give a proof of the existence of generalized definable locally compact models for arbitrary approximate subgroups via an application of topological dynamics in model theory. Our construction is simpler and shorter than the original one…
We present a general scheme how to construct a substitution rule for generating $d$-dimensional analogues of the paperfolding structures. This substitution is proven to be primitive, so that the translation action on the hull forms a…
We consider complex manifolds that admit actions by holomorphic transformations of classical simple real Lie groups and classify all such manifolds in a natural situation. Under our assumptions, which require the group at hand to be…
It is now very known how the subprojectivity of modules provides a fruitful new unified framework of the classical projectivity and flatness. In this paper, we extend this fact to the category of complexes by generalizing and unifying…
A global representation is a compatible collection of representations of the outer automorphism groups of the groups belonging to some collection of finite groups $\mathscr{U}$. Global representations assemble into an abelian category…
In this paper some reflections on the concept of transition are presented: groupoids are introduced as models for the construction of a ``generalized logic'' whose basic statements involve pairs of propositions which can be conditioned. In…
This is a survey article written for the Jahresberichte der DMV. Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and…
We introduce a natural notion of holomorphic map between generalized complex manifolds and we prove some related results on Dirac structures and generalized Kaehler manifolds.
We exploit the theory of $\infty$-stacks to provide some basic definitions and calculational tools regarding stratified homotopy theory of stratified topological stacks.
We generalize the dual notions of "expansion" and "collapse" so they can be applied to arbitrary metric spaces. We also expand the theory to allow for infinitely many such moves. Those tools are then employed to prove a variety of…
We introduce the notion of families of n-marked smooth rational tropical curves over smooth tropical varieties and establish a one-to-one correspondence between (equivalence classes of) these families and morphisms from smooth tropical…
Given a finite state space E, we build a universal dilation for all possible discrete time Markov chains on E, homogeneous or not: we introduce a second system (an ``environment'') and a deterministic invertible time-homogeneous global…
In this paper, we define a new structure analogous to group, called partial group. This structure concerns the partial stability by the composition inner law. We generalize the three isomorphism theorems for groups to partial groups.
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…
Complex systems have motivated continuing interest from the scientific community, leading to new concepts and methods. Growing systems represent a case of particular interest, as their topological, geometrical, and also dynamical properties…
We introduce the notion of evolution on sets and study several sets endowed with this structure and obtain some results about this new notion.
Many moduli spaces are constructed as quotients of group actions; this paper surveys the classical theory, as well as recent progress and applications. We review geometric invariant theory for reductive groups and how it is used to…
We construct classifying spaces for discrete and compact Lie groups, with the property that they are topological groups and complete metric spaces in a natural way. We sketch a program in view of extending these constructions.
Topos properties of the category of covering groupoids over a fixed groupoid are discussed. A classification result for connected covering groupoids over a fixed groupoid analogous to the fundamental theorem of Galois theory is given.