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Coherence theorems for covariant structures carried by a category have traditionally relied on the underlying term rewriting system of the structure being terminating and confluent. While this holds in a variety of cases, it is not a…
Simple drawings are drawings of graphs in which any two edges intersect at most once (either at a common endpoint or a proper crossing), and no edge intersects itself. We analyze several characteristics of simple drawings of complete…
This paper is a short introduction to the theory of tangles, both in graphs and general connectivity systems. An emphasis is put on the correspondence between tangles of order k and k-connected components. In particular, we prove that there…
We present a graph theory-based method to characterise flow defects and structural shifts in condensed matter. We explore the connection between dynamical properties, particularly the recently introduced concept of ''softness'', and…
The collection of reflecting hyperplanes of a finite Coxeter group is called a reflection arrangement and it appears in many subareas of combinatorics and representation theory. We focus on the problem of counting regions of reflection…
We introduce a new combinatorial abstraction for the graphs of polyhedra. The new abstraction is a flexible framework defined by combinatorial properties, with each collection of properties taken providing a variant for studying the…
Symmetry breaking for graphs and other combinatorial objects is notoriously hard. On the one hand, complete symmetry breaks are exponential in size. On the other hand, current, state-of-the-art, partial symmetry breaks are often considered…
Any solid object can be decomposed into a collection of convex polytopes (in short, convexes). When a small number of convexes are used, such a decomposition can be thought of as a piece-wise approximation of the geometry. This…
Entanglement is a complexity measure of digraphs that origins in fixed-point logics. Its combinatorial purpose is to measure the nested depth of cycles in digraphs. We address the problem of characterizing the structure of graphs of…
Semigraphoids are combinatorial structures that arise in statistical learning theory. They are equivalent to convex rank tests and to polyhedral fans that coarsen the reflection arrangement of the symmetric group. We resolve two problems on…
The paper surveys highlights of the ongoing program to classify discrete polyhedral structures in Euclidean 3-space by distinguished transitivity properties of their symmetry groups, focussing in particular on various aspects of the…
Twisting process for quantum linear spaces is defined. It consists in a particular kind of globally defined deformations on finitely generated algebras. Given a quantum space (A_1,A), a multiplicative cosimplicial quasicomplex C[A_1] in the…
Ubiquitous geometric objects can be precisely and efficiently represented as polyhedra. The transformation of a polyhedron into a vector, known as polyhedra representation learning, is crucial for manipulating these shapes with mathematical…
The theory of finite automata concerns itself with words in a free monoid together with concatenation and without further structure. There are, however, important applications which use alphabets which are structured in some sense. We…
Given the set of paths through a digraph, the result of uniformly deleting some vertices and identifying others along each path is coherent in such a way as to yield the set of paths through another digraph, called a \emph{path abstraction}…
Tetravalent graphs admitting a half-arc-transitive subgroup of automorphisms, that is a subgroup acting transitively on its vertices and its edges but not on its arcs, are investigated. One of the most fruitful approaches for the study of…
In this thesis we give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. In his seminal book, Connes constructs a map from the equivariant cohomology of a manifold carrying the…
We describe symmetry structure of a general singular theory (theory with constraints in the Hamiltonian formulation), and, in particular, we relate the structure of gauge transformations with the constraint structure. We show that any…
The usual coherence theorem of MacLane for categories with multiplication assumes that a certain pentagonal diagram commutes in order to conclude that associativity isomorphisms are well defined in a certain practical sense. The practical…
Graphs, and graph transformation systems, are used in many areas within Computer Science: to represent data structures and algorithms, to define computation models, as a general modelling tool to study complex systems, etc. Research in term…