Related papers: On Targeted Complexity of Discrete Motion

Given a compact metric space $X$, we associate to it an inverse sequence of finite $T_0$ topological spaces. The inverse limit of this inverse sequence contains a homeomorphic copy of $X$ that is a strong deformation retract. We provide a…

Topological data analysis can extract effective information from higher-dimensional data. Its mathematical basis is persistent homology. The persistent homology can calculate topological features at different spatiotemporal scales of the…

We design a motion planning algorithm to coordinate the movements of two robots along a figure eight track, in such a way that no collisions occur. We use a topological approach to robot motion planning that relates instabilities in motion…

We introduce a quasilocal version of holographic complexity adapted to `terminal states' such as spacelike singularities. We use a modification of the action-complexity ansatz, restricted to the past domain of dependence of the terminal…

In this paper we define and study for a finite partially ordered set P a class of simplicial complexes on the set P_r of r-element multichains from P. The simplicial complexes depend on a strictly monotone function from [r] to [2r]. We show…

We define and develop a homotopy invariant notion for the sequential topological complexity of a map $f:X\to Y,$ denoted $TC_{r}(f)$, that interacts with $TC_{r}(X)$ and $TC_{r}(Y)$ in the same way Jamie Scott's topological complexity map…

In this paper we study the complex symmetry in the several variable Fock space by using the techniques of weighted composition operators and semigroups. We characterize unbounded weighted composition operators that are (real) complex…

We introduce a new `geometric realization' of an (abstract) simplicial complex, inspired by probability theory. This space (and its completion) is a metric space, which has the right (weak) homotopy type, and which can be compared with the…

In this paper we prove that a simplicial map of finite-dimensional locally finite simplicial complexes has contractible point inverses if and only if it is an $\epsilon$-controlled homotopy equivalence for all $\epsilon>0$ if and only if…

In an earlier paper of Cadek, Vokrinek, Wagner, and the present authors, we investigated an algorithmic problem in computational algebraic topology, namely, the computation of all possible homotopy classes of maps between two topological…

We show how locally smooth actions of compact Lie groups on a manifold $X$ can be used to obtain new upper bounds for the topological complexity $\TC(X)$, in the sense of Farber. We also obtain new bounds for the topological complexity of…

There are two rather distinct approaches to Morse theory nowadays: smooth and discrete. We propose to study a real valued function by assembling all associated sections in a topological category. From this point of view, Reeb functions on…

This paper establishes the separation of complexity classes $\mathbf{P}$ and $\mathbf{NP}$ through a novel homological algebraic approach grounded in category theory. We construct the computational category $\mathbf{Comp}$, embedding…

This paper lays the foundations of an approach to applying Gromov's ideas on quantitative topology to topological data analysis. We introduce the "contiguity complex", a simplicial complex of maps between simplicial complexes defined in…

In this paper we introduce a method which allows us to study properties of the random uniform simplicial complex. That is, we assign equal probability to all simplicial complexes with a given number of vertices and then consider properties…

For a finite simplicial complex K and a CW-pair (X,A), there is an associated CW-complex Z_K(X,A), known as a polyhedral product. We apply discrete Morse theory to a particular CW-structure on the n-sphere moment-angle complexes Z_K(D^{n},…

Let $K$ be a finite simplicial complex. We prove that the normalized expected Betti numbers of a random subcomplex in its $d$-th barycentric subdivision $\text{Sd}^d (K)$ converge to universal limits as $d$ grows to $+ \infty$. In…

Simplicial synchronization reveals the role that topology and geometry have in determining the dynamical properties of simplicial complexes. Simplicial network geometry and topology are naturally encoded in the spectral properties of the…

We introduce $k$-robust clique complexes, a family of simplicial complexes that generalizes the traditional clique complex. Here, a subset of vertices forms a simplex provided it does not contain an independent set of size $k$. We…

We introduce the notion of a template for discrete Morse theory. Templates provide a memory efficient approach to the computation of homological invariants (e.g., homology, persistent homology, Conley complexes) of cell complexes. We…