Overview: Topology, algebraic and differentiable, is increasingly important for the theory and practice of data analysis as it seeks to formalize and exploit the scale- and parametrization-invariant settings, highly relevant in high-dimensional, heterogeneous settings. As the area grows, so does the variety of tools and methods it exploits.
Time and place: W1-C-512, 6.10-14, 13-15:00.
Keywords: Applied topology, data analysis, aggregation, topological signal processing, persistent homology.
- Refresher on topology
- homotopy and homotopy equivalence;
- simplicial sets and maps; simplicial approximations;
- nerve lemma and diagrams of spaces.
- first applications: Dowker theorem and Netflix complex.
- Topological inference:
- Approximations via Cech spaces on random samples (Niyogi-Smale-Weinberger).
- Approximations with Vietoris-Rips complexes (Hausmann, Latschev).
- Persistent homology theory.
- Definitions; algorithms; stability theorem.
- Persistent diagrams as a tool to characterize functions. Properties of generic persistent diagrams.
- Structure of persistent diagrams for Brownian motions.
- Variations on persistence theme.
- Refresher on generic smooth functions and Sard lemma. Singularities of mappings between manifolds. Morse functions; Whitney embedding theorem.
- Reeb graphs and spaces. Merge trees.
- Persistence for Morse functions. Applications in large deviations.
- Biparametric persistence for functions on manifolds.
- Euler characteristic calculus.
- Information fusion in topological sensor networks.
- Topological signal processing; deconvolution.
- Minkowski functionals and generalizations of Hadwiger’s theorem.
- general problem of aggregation; spaces with averages. Averaging in spaces of non-positive curvature.
- Topology of aggregation: Eckmann’s theorem..
- Arrow impossibility theorem; Chichilnisky-Heal model; simplicial models and Arrow’s theorem.
- Topological tools of analyzing the multidimensional time series.
Prerequisites: Basic familiarity with manifolds and elements of algebraic topology will be useful.
Evaluation: Take home test.
Textbooks and reference materials: see lecture outlines.
Office hours: by appointment.