Related papers: Framed combinatorial topology
Classical mathematics are founded within set theory, but sets don't have \emph{symmetries}. We conjecture that if we allow sets with symmetries, then many problems such as \emph{Mirror symmetry} or \emph{Homological mirror symmetry} can be…
Factorization algebras are local-to-global objects living on manifolds, and they arise naturally in mathematics and physics. Their local structure encompasses examples like associative algebras and vertex algebras; in these examples, their…
Using a notation of corner between edges when graph has a fixed rotation, i.e. cyclical order of edges around vertices, we define combinatorial objects - combinatorial maps as pairs of permutations, one for vertices and one for faces.…
Topological data analysis is a relatively new branch of machine learning that excels in studying high dimensional data, and is theoretically known to be robust against noise. Meanwhile, data objects with mixed numeric and categorical…
For the classical mind, quantum mechanics is boggling enough; nevertheless more bizarre behavior could be imagined, thereby concentrating on propositional structures (empirical logics) that transcend the quantum domain. One can also…
We generalize the classical probability frame by adopting a wider family of random variables that includes nondeterministic ones. The frame that emerges is known to host a ''classical'' extension of quantum mechanics. We discuss the notion…
Combinatorial design theory studies set systems with certain balance and symmetry properties and has applications to computer science and elsewhere. This paper presents a modular approach to formalising designs for the first time using…
The quantum mechanical formalism doesn't support our intuition, nor does it elucidate the key concepts that govern the behaviour of the entities that are subject to the laws of quantum physics. The arrays of complex numbers are kin to the…
With any state of a multipartite quantum system its separability polytope is associated. This is an algebro-topological object (non-trivial only for mixed states) which captures the localisation of entanglement of the state. Particular…
In this paper, we introduce the foundation of a fractal topological space constructed via a family of nested topological spaces endowed with subspace topologies, where the number of topological spaces involved in this family is related to…
An objective of the theory of combinatorial groupoids is to introduce concepts like "holonomy", "parallel transport", "bundles", "combinatorial curvature" etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes,…
While topological data analysis has emerged as a powerful paradigm for structural inference, its foundational tools, notably persistent homology and the persistent Laplacian, are frequently insensitive to localized structural fluctuations…
Random intersection graphs model networks with communities, assuming an underlying bipartite structure of groups and individuals, where these groups may overlap. Group memberships are generated through the bipartite configuration model.…
Correlation clustering is a widely-used approach for clustering large data sets based only on pairwise similarity information. In recent years, there has been a steady stream of better and better classical algorithms for approximating this…
Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a…
Interactions in complex systems are widely observed across various fields, drawing increased attention from researchers. In mathematics, efforts are made to develop various theories and methods for studying the interactions between spaces.…
We study tilings of the plane that combine strong properties of different nature: combinatorial and algorithmic. We prove existence of a tile set that accepts only quasiperiodic and non-recursive tilings. Our construction is based on the…
These lecture notes cover 13 sessions and are presented as an e-print, intended to evolve over time. Quantum invariants do more than distinguish topological objects; they build bridges between topology, algebra, number theory and quantum…
A basic assumption of tiling theory is that adjacent tiles can meet in only a finite number of ways, up to rigid motions. However, there are many interesting tiling spaces that do not have this property. They have "fault lines", along which…
For groups of a topological origin, such as braid groups and mapping class groups, an important source of interesting and highly non-trivial representations is given by their actions on the twisted homology of associated spaces; these are…