相关论文: Persistence and Sheaves
We give an algorithm to determine whether a kernel sheaf over a smooth projective curve over an algebraically closed field is semistable. The algorithm uses symmetric powers to make destabilizing subbundles visible as global sections.
Through the theory of Lie bi-algebroids and generalized complex structures, one could define a cohomology theory naturally associated to a holomorphic Poisson structure. It is known that it is the hypercohomology of a bi-complex such that…
In many applications concerning the comparison of data expressed by $\mathbb{R}^m$-valued functions defined on a topological space $X$, the invariance with respect to a given group $G$ of self-homeomorphisms of $X$ is required. While…
We study the local and global intersection cohomology of the intersection of two Schubert varieties in a flag manifold. In this version some new references are added.
The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global…
We investigate the dynamics of cellular solidification patterns using three-dimensional phase-field simulations. The cells can organize into stable hexagonal patterns or exhibit unsteady evolutions. We identify the relevant secondary…
Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and…
Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations…
This paper introduces an unsupervised method to estimate the class separability of text datasets from a topological point of view. Using persistent homology, we demonstrate how tracking the evolution of embedding manifolds during training…
We introduce in this paper a hypercohomology version of the resonance varieties and obtain some relations to the characteristic varieties of rank one local systems on a smooth quasi-projective complex variety $M$, see Theorem (3.1) and…
Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove…
This paper presents a mathematically rigorous framework for brain-inspired representation learning founded on the interplay between persistent topological structures and cohomological flows. Neural computation is reformulated as the…
The main aim of this paper is to explore the ideas of persistent homology and extended persistent homology, and their stability theorems, using ideas from [Bubenik and Scott, 2014; Cohen-Steiner, Edelsbrunner, and Harer, 2007; and…
A multicomplex structure is defined from an ordered lattice of multigraphs. This structure will help us to observe the features of Persistent Homology in this context, its interaction with the ordering and the repercussions of the process…
We define a deformation of our earlier link homologies for fundamental representations of sl_m. The deformed homology of a link is isomorphic to the deformed homology of the disjoint union of its components. Moreover, there exists a…
Gerstenhaber and Schack ([GS]) developed a deformation theory of presheaves of algebras on small categories. We translate their cohomological description to sheaf cohomology. More precisely, we describe the deformation space of (admissible)…
We prove existence of reflexive sheaves on singular surfaces and threefolds with prescribed numerical invariants and study their moduli.
This paper provides an overview of the applications of sheaf theory in deep learning, data science, and computer science in general. The primary text of this work serves as a friendly introduction to applied and computational sheaf theory…
We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This in particular gives a conceptual explanation of the appearance of graph cohomology…
In this paper, we propose Quasi Zigzag Persistent Homology (QZPH) as a framework for analyzing time-varying data by integrating multiparameter persistence and zigzag persistence. To this end, we introduce a stable topological invariant that…