Lectures

Week 0: Introduction

An introduction to the course including topics to be covered, expectations of students, basics of topology, and linear algebra review.

  1. Lecture part 1, course overview
  2. Lecture part 2, intro to topology
  3. Slides
  4. Recitation part 1, vector spaces review
  5. Recitation part 2, linear transformations review
  6. Recitation part 3, quotient spaces review
  7. Recitation part 4, basic linear algebra in MATLAB

Week 1: Topology

The basics of topology including topological spaces, continuous maps, and metric spaces.

  1. Lecture part 1, topological spaces
  2. Lecture part 2, continuous maps
  3. Slides
  4. Recitation part 1, intro to metric spaces
  5. Recitation part 2, metric spaces in practice

Exercise set 1

Week 2: Simplicial Complexes

Simplicial complexes, simplicial maps, filtrations, and computations.

  1. Lecture part 1, simplicial complexes
  2. Lecture part 2, simplicial maps
  3. Lecture part 3, Rips and Cech complexes
  4. Slides
  5. Recitation part 1, simplicial complexes and examples
  6. Recitation part 2, intro to JavaPlex and simplicial complexes in MATLAB

Week 3: Simplicial Homology

The theory and computation of simplicial homology.

  1. Lecture part 1, simplicial homology
  2. Lecture part 2, betti numbers
  3. Lecture part 3, computing homology
  4. Slides
  5. Recitation part 1, review and examples
  6. Recitation part 2, homology with different coefficients
  7. Recitation part 3, simplicial homology in JavaPlex

Week 4: Persistent Homology

  1. Lecture part 1, persistent homology
  2. Lecture part 2, the persistence algorithm
  3. Recitation part 1
  4. Recitation part 2

Week 5: Speeding Up Persistent Homology

  1. Lecture part 1, speeding up persistent homology
  2. Lecture part 2, subsampling
  3. Recitation part 1
  4. Recitation part 2

Week 6: Theoretical Foundations of Persistent Homology

  1. Lecture part 1, the algebraic decomposition theorem
  2. Lecture part 2, the bottleneck stability theorem
  3. Recitation part 1
  4. Recitation part 2

Week 7: Extensions of Persistent Homology

  1. Lecture part 1, multiparameter persistent homology
  2. Lecture part 2, invariants for multiparameter persistent homology
  3. Recitation part 1
  4. Recitation part 2
  5. Recitation part 3

Week 8: Visualization and Interpretation

  1. Lecture
  2. Recitation

Week 9: Neuroscience Background

  1. Lecture place cells, place fields, and the nerve lemma