Related papers: Quantifying the structural stability of simplicial…
We say that an algorithm is stable if small changes in the input result in small changes in the output. This kind of algorithm stability is particularly relevant when analyzing and visualizing time-varying data. Stability in general plays…
Given a manifold with boundary, one can consider the space of subsurfaces of this manifold meeting the boundary in a prescribed fashion. It is known that these spaces of subsurfaces satisfy homological stability if the manifold has at least…
We introduce the notion of doubling and r-tupling for simplicial complexes, a notion reminiscent to that of matching complexes in graph theory. We prove a connectivity result for such complexes and relate r-tupling to stabilizing r times…
We establish a strong, geometric lower bound on the (sequential) topological complexity of the unordered configuration spaces of a general graph. As an application, we show that, for most graphs, the topological complexity eventually…
We prove a homological stability theorem for congruence subgroups of symplectic groups. From this theorem, we deduce a generalization of a theorem of Borel showing that certain homology groups of a congruence subgroup do not depend on the…
Robustness of higher-order networks is often quantified by the instantaneous smallest positive eigenvalue of the Hodge $1$-Laplacian under simplex deletion. We show that this observable is generically ill-defined: along a deletion…
Associated to every group with a weak spherical Tits system of rank n+1 with an appropriate rank n subgroup, we construct a relative spectral sequence involving group homology of Levi subgroups of both groups. Using the fact that such Levi…
We generalize the stable graph regularity lemma of Malliaris and Shelah to the case of finite structures in finite relational languages, e.g., finite hypergraphs. We show that under the model-theoretic assumption of stability, such a…
We illustrate how to extend the concept of structural stability through applying it to the front propagation speed selection problem. This consideration leads us to a renormalization group study of the problem. The study illustrates two…
Persistent homology is constrained to purely topological persistence while multiscale graphs account only for geometric information. This work introduces persistent spectral theory to create a unified low-dimensional multiscale paradigm for…
Given a filtration function on a finite simplicial complex, stability theorem of persistent homology states that the corresponding barcode is continuous with respect to changes in the filtration function. However, due to the discrete…
Symmetries are ubiquitous in network systems and have profound impacts on the observable dynamics. At the most fundamental level, many synchronization patterns are induced by underlying network symmetry, and a high degree of symmetry is…
We show that the homology of strata of abelian differentials stabilizes in a range where the number of simple zeros is large relative to the homological degree. In this range, we show that the rational cohomology agrees with the restriction…
We elaborate on the interpretation of some mixed finite element spaces in terms of differential forms. First we develop a framework in which we show how tools from algebraic topology can be applied to the study of their cohomological…
In the study of dynamical systems on networks/graphs, a key theme is how the network topology influences stability for steady states or synchronized states. Ideally, one would like to derive conditions for stability or instability that…
Motivated by Xia-Zhou's recent work on applying symmetry groups to the N-body problem, we will study relative equilibria of the equilateral triangle and the square configurations under $\alpha$-homogeneous and quasi-homogeneous potentials…
Persistent homology is a topological data analysis tool that has been widely generalized, extending its scope beyond the field of topology. Among its extensions, steady and ranging persistence were developed to study a wide variety of graph…
Hypergraphs and simplical complexes both capture the higher-order interactions of complex systems, ranging from higher-order collaboration networks to brain networks. One open problem in the field is what should drive the choice of the…
In this paper we present a new proof of the homological stability of the moduli space of closed surfaces in a simply connected background space $K$, which we denote by $S_g (K)$. The homology stability of surfaces in $K$ with an arbitrary…
The computation of homology groups for evolving simplicial complexes often requires repeated reconstruction of boundary operators, resulting in prohibitive costs for large-scale or frequently updated data. This work introduces MMHM, a…