MATH 7375 - Topics in Topology: Topological Methods for the Analysis of Data

Course description: Topology, much like geometry, is concerned with studying the shape of mathematical spaces; these include surfaces, collections of functions and even more complicated objects. Over the last decade some of these same techniques have found applications to problems in data science, in areas such as chemistry, neuroscience and robotics. In this course, we will introduce some of the main tools powering this rapidly-growing field (e.g., persistent homology and cohomology), we will present several examples of applications (e.g., to computer vision, time series analysis and computational biology), as well as some of the theoretical open problems in the field. The course will include an independent research component.

Major topics:

Unit I: Background

  1. Spaces: Metric, simplicial, topological, manifolds, homeomorphisms and homotopies

  2. Reconstructing spaces with positive reach from Hausdorff-close samples

  3. Homology: induced homomorphisms, homotopy invariance and computation

  4. Application: Clustering

Unit II: Persistence and stability in topological data analysis

  1. Persistence: Intuition, Definition and Computation  

  2. Clustering revisited

  3. The Stability of Persistence

  4. Topological inference

  5. Further directions: Quivers, Cohomology, Eilenberg-MacLane coordinates

Unit III: Applications

  1. Time series analysis: recurrence in biological systems, chatter detection in mechanical systems and video analysis

  2. Machine learning with persistence diagrams

  3. Topological dimensionality reduction

References:

  1. Elements of algebraic topology, by James R. Munkres, Vol. 2. Reading: Addison-Wesley, 1984.

  2. Abstract Algebra, Third Edition, by David S. Dummit and Richard M. Foote, 2003.

  3. Persistence theory: From quiver representations to data analysis, by Steve Y. Oudot, Mathematical Surveys and Monographs, Vol. 209, American Mathematical Society, 2015.

  4. Bulletin of the AMS Book review: Elementary Applied Topology (R. Grhist), and Persistence Theory (S. Oudot), by Jose A. Perea, BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 57, Number 1, January 2020, Pages 153–159.

Syllabus: pdf   

GitHub: link

Project: format, ideas

Homeworks:

  • Homework 1 [Due: 02/12/2023]: Please follow the installation instructions and complete the associated juPyter notebook

  • Homework 2 [Due: 3/29/2023]: Please complete and submit a pdf copy of the juPyter notebook from Demo 3.

Lectures:

  • Lecture 1 [01/09/2023] pdf: Administrative details, what is the course about? Intro to Topological Data Analysis

  • Lecture 2 [01/11/2023] pdf: Metric spaces and generalizations (extended, pseudo, etc), Examples (p-norms, Hausdorff distance, angular distance, shortest path distance) and applications (e.g., computer graphics, computational chemistry, ISOMAP) and the topology inference problem.

  • Lecture 3 [01/18/2023] pdf: Topology of metric spaces (openness, isometry, continuity), topological spaces and homeomorphisms (e.g., invariance of domain)

  • Lecture 4 [01/23/2023] pdf: New topologies from old (subspace, product and quotient topologies), homotopies and homotopy equivalences (the circle is not contractible), application: 2D Brouwer’s fixed point theorem.

  • Lecture 5 [01/25/2023] pdf: Compactness (Heine-Borel, Hawaiian earrings, topologist’s sine curve), the medial axis, the reach, embedded manifolds, reconstructing the homotopy type of manifolds with positive reach via off-sets (i.e., Niyogi+Smale+Weinberger).

  • Lecture 6 [01/30/2023] pdf: Simplicial complexes (Rips, Nerve, Cech), their geometric realization and the Nerve Lemma (when off-sets have the same homotopy type as the Cech complex)

  • Lecture 7 [02/01/2023] pdf: The topology of Cech vs Rips (the Rips complex of the circle), homotopy type determines number of path-connected components, edge-path connected components in simplicial complexes.

  • Lecture 8 [02/06/2023] pdf: Singular chains, cycles, boundaries and induced chain maps.

  • Lecture 9 [02/08/2023] pdf: Singular homology, induced linear maps, H_0 measures path-connected components, homotopy invariance of singular homology.

  • Lecture 10 [02/13/2023] pdf, demo: Clustering — Computational Activity. k-Means, single linkage, DBSCAN*, persistable, UBER data (Luis Scoccola).

  • Lecture 11 [02/15/2023] pdf: Simplicial homology, computing simplicial homology via matrix reduction, isomorphism between simplicial and singular homology.

  • Lecture 12 [02/22/2023] pdf, demo: Examples - Computing simplicial homology (n-sphere, torus, projective plane, Klein bottle), Betti numbers, Euler characteristic and Euler curves. Computational Activity: classification of 3D shapes with Euler Curves (Matt Piekenbrock)

  • Lecture 13 [02/27/2023] pdf: Persistent vector spaces (e.g., from the homology of sublevel sets, offsets and Rips complexes), the dimension and rank invariants, direct sums and indecomposables, interval persistent vector spaces.

  • Lecture 14 [03/01/2023] pdf: Intervals are indecomposable (and converse true if indexed over a totally ordered set), pointwise finite implies unique decomposition as direct sum of indecomposables, the Barcode: pointwise finite over totally ordered decomposes uniquely as a sum of intervals.

  • Lecture 15 [03/13/2023] pdf: Interval decomposition = compatible bases, filtrations, the elder rule

  • Lecture 16 [03/15/2023] pdf, demo: Why the elder rule works, the matrix reduction algorithm. Computational activity: Computing persistent homology with Ripser

  • Lecture 17 [03/20/2023] pdf: Computing Rips persistence diagrams, the stability of persistence — The Gromov-Hausdorff, interleaving and bottleneck distances; the Isometry theorem.

  • Lecture 18 [03/22/2023] : Live invited presentations

    1. 1. [slides] Density aware clustering with barcodes and degree-Rips (Luis Scoccola)

    2. 2. [slides] Topological Time Series Analysis (Jose Perea)

    3. 3. [slides] Topological Synthesis and Analysis of Multimedia Data (Chris Tralie).

  • Lecture 19 [03/27/2023]: pdf, demo: Interpretation and consequences of the stability theorem. Computational activity: Stability in action — Sampling (random vs maxmin), dealing with noise.

  • Lecture 20 [03/29/2023] slides: Machine learning with persistence diagrams (Kernels, persistence features, persistence images, landscapes, template functions)