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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.

…

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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)…

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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.…

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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 …

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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 …

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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

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 …

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:

- Three cyclists start and end in the same formation, A>B>C. During the race, they never

- Consider the language \(L\) consisting of all words in alphabet a,b,c having no subwords aaa. Find the generating function

$$

f(x)=\sum |L_n| z^n,

$$

where \(L_n\) is the set of words of lengths \(n\) in \(L\). Estimate how fast \(\#

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\)

- Consider the closed cycle \(\gamma:[0,1]\to\Real^2\) shown on the left (we assume that \(\gamma(0)=\gamma(1)\) is the dot on the positive \(x\)-axis). Define the functions

\[

I_-(t)=\int_0^t \left(\frac{(x+1)dy -ydx}{(x+1)^2+y^2}.\dot{\gamma}(t)\right)dt,\quad I_+(t)=\int_0^t \left(\frac{(x-1)dy -ydx}{(x-1)^2+y^2}.\dot{\gamma}(t)\right)dt,

\]

given as partial integrals over the

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