Ling Zhou (1/21/22): Persistent homotopy groups of metric spaces
In this talk, I will quickly overview previous work on discrete homotopy groups by Plaut et al. and Barcelo et al., and work blending homotopy groups with persistence, including those by Frosini and Mulazzani, Letscher, Jardine, Blumberg and Lesnick, and by Bantan et al. By capturing both
From playlist Vietoris-Rips Seminar
Ippei Obayashi (3/30/22): Stable volumes for persistent homology
Persistent homology is a powerful tool to characterize the shape of data quantitatively using topology. A persistence diagram (or barcode) is the output of persistent homology. The diagram is a scatter plot on the X-Y plane, and each point on the diagram called a birth-death pair correspon
From playlist AATRN 2022
An introduction to persistent homology
Title: An introduction to persistent homology Venue: Webinar for DELTA (Descriptors of Energy Landscape by Topological Analysis Abstract: This talk is an introduction to applied and computational topology, in particular as related to the study of energy landscapes arising in chemistry. W
From playlist Tutorials
Ling Zhou (8/30/21): Other Persistence Invariants: homotopy and the cohomology ring
In this work, we study both the notions of persistent homotopy groups and persistent cohomology rings. In the case of persistent homotopy, we pay particular attention to persistent fundamental groups for which we obtain a precise description via dendrograms, as a generalization of a simila
From playlist Beyond TDA - Persistent functions and its applications in data sciences, 2021
Ling Zhou (5/10/22): Persistent homotopy groups of metric spaces
By capturing both geometric and topological features of datasets, persistent homology has shown its promise in applications. Motivated by the fact that homotopy in general contains more information than homology, we study notions of persistent homotopy groups of compact metric spaces, toge
From playlist Bridging Applied and Quantitative Topology 2022
Persistent homology algorithm: An Example [Henry Adams]
I give a brief introduction to the persistent homology algorithm by running it on an example. My goal is not to give a complete description of the algorithm, but instead to show how some aspects of the algorithm work. I also hope to explain some of the geometric intuition you should have i
From playlist Tutorial-a-thon 2021 Fall
Michael Kerber (12/08/2021): Multi-Parameter Persistent Homology is Practical
Abstract: Multi-parameter persistent homology is an active research branch of topological data analysis. Early work has mainly focused on the theoretical part of the area, leaving the links to application area underdeveloped. One reason for this imbalance is the difficulty of computing the
From playlist AATRN 2021
Henry Adams (6/2/20): From persistent homology to machine learning
Title: From persistent homology to machine learning Abstract: I will give an overview of a variety of ways to turn persistent homology output into input for machine learning tasks, including a discussion of the stability and interpretability properties of these methods. Persistent homolog
From playlist SIAM Topological Image Analysis 2020
Žiga Virk (9/25/19): Geometric interpretation of persistence
Title: Geometric interpretation of persistence Abstract: Given a reasonably nice metric space X, its filtration by complexes and the corresponding persistent homology provide a multi-scale representation of X. At small scales the complexes usually reconstruct the homotopy type of the spac
From playlist AATRN 2019
Benjamin Schweinhart (4/3/18): Persistent homology and the upper box dimension
We prove the first results relating persistent homology to a classically defined fractal dimension. Several previous studies have demonstrated an empirical relationship between persistent homology and fractal dimension; our results are the first rigorous analogue of those comparisons. Spe
From playlist AATRN 2018
Persistent matchmaking - Uli Bauer
Workshop on Topology: Identifying Order in Complex Systems Topic: Persistent matchmaking Speaker: Uli Bauer Affiliation: Technical University of Munich Date: March 5, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
Henry Adams (5/3/22): Topology in Machine Learning
Abstract: How do you "vectorize" geometry, i.e., extract it as a feature for use in machine learning? One way is persistent homology, a popular technique for incorporating geometry and topology in data analysis tasks. I will survey applications arising from materials science, computer visi
From playlist Tutorials
Jose Perea (5/2/21): Quasiperiodicity and Persistent Kunneth Theorems
A signal is said to be quasiperiodic if its constitutive frequencies are linearly independent over the rationals. With appropriate parameters, the sliding window embedding of such a function can be shown to be dense in a torus of dimension equal to the number of independent frenquencies. I
From playlist TDA: Tutte Institute & Western University - 2021
Ulrich Bauer (4/6/22): Persistence in functional topology
I will illustrate the central role and the historical development of persistent homology beyond applied topology, connecting recent developments in persistence theory with classical results in critical point theory and the calculus of variations. Presenting recent joint work with M. Schmah
From playlist AATRN 2022
The Optimality of the Interleaving Distance on Multidimensional... Modules - Michael Lesnick
Michael Lesnick Stanford University; Member, School of Mathematics, IAS March 6, 2013 Persistent homology is a central object of study in applied topology. It offers a flexible framework for defining invariants, called barcodes, of point cloud data and of real valued functions. Many of the
From playlist Mathematics
Interactive visualization of 2-D persistence modules - Lesnick
Michael Lesnick Columbia University November 7, 2015 In topological data analysis, we often study data by associating to the data a filtered topological space, whose structure we can then examine using persistent homology. However, in many settings, a single filtered space is not a rich en
From playlist Mathematics
Vanessa Robins (9/8/22): The Extended Persistent Homology Transform for Manifolds with Boundary
The Persistent Homology Transform (PHT) is a topological transform introduced by Turner, Mukherjee and Boyer in 2014. Its input is a shape embedded in Euclidean space; then to each unit vector the transform assigns the persistence module of the height function over that shape with respect
From playlist AATRN 2022