We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices, and, in most detail, the… (More)
A key requirement facing organisms acting in uncertain dynamic environments is the real-time estimation and prediction of environmental states, based on which effective actions can be selected. While it is becoming evident that organisms employ exact or approximate Bayesian statistical calculations for these purposes, it is far less clear how these putative… (More)
It is becoming increasingly evident that organisms acting in uncertain dynamical environments often employ exact or approximate Bayesian statistical calculations in order to continuously estimate the environmental state, integrate information from multiple sensory modalities, form predictions and choose actions. What is less clear is how these putative… (More)
We study the homology of simplicial complexes built via deter-ministic rules from a random set of vertices. In particular, we show that, depending on the randomness that generates the vertices, the homology of these complexes can either become trivial as the number n of vertices grows, or can contain more and more complex structures. The different… (More)
 Lior Aronshtam and Nathan Linial. The threshold for d-collapsibility in random complexes*.ishing of top homology in random simplicial complexes.  Omer Bobrowski and Matthew Kahle. Topology of random geometric complexes: a survey.  Omer Bobrowski and Sayan Mukherjee. The topology of probability distributions on manifolds.
Acknowledgement First and foremost, I would like to express my deepest gratitude to my advisor, Professor Robert Adler, for his amazing dedication in guiding and inspiring me through my PhD, while showing full confidence in me and letting me build my academic independence; for always encouraging, opening every possible door for me, and giving me much more… (More)