- 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)