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We study orbit spaces of generalized gradient vector fields for Morse functions. Typically, these orbit spaces are non-Hausdorff. Nevertheless, they are quite structured topologically and are amenable to study. We show that these orbit…
Monotone vector fields were introduced almost 40 years ago as nonlinear extensions of positive definite linear operators, but also as natural extensions of gradients of convex potentials. These vector fields are not always derived from…
After surveying classical notions of PL topology of the Seventies, we clarify the relation between Morse theory and its discretization by Forman. We show that PL handles theory and discrete Morse theory are equivalent, in the sense that…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
This paper considers the problem of embedding directed graphs in Euclidean space while retaining directional information. We model a directed graph as a finite set of observations from a diffusion on a manifold endowed with a vector field.…
In this paper, we prove that discrete Morse functions on digraphs are flat Witten-Morse functions and Witten complexes of transitive digraphs approach to Morse complexes. We construct a chain complex consisting of the formal linear…
Our objective is to develop a stratified Morse theory with tangential conditions. We define a continuous strata-wise smooth Morse function on an abstract stratified space by using control conditions and radiality assumptions on the gradient…
A large class of real $3$-dimensional nilpotent polynomial vector fields of arbitrary degree is considered. The aim of this work is to present general properties of the discrete and continuous dynamical systems induced by these vector…
The Morse complex $\mathcal{M}(\Delta)$ of a finite simplicial complex $\Delta$ is the complex of all gradient vector fields on $\Delta$. In this paper we study higher connectivity properties of $\mathcal{M}(\Delta)$. For example, we prove…
We construct a space of vector fields that are normal to differentiable curves in the plane. Its basis functions are defined via saddle point variational problems in reproducing kernel Hilbert spaces (RKHSs). First, we study the properties…
A great open problem is: can one learn the topology of the non-smooth path spaces with an L2 Hodge-deRham theory This one hopes to establish through a suitable complex of differential forms. Since the space is a Banach manifolds, and the…
We discuss different generalizations of the classical notion of the index of a singular point of a vector field to the case of vector fields or 1-forms on singular varieties, describe relations between them and formulae for their…
A novel Neural Network architecture is proposed using the mathematically and physically rich idea of vector fields as hidden layers to perform nonlinear transformations in the data. The data points are interpreted as particles moving along…
Various applications ranging from robotics to climate science require modeling signals on non-Euclidean domains, such as the sphere. Gaussian process models on manifolds have recently been proposed for such tasks, in particular when…
Theorems on the existence of vector fields with given sets of Indexes of isolated Singular points are proved for the cases of closed manifolds, pairs of manifolds, manifolds with boundary, and gradient fields. It is proved that, on a…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
We consider quotients of complex Stiefel manifolds by finite cyclic groups whose action is induced by the scalar multiplication on the corresponding complex vector space. We obtain a description of their tangent bundles, compute their mod p…
We introduce a classical field theory based on a concept of extended causality that mimics the causality of a point-particle Classical Mechanics by imposing constraints that are equivalent to a particle initial position and velocity. It…
Computing homology and cohomology is at the heart of many recent works and a key issue for topological data analysis. Among homological objects, homology generators are useful to locate or understand holes (especially for geometric…
In this paper we address the following questions: (i) Let $C\subset \mathbb C^2$ be an orbit of a polynomial vector field which has finite total Gaussian curvature. Is $C$ contained in an algebraic curve? (ii) What can be said of a…