Primoz Skraba: Euler Characteristic Surfaces
Euler characteristic curves have been used in a number of different applications from fMRI's to classification of stone tools. Given a space with a real valued function, Euler curves assign to each value of the function the Euler characteristic of the preimage of the super (sub)-level set. In the case of random functions (such as Gaussian random fields), we usually consider the super level sets, also called excursion set. This is analogous to persistence barcodes or diagrams which consider the evolution of homology groups over varying excursion sets. For one dimensional functions, this is well understood.
Omer Bobrowski: Random Geometric Complexes and TDA
Topological Data Analysis (TDA) broadly refers to the use of concepts from mathematical topology to analyze data and networks. In the past decade a variety of powerful topological tools has been introduced, and were proven useful for applications in various fields (e.g. shape analysis, signal processing, neuroscience, genomic research). In this sequel talk, we will demonstrate how the theory of random geometric complexes can be used in TDA. In particular, we will discuss the analysis of ״topological noise״ that appears in topological inference problems. This study is highly useful for developing statistical methods such as filtering and hypothesis testings in TDA. We will also discuss recent developments in the study of persistent homology of random geometric complexes.
Pawel Dlotko: Computational topology with Gudhi library
In this talk I will present a Gudhi library (http://gudhi.gforge.inria.fr/): one of the most complete software package for computational topology. I will introduce Rips, Alpha and Withess complexes and show how to construct them given a point clouds. Later we will discuss cubical complexes and the computations that can be done with them. If time permits we will also discuss various algorithms for manifold reconstruction. At the end we will discuss Gudhi_stat, a statistical library in Gudhi. Based on simple examples we will perform various statistical operations on persistence diagrams.
Claudia Colonnello: Early Prediction of Silo Collapse
Cylindrical thin-walled silos are ubiquitous as industrial storage facilities for granular materials. Contact interactions with the grains may produce the deformation of the shell that conforms the wall of the silo. In particular, when the filling height of the silo is large enough, an intricate pattern of indentations develops on the wall of the silo during the gravity-driven discharge of the granular material. Moreover, when the filling height exceeds a certain critical value, an irreversible deformation of the wall occurs, that often concludes with the catastrophic collapse of the structure. The understanding and efficient prevention of this phenomenon is a challenging problem, as the elastic shell is coupled to the complex dynamics of the granular material. We perform laboratory experiments using silos made of paper and acquire sequences of digital images showing the evolution of the deformation pattern for different values of the initial filling height. The instantaneous geometric characterization of these patterns cannot distinguish in advance those processes in which the deformation will become irreversible. However, using the methods of algebraic topology we find that the dynamics of the deformation process changes significantly as the initial filling height is increased above the critical threshold. We study the implications of our results on the nature of the interaction that causes the deformation. Moreover, our results allow for an early prediction of the collapse of the silo, that can be useful for the development of an early warning system.
Trinh Khanh Duy: Law of Large Numbers for Persistence Diagrams
For a stationary point process on Euclidean space, we restrict it on finite windows and consider the persistence diagram of the restricted one. As the window size tends to infinity, we prove the strong law of large numbers for persistence diagrams. An analogous result also holds for binomial point process, an i.i.d. sampling from a continuous distribution on Euclidean space. The talk is based a joint work with Yasuaki Hiraoka and Tomoyuki Shirai (arXiv:1606.06518 and arXiv:1612.08371).