1. Lecture 10: The Future May 17, 2019
Gunnar Carlsson
Stanford University

2. Mapper Stability Measures
Use more flexible coverings, Voronoi, Alpha Complex, Witness
Different clustering strategies
Persistent homology to select useful topological models
Measures of how well Mapper represents a metric space

3. Deep Learning Better features for images
Better understanding of text, especially word embeddings
Explore the effect of architectures on various things, especially generalization
Find more methods for incorporating topology of feature space

4. Vectorization
Complete analysis of tropical functions on one dimensional persistent diagrams
Extend tropical functions to multidimensional case

5. Generalized Persistence
Find better ways to represent multidimensional persistence
Find descriptions of multidimensional persistence that include both ordinary persistence directions as well as zig-zag ones
Use 2D persistence to develop a tool that can be used easily which uses superlevel set persistence for density and distance scale

6. Stochastic Aspects
Develop persistence profiles for metric measure spaces
Prove stability theorems for them in presence of “Gaussian noise”

7. Formal Properties of Persistence
Künneth formula
Improve Mayer-Vietoris

8. Complement Problems Parametrized embedding calculus
Unstable Adams spectral sequence for parametrized situation
Adapt to create initial guesses for optimization problems