Applied Topology Videos, Spring 2020
David B. Damiano
These videos were recorded in Spring 2020 for MATH 363, Topics in Topology following the College's move to remote learning.
These form an introduction to topological data analysis for undergraduates. The prerequisite was a theory-based course in linear algebra.
The videos are recordings of white board lectures. PDFs of the white boards are provided below.
The videos can be found on YouTube at Applied Topology - David Damiano -2020.
The video link is to the YouTube channel for the Applied Algebraic Topology Research Network, which has graciously agreed to host these videos.
Thanks go to the co-organizers of the AARTN, Henry Adams, Colorado State University, Sara Kalisnik, Bentley University, and Elchanan Solomon, Duke University.
- Quick Review Part A Complexes and Filtrations
This lecture provides an intuitive review of simplicial complexes and filtrations. The definition and several examples of simplicial complexes and filtrations are provided. The example filtrations are given by an explicit construction, by skeleta, and by distance (Rips construction).
- Quick Review Part B Complexes and Filtrations
This lecture introduces simplicial homology mod 2 using mod 2 vector spaces. The chain vector spaces, boundary homomorphism, boundaries, cycles and homologies are presented and examples are provided. The homology vector spaces in each dimension are the vector space quotients of the vector space of cycles by the vector space of boundaries.
- Quick Review Part C Complexes and Filtrations
This lecture demonstrates how to represent the boundary map between vector spaces of chains as a matrix. Determining a basis for homology then becomes a problem in linear algebra. These calculations are demonstrated on a two dimensional simplicial complex.
- Intuitive Persistence
This lecture introduces the persistent homology of a filtration of a simplicial complex. In each dimension inclusions of the filtration subcomplexes induce a sequence of maps on the homology vector spaces of the given dimension of the subcomplexes. Tracking cycles from their birth to death in this sequence of inclusions leads to the definition of persistence and the construction of persistence barcodes.
- Persistence Diagrams
In this lecture we show how to construct the persistence diagram from a barcode. Persistence barcodes and diagrams are the primary diagrammatic
representations of persistence. Here we demonstrate the construction on a small simplicial complex.
- Bottleneck_Distance
The bottleneck distance between persistence diagrams is a critical tool in persistent homology. It relies on partial matches between persistence diagrams. Optimal partial matchings are used to define the bottleneck distance between diagrams. In this lecture we describe the bottleneck distance, apply it to a simple example, and show that it defines a metric on the space of diagrams.
- Bottleneck Triangle Inequality
In this lecture we prove the triangle inequality for the bottleneck distance. This completes the argument that the bottleneck distance is a metric on
the set of persistence diagrams.
- Stability Theorem
After the theoretical justification for the construction of persistence diagrams, the stability theorem is the most important result in the theory of
persistence. Intuitively it states that small changes or perturbations in the construction of a simplicial complex and a filtration thereof results in
small changes in its persistence diagram as measured by the bottleneck distance. We should think of this as continuity result about the persistence construction.
- TDA and Statistics I
In this lecture we address the question of whether the collection of persistence diagrams contains a well-defined mean or average of a pair of diagrams. This is motivated by a desire to apply statistical analyses to persistence diagrams. As it turns out, there are simple examples that demonstrate the impossibility of this task. We consider one such example in this lecture.
- TDA and Statistics II
In this lecture we present persistence landscapes, a construction developed by Peter Bubenik. A persistence landscape is a finite sequence of non-increasing real valued function defined on the non-negative real axis. It is possible to define a unique mean of persistence landscapes resolving the problem posed in the previous lecture.
- TDA and Statistics III
In this lecture we introduce the norm of a persistence landscape and Bubenik's interpretation of the mean of a finite set of persistence landscapes. These are important constructions which justify the use of peristence landscapes.
- Introduction to Cubical Complexes and Persistence
In this lecture we introduce cubical complexes and cubical homology. An important application of persistent homology is to the analysis of digital images. However, rather than use simplicial complexes, it is more efficient to use the natural grid structure of a digital image to construct filtrations and utilize the grid structure to compute persistent
homology. These complexes are called cubical complexes and the homology theory is called cubical homology.
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