Related papers: Path space forms and surface holonomy
We construct two algebraic versions of homotopy theory of rational disconnected topological spaces, one based on differential graded commutative associative algebras and the other one on complete differential graded Lie algebras. As an…
Hyperspaces form a powerful tool in some branches of mathematics: lots of fractal and other geometric objects can be viewed as fixed points of some functions in suitable hyperspaces - as well as interesting classes of formal languages in…
This paper gives an introduction to the homotopy theory of quasi-categories. Weak equivalences between quasi-categories are characterized as maps which induce equivalences on a naturally defined system of groupoids. These groupoids…
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 algebraic varieties parametrized by topological spaces and enlarge the domains of Lawson homology and morphic cohomology to this category. We prove a Lawson suspension theorem and splitting theorem. A version of Friedlander-Lawson…
The paper is devoted to introduce some notions extending the unique path lifting property from a homotopy viewpoint and to study their roles in the category of fibrations. First, we define some homotopical kinds of the unique path lifting…
We consider the problem of inferring the topology of a network using the measurements available at the end nodes, without cooperation from the internal nodes. To this end, we provide a simple method to obtain path interference which…
We consider Euclidean path integrals with higher derivative actions, including those that depend quadratically on acceleration, velocity and position. Such path integrals arise naturally in the study of stiff polymers, membranes with…
From descent theory to higher geometry, the idea of gluing has been embedded in many elegant and powerful techniques, proving instrumental for the solution of many problems. In this paper, we introduce a framework that allows to link…
In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…
While many classical traffic models treat the spatial extension of streets continuously or by discretization into cells of a certain length, we will subdivide roads into comparatively long homogeneous road sections of constant capacity with…
Many hypergeometric differential systems that arise from a geometric setting can be endowed with the structure of mixed Hodge modules. We generalize this fundamental result to the tautological systems associated to homogeneous spaces by…
We propose a practical computational framework for detecting structural changes in parameter-dependent topological data. In many applications, such as time-series data analysis, anomaly detection, and monitoring of systems under changing…
Deep networks learn internal representations whose geometry--how features bend, rotate, and evolve--affects both generalization and robustness. Existing similarity measures such as CKA or SVCCA capture pointwise overlap between activation…
The main purpose of this article is to introduce a comprehensive, unified theory of the geometry of all connections. We show that one can study a connection via a certain, closely associated second-order differential equation. One of the…
For a pointed topological space $X$, we use an inductive construction of a simplicial resolution of $X$ by wedges of spheres to construct a "higher homotopy structure" for $X$ (in terms of chain complexes of spaces). This structure is then…
We develop a new perspective on the discretization of the phase space structure of gravity in 2+1 dimensions as a piecewise-flat geometry in 2 spatial dimensions. Starting from a subdivision of the continuum geometric and phase space…
Multipath cohomology is a cohomology theory for directed graphs, which is defined using the path poset. The aim of this paper is to investigate combinatorial properties of path posets, and to provide computational tools for multipath…
We consider commuting pairs of holomorphic endomorphisms of P^2 with disjoint sequence of iterates. The remaining case to be studied is when their degrees coincide after some number of iterations. We show in this case that they are either…
We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial…