Related papers: Interaction homotopy and interaction homology
We construct open-closed superstring interactions based on the open-closed homotopy algebra structure. It provides a classical open superstring field theory on general closed-superstring-field backgrounds described by classical solutions of…
Systems of cooperation and interaction are usually studied in the context of real or complex vector spaces. Additional insight, however, is gained when such systems are represented in vector spaces with multiplicative structures, i.e., in…
A major challenge for autonomous vehicles is handling interactive scenarios, such as highway merging, with human-driven vehicles. A better understanding of human interactive behaviour could help address this challenge. Such understanding…
We study several different notions of algebraicity in use in stable homotopy theory and prove implications between them. The relationships between the different meanings of algebraic are unexpectedly subtle, and we illustrate this with…
A topology is introduced on spaces of Legendrian submanifolds and groups of contactomorphisms. The definition is motivated by the Alexandrov topology in Lorentz geometry.
We present an algebraic framework for interacting extended quantum systems to study complex phenomena characterized by the coexistence and competition of different states of matter. We start by showing how to connect different…
In this text we expose basic cases of some fundamental ideas and methods of topology. Namely, of homotopy, degree, fundamental group, covering, Whitehead invariant, etc. This is done by considering the elementary example: closed polygonal…
The question of whether a given H-space X is, up to homotopy, a loop space has been studied from a variety of viewpoints. Here we address this question from the aspect of homotopy operations, in the classical sense of operations on homotopy…
A convenient algebraic structure to describe some forms of dynamics of two hamiltonian systems with nonpotential (magnetic--type) interaction is considered. An algebraic mechanism of generation of such dynamics is explored on simple "toy"…
We define and develop a homotopy invariant notion for the topological complexity of a map $f:X \to Y$, denoted TC($f$), that interacts with TC($X$) and TC($Y$) in the same way cat($f$) interacts with cat($X$) and cat($Y$). Furthermore,…
Intersection homology is a topological invariant which detects finer information in a space than ordinary homology. Using ideas from classical simple homotopy theory, we construct local combinatorial transformations on simplicial complexes…
Homotopy is an important feature of associative and Jordan algebraic structures: such structures always come in families whose members need not be isomorphic among other, but still share many important properties. One may regard homotopy as…
This is a survey of knot contact homology, with an emphasis on topological, algebraic, and combinatorial aspects.
Graph-based collaborative filtering is capable of capturing the essential and abundant collaborative signals from the high-order interactions, and thus received increasingly research interests. Conventionally, the embeddings of users and…
Persistent homology is a powerful tool for characterizing the topology of a data set at various geometric scales. When applied to the description of molecular structures, persistent homology can capture the multiscale geometric features and…
Topological techniques have become a popular tool for studying information flows in neural networks. In particular, simplicial homology theory is used to analyze how cognitive representations of space emerge from large conglomerates of…
The monograph summarizes the author's results on the geometry of anholonomic and locally anisotropic interactions, published in J. Math. Phys., Nucl. Phys. B, Ann. Phys. (NY), JHEP, Rep. Math. Phys., Int. J. Theor. Phys. and in some former…
Here we shall consider the topology and dynamics associated to a wide class of matchbox manifolds, including a large selection of tiling spaces and all minimal matchbox manifolds of dimension one. For such spaces we introduce topological…
In this work, we introduce a novel approach based on algebraic topology to enhance graph convolution and attention modules by incorporating local topological properties of the data. To do so, we consider the framework of sheaf neural…
We examine "dynamical similarities" in the Lagrangian framework. These are symmetries of an intrinsically determined physical system under which observables remain unaffected, but the extraneous information is changed. We establish three…