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A Hom-group G is a nonassociative version of a group where associativity, invertibility, and unitality are twisted by a map \alpha: G\longrightarrow G. Introducing the Hom-group algebra KG, we observe that Hom-groups are providing examples…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
Crystal structures are characterized by atomic bases within a primitive unit cell that repeats along a regular lattice throughout 3D space. The periodic and infinite nature of crystals poses unique challenges for geometric graph…
Persistent (co)homology is a central construction in topological data analysis, where it is used to quantify prominence of features in data to produce stable descriptors suitable for downstream analysis. Persistence is challenging to…
One of the most powerful techniques to study protein structures is to look for recurrent fragments (also called substructures or spatial motifs), then use them as patterns to characterize the proteins under study. An emergent trend consists…
Topological transforms are parametrized families of topological invariants, which, by analogy with transforms in signal processing, are much more discriminative than single measurements. The first two topological transforms to be defined…
Graph spectra are an important class of structural features on graphs that have shown promising results in enhancing Graph Neural Networks (GNNs). Despite their widespread practical use, the theoretical understanding of the power of…
Graphs serve as powerful tools for modeling pairwise interactions in diverse fields such as biology, material science, and social networks. However, they inherently overlook interactions involving more than two entities. Simplicial…
A number of machine learning tasks entail a high degree of invariance: the data distribution does not change if we act on the data with a certain group of transformations. For instance, labels of images are invariant under translations of…
This is a chapter in an upcoming book on aperiodic order. We go over different versions of tiling cohomology (\v Cech, pattern-equivariant, PV, quotient) with emphasis on the inverse limit constructions used to compute these cohomologies.…
This paper introduces the notion of involution module, the first generalization of the modular decomposition of 2-structure which has a unique linear-sized decomposition tree. We derive an O(n^2) decomposition algorithm and we take…
The isomorphism problem for centrally nilpotent loops can be tackled by methods of cohomology. We develop tools based on cohomology and linear algebra that either lend themselves to direct count of the isomorphism classes (notably in the…
We introduce a new homological machine for the study of secondary geometric invariants. The objects, called spark complexes, occur in many areas of mathematics. The theory is applied here to establish the equivalence of a large family of…
For any type of fundamental groupoid scheme, we construct an algebraic cohomology theory for varieties with coefficients in the base field. This is a minor variant of \'etale cohomology, involving neither de Rham complexes nor…
We motivate and survey the theory of BPS invariants of categories and BPS cohomology of stacks, indicating applications in enumerative geometry and representation theory, as well as recent advances.
We consider complete graphs with edge weights and/or node weights taking values in some set. In the first part of this paper, we show that a large number of graphs are completely determined, up to isomorphism, by the distribution of their…
Higher order group cohomology is defined and first properties are given. Using modular symbols, an Eichler-Shimura homomorphism is constructed mapping spaces of higher order cusp forms to higher order cohomology groups.
In this paper we introduce a novel family of attributed graphs for the purpose of shape discrimination. Our graphs typically arise from variations on the Mapper graph construction, which is an approximation of the Reeb graph for point cloud…
We give a combinatorial description (including explicit differential-form bases) for the cohomology groups of the space of n distinct nonzero complex numbers, with coefficients in rank-one local systems which are of finite monodromy around…
This manuscript develops a geometric approach to ordinary cohomology of smooth manifolds, constructing a cochain complex model based on co-oriented smooth maps from manifolds with corners. Special attention is given to the pull-back product…