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 1, June 10.

Lecture 2, June 11.

Lecture 3, June 12.

Lecture 4, June 13.

Lecture 5, June 14.

Evaluation: Take home test.

Textbooks and reference materials: see lecture outlines.

Office hours: by appointment.