Lecture Topics

Exploratory data analysis, mathematical modeling. Topological modeling as extension of both. Why topology? Functionality and justifiability of models. Hypothesis generation. Requirements for artificial intelligence.
Simplicial complexes, nerves, homology and homotopy as counting problems, homeomorphism, homotopy equivalence, Borel-Moore homology, filtered complexes, adapting homology to particular shape analytic tasks.
Machine Learning
Standard machine learning methodology, PCA, MDS, SVM, other classifiers, neural nets, clustering theory. How these methods are used in artificial intelligence.
Finite Metric Spaces to Geometry
Complex constructors for finite metric spaces. Cech, Vietoris-Rips, alpha shapes, witness complex, Mapper. Definition of persistent homology. Classification theorem for persistence vector spaces. Zig-zag and multidimensional persistence.
Theory and Functionality of the Mapper Methodology
Segmentation, hot spot analysis, explanations and confirmatory analysis, functoriality and split highlighting, feature space modeling, numerous examples.
Applications of Persistent Homology
Statistics of image patches, viral evolution, neuroscience applications (including Curto et al, Hess et al, Saggar et al), materials science applications (MacPherson et al, Hiraoka et al). Designing barcodes for shape recognition tasks
Local to Global Methods
Adaptation of local-to global methods from topology to point cloud situation. Tree theorem for persistent homology. Persistent homology and circular coordinates. Iterative methods for TDA with examples from alternate methods of imaging. Künneth formulas.
Vectorizing Persistence Bar Codes
Application of Guowei Wei to drug discovery. Algebraic and metric structures on sets of barcodes. Persistence landscapes and persistence images. S. Kalisnik’s work on tropical structures on bar codes. Stability theory for persistent homology.
Theory of Clustering
Clustering as one of many discretization tasks, philosophy of clustering, classification theorem of G. C. and Memoli, clustering for directed networks.
Algorithms as Data Sources
Neural nets, deep learning, connections to TDA, examples.