Related papers: Persistent Local Systems of Periodic Spaces
In this paper we study Holder continuous linear cocycles over transitive Anosov diffeomorphisms. Under various conditions of relative pinching we establish properties including existence and continuity of measurable invariant sub-bundles…
Persistent homology is an important methodology in topological data analysis which adapts theory from algebraic topology to data settings. Computing persistent homology produces persistence diagrams, which have been successfully used in…
Most algorithms for computing persistent homology do so by tracking cycles that represent homology classes. There are many choices of such cycles, and specific choices have found different uses in applications. Although it is known that…
Phenomenological (P-type) bifurcations are qualitative changes in stochastic dynamical systems whereby the stationary probability density function (PDF) changes its topology. The current state of the art for detecting these bifurcations…
We present theoretical and experimental results probing the rich topological structure of arbitrarily disordered finite tight binding Hamiltonians with chiral symmetry. We extend the known classification by considering the topological…
Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically -- as long as the algorithm does not…
We reconsider the classical problem of the continuation of degenerate periodic orbits in Hamiltonian systems. In particular we focus on periodic orbits that arise from the breaking of a completely resonant maximal torus. We here propose a…
Persistent homology (PH) is a relatively new field in applied mathematics that studies the components and shapes of discrete data. In this work, we demonstrate that PH can be used as a universal framework to identify phases in spin models,…
The rings of linear continuous operators on the topological spaces of $\mathfrak{G}$-zero maps were described, where $\mathfrak{G}$ is a filter on a set with an involution. This applies to modules of formal series with well ordered support…
Persistent homology is a method for computing the topological features present in a given data. Recently, there has been much interest in the integration of persistent homology as a computational step in neural networks or deep learning. In…
In this paper, we give lower bounds for the homology of the fibers of a map to a manifold. Using new sheaf theoretic methods, we show that these lower bounds persist over whole open sets of the manifold, and that they are stable under…
Understanding the structure of high-dimensional data is fundamental to neuroscience and other data-intensive scientific fields. While persistent homology effectively identifies basic topological features such as "holes," it lacks the…
Local Hochschild, cyclic Homology and K-theory were introduced by N. Teleman in [10] with the purpose of unifying different settings of the index theorem. This paper is one of the research topics announced in [10], {\S}10. The definition of…
To understand an aperiodic tiling (or a quasicrystal modeled on an aperiodic tiling), we construct a space of similar tilings, on which the group of translations acts naturally. This space is then an (abstract) dynamical system. Dynamical…
Given a graded $E_1$-module over an $E_2$-algebra in spaces, we construct an augmented semi-simplicial space up to higher coherent homotopy over it, called its canonical resolution, whose graded connectivity yields homological stability for…
Aperiodic point sets (or tilings) which can be obtained by the method of cut and projection from higher dimensional periodic sets play an important role for the description of quasicrystals. Their topological invariants can be computed…
We compute rationally the topological (complex) K-theory of the classifying space BG of a discrete group provided that G has a cocompact G-CW-model for its classifying space for proper G-actions. For instance word-hyperbolic groups and…
Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework…
Locally symmetric spaces like $SL(n,\mathbb Z)\backslash SL_n(\mathbb R)/SO(n)$ contain immersed compact flat manifolds of dimension equal to the real rank. We give a lower bound for the contribution of these cycles to the homology of…
In this paper we prove a recursive characterisation of generic rigidity for frameworks periodic with respect to a partially variable lattice. We follow the approach of modelling periodic frameworks as frameworks on a torus and use the…