**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.

**Evaluation**: Take home test.

**Textbooks** and reference materials: see lecture outlines.

**Office hours:** by appointment.