Archive | work

RSS feed for this section

day 10

Social choice theory, reminder.

Topological version of social choice theory. Chichilnisky theory of manipulability. Arrow’s impossibility theorem as a topological result.

Topological aspects of clustering. Kleinberg’s impossibility theorem. Carlsson-Memoli’s characterization of simple link clustering.…

Read full story Comments { 0 }

day 9

\(\def\Real{\mathbb{R}}\)

Data aggregation. Averaging. Examples.

Axioms: anonymity, unanimity.

More topology: Kunneth formula.

Nonexistence of averaging on spheres.

General result (Eckmann): averagings of all orders on tame topological spaces exists only when the space is contractible.…

Read full story Comments { 0 }

day 8

\(\def\Real{\mathbb{R}\)

Computing persistent homologies: positive and negative cells, reduced matrices etc.

Stability.

Jitter in persistent homology as “texture” descriptor, see e.g. this or this.

Fundamental properties of jitter.

 …

Read full story Comments { 0 }

day 7

\(\def\Real{\mathbb{R}} \)

Basic ideas in topological inference. Recovering the homologies of a model from a sample.

Filtrations. Persistent homologies. Examples.

(Based on books by Oudot and Edelsbrunner-Harer)…

Read full story Comments { 0 }

day 6

\(\def\Real{\mathbb{R}} \)

Software tools for computing homologies – examples (Eirene, PHat, Perseus…).

Examples of topological data analysis: is internet sampled from hyperbolic plane?

Persistence’ algebraic metaphor: quivers and Gabriel’s theorem.…

Read full story Comments { 0 }

day 5

\(\def\Real{\mathbb{R}} \)

Topological inference

From data to simplicial complexes, A: Dowker’s theorem

From data to simplicial complexes, B: Čech and Vietoris-Rips complexes

Approximation results: finite samples from a manifold approximate the manifold well, in case of Čech complexes, thanks to …

Read full story Comments { 0 }

day 4

\(\def\Real{\mathbb{R}}\)

Homology.

Simplicial complexes; definition of simplicial homology. Basic properties. Functoriality.
Geometric realizations; simplicial approximations.
Homotopy invariance.

Euler calculus.

Additivity; Euler characteristic as measure. Integrals with respect to Euler characteristics.
Properties: linearity, Fubini theorem.
Classes of admissible sets.
Applications in …

Read full story Comments { 0 }

day 3

\(\def\Real{\mathbb{R}}\)

Thom’s strong transversality theorem. Jets.

Examples: codimension of set of points of given corank.Typical singularities of visible contours.

Taken’s embedding theorem.

Short survey of geometric dimensionality reduction tools in data analysis.

  • Model-based dimensionality reduction tools.
    Setup: a point cloud
Read full story Comments { 0 }

day 2

We will address essential tools in differential topology:

Partition of unity, using it to embed a manifold into a (high-dimensional) Euclidean space.

First applications of these tools to data analysis will be discussed (mapping of the terrain using local beacon …

Read full story Comments { 0 }

day 1

We will be covering the basic notions: topological spaces, compactness, homeomorphisms, homotopy,
and introduce the most important class of models for data analysis: manifolds.

Homework:

  1. Three cyclists start and end in the same formation, A>B>C. During the race, they never
Read full story Comments { 0 }