- Day 1. Introduction to Homology and Persistent Homology
- Lecture 1. Motivation and Basic Constructions – The shape of data, simplicial and cubical complexes, homology, persistent homology, persistence diagrams, Cech complexes, Vietoris-Rips complexes, digital images
- Lecture 2. Foundational Results – The persistence algorithm, Wasserstein distance, stability, flavors of persistence (e.g. zigzag, multiparameter)
- Day 2. Mathematics of Persistent Homology
- Lecture 3. Algebra of Persistence Modules – Commutative algebra, representations of quivers, graded modules, algebraic stability
- Lecture 4. Geometry and Combinatorics – Geometric stability, Möbius inversion, coarse geometry
- Day 3. Statistics and Machine Learning
- Lecture 5. Statistics – Hilbert spaces, kernels, persistence landscapes, averages, variance, hypothesis testing, permutation tests, principal component analysis, subsampling
- Lecture 6. Machine Learning – Classification, regression, support vector machines, deep learning, multilayer perceptrons, convolutional neural networks, topological loss, topological layers
- Day 4. Applications and Software
- Lecture 7. Applications – Preprocessing, mathematical encoding of data, time series, case studies
- Lecture 8. Software and Algorithms – Computational advances, guide to current software
- Day 5. Advanced Topics and Current Research Problems
- Lecture 9. Multiparameter Persistent Homology – Theory, algorithms, software, open problems
- Lecture 10. Mathematics of Persistent Homology – Graded persistence diagrams, virtual persistence diagrams, categorical stability, universal constructions, Cerf theory, open problems”
All lectures take place in the STEAM Center, First Floor, Main Room (Building 43 in the Campus Map, Address: 1302 N Patterson St, Valdosta, GA 31601)
Lab sessions take place in Odum Library, Room 3270 (Building 29 in the Campus Map)