Minicourse: Algebraic and Differential Topology in Data Analysis (ADTDA)

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.

Course plan:

• 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.
• Aggregation
• 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.

Lecture 2, June 11.

Evaluation: Take home test.

Textbooks and reference materials: see lecture outlines.

Office hours: by appointment.