Related papers: Curvature Sets Over Persistence Diagrams
We characterize structures such as monotonicity, convexity, and modality in smooth regression curves using persistent homology. Persistent homology is a key tool in topological data analysis that detects higher-dimensional topological…
We study the Abramovich--Vistoli moduli space of genus zero orbifold stable maps to [Sym^2 P^2], the stack symmetric square of P^2. This compactifies the moduli space of stable maps from hyperelliptic curves to P^2, and we show that all…
Sets of invariant measures are considered for continuous maps of a metric compact set. We take Kantorovich metric to calculate distance between measures and Hausdorff metrics to calculate distance between compact sets. Consider the function…
The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space.…
Polymers are large macromolecules composed of repeating structural units known as monomers and are widely applied in fields such as energy storage, construction, medicine, and aerospace. However, existing graph neural network methods,…
This paper is the second part of a curvature-based program for rigid-body dynamics on SU2. In Part I, Curvature-Driven Dynamics on S3: A Geometric Atlas, we introduced the inertial curvature field Kgeo associated with a left-invariant…
We propose a variation of the classical Hilbert scheme of points - the double nested Hilbert scheme of points - which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth…
We develop a framework for algorithms finding the diameter in graphs of bounded distance Vapnik-Chervonenkis dimension, in (parameterized) subquadratic time complexity. The class of bounded distance VC-dimension graphs is wide, including,…
In this paper we study pseudo-Riemannian spaces with a degenerate curvature structure i.e. there exists a continuous family of metrics having identical polynomial curvature invariants. We approach this problem by utilising an idea coming…
Persistence diagrams are one of the main tools in the field of Topological Data Analysis (TDA). They contain fruitful information about the shape of data. The use of machine learning algorithms on the space of persistence diagrams proves to…
For two-parameter families of dissipative twist maps, we investigate the dynamics of invariant graphs as well as the thresholds for their existence and breakdown. Our main results are as follows: (1) For arbitrarily small $C^r$…
Scientific time series often encode predictive geometric structure, including connectivity, cycles, shell-like geometry, directional changes, and nonlinear neighborhoods, that standard dot-product attention does not explicitly represent. We…
Graph Sparsification aims at compressing large graphs into smaller ones while preserving important characteristics of the input graph. In this work we study Vertex Sparsifiers, i.e., sparsifiers whose goal is to reduce the number of…
The stability of persistence diagrams is among the most important results in applied and computational topology. Most results in the literature phrase stability in terms of the bottleneck distance between diagrams and the $\infty$-norm of…
We generalize Milnor link invariants to all types of surface-links in $4$--space (possibly with boundary). This is achieved by using the notion of cut-diagram, which is a 2-dimensional generalization of Gauss diagrams, associated to…
Persistence diagrams, the most common descriptors of Topological Data Analysis, encode topological properties of data and have already proved pivotal in many different applications of data science. However, since the (metric) space of…
Multiparameter persistent homology has emerged as a powerful generalization of topological data analysis, capable of encoding multivariate filtrations. However, the algebraic complexity of multiparameter persistence modules, marked by wild…
Let $G$ be a group acting properly and by isometries on a metric space $X$; it follows that the quotient or orbit space $X/G$ is also a metric space. We study the Vietoris-Rips and \v{C}ech complexes of $X/G$. Whereas (co)homology theories…
Splitting invariants describe how a plane curve "splits" by the pull-back under a Galois cover over the projective plane whose branch locus contains no component of the plane curve. They enable us to distinguish the embedded topology of…
We introduce the notion of matrices graph, defining continued fraction algorithms where the past and the future are almost independent. We provide an algorithm to convert more general algorithms into matrices graphs. We present an algorithm…