The state of granular media can be represented by a persistence diagram. This representation provides an interesting insight into the physical properties of the granular media as demonstrated on a system undergoing compression. Time evolution of the system can be seen as a curve in the space of persistence diagrams. Different notions of distance in this space provide a useful tool for understanding the dynamic. In particular the compressed systems (viewed as a discrete dynamical system) exhibit a few different regimes where dynamics changes from fast to slow. Dependence of the system on its previous state is strongly affected by the sampling rate. We conclude the talk by addressing the problem of determining the 'appropriate' sampling rate.
The theory of multiparameter persistent homology was initially developed in the discrete setting of filtered simplicial complexes. Stability of persistence was proved for topological spaces filtered by continuous vector-valued functions, that is, for continuous data. This talk aims to provide a bridge between the continuous setting, where stability properties hold, and the discrete setting, where actual computations are carried out. The existence of this bridge is not obvious due to the phenomenon of structural gap between the two settings, called topological aliasing, which appears in the multiparameter case when using the standard piecewise linear interpolation of the discrete model. We solve the problem by introducing an adapted axis-wise linear interpolation and develop a stability preserving method for comparing rank invariants of vector functions obtained from discrete data. These advances support the choice of multiparameter persistent homology as a tool for shape comparison in computer vision. This is a joint work with M. Ethier, P. Frosini, and C. Landi.
We study the topology of random Cech complexes, generated by $n$ iid points in a Euclidean space, and a radius $r$. In particular, we are interested in the limiting behavior of the Betti numbers of such complexes, as $n\to\infty$ and $r\to 0$. We consider different cases in which samples are generated by either pure noise, a probability distribution on a close manifold, or a combination of both. The study of the Betti numbers can be done directly, or via a Morse-theoretic approach, by counting critical points of distance functions. The results show that the Betti numbers of a random complex exhibit a different limiting behavior, depending on both the support of the underlying distribution and the rate in which $r \to 0$. In this talk, we will present the known results to date for each of these limiting phenomena. In addition, we will discuss how these results could be applied to the problem of recovering the topology of a hidden manifold from a finite set of samples.
Problems in applied topology often require computation of an optimal (in some sense) representative among shapes satisfying particular topological constraints. In many cases this is a very challenging, if not infeasible task. The aim of this talk is to show that in some cases the needed optimality is a typical property, meaning that a representative chosen uniformly at random will be close to optimal with an overwhelming probability. In particular, we consider the space of closed polygonal chains which represent a fixed free homotopy class in a (multi-)punctured plane and have a large number of small edges. By employing tools from large deviation theory we show that the uniform measure on this space is sharply concentrated around representatives whose length is arbitrarily close to the length infimum. As a consequence, sampling from the stationary distribution of a Markov process (using, say, the Metropolis-Hastings algorithm) gives us a simple way to approximate the minimal length representative in a free homotopy class. This result has an important application in multi-agent systems, as it allows the agents to move towards the optimal configuration without any control.