Overview: This topics course is a broad introduction to the field of applied topology. We will first give a brief and intuitive introduction to topology, including homotopy equivalent spaces, homology groups, and homotopy groups. We next move to the realm of data analysis: given only a dataset, i.e. a finite sampling from a space, what can we say about the space’s shape (which may be reflective of patterns within the data)? The main technique is persistent homology; we describe its theoretical underpinnings, initial algorithms for its computation, how it has been used on real-life data, and coding examples. The two main flavors of persistent homology are sublevelset persistent homology for real-valued functions (closely related to Morse theory) and Čech, alpha, or Vietoris-Rips persistent homology for point cloud data. Both flavors are stable, i.e., 1-Lipschitz. We will also introduce zigzag persistent homology, sensor networks, mapper and Reeb graphs, multiparameter persistence (briefly), clustering (hierarchical clustering, k-means), dimensionality reduction (PCA, multidimensional scaling, Laplacian eigenmaps), machine learning (support vector machines), the integration of topology and machine learning, etc.
Goals: Students will become fluent with the main ideas of applied topology, will be exposed to current research trends, and will be able to communicate these ideas to others. Applied topology is an interdisciplinary area combining theory, algorithms, and applications; students will be exposed to all three while given the flexibility to focus on topics of interest. Students will ground their knowledge by writing software.
Syllabus: Coming this summer.
Notes
Homework
Schedule
| Date | Class Topic | Remark |
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| Aug 22 | Course overview | Henry in Chicago |
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| Aug 25 | Course overview | |
| Aug 27 | ||
| Aug 29 | ||
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| Sep 1 | Labor Day | |
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| Sep 12 | ||
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| Sep 26 | ||
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| Oct 13 | ||
| Oct 15 | ||
| Oct 17 | No class: Homecoming | |
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| Nov 21 | ||
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| Thanksgiving | ||
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| Dec 1 | ||
| Dec 3 | ||