# Archive | work

## 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.
Applications in …

## 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

## 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 …

## 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

## problems for review

1. 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 $$\# ## midterm 2 \(\def\pd{\partial} \def\Real{\mathbb{R}}$$

1. 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

## Stokes and relatives

• The (general) Stokes Theorem: this unique theorem puts
Green’s/Gauss/Stokes Theorems (from intermediate calculus)
under one umbrella. Let $$C$$ be a polyhedron with boundary $$\partial C$$. Then
$${\int_{\partial C} \omega = \int_C d\omega }$$
• Example: integrate the form $$\omega = • ## solutions for problems on vector analysis and differential forms Solutions Exercise: Assume that 4 vectors in \(\mathbb{R}^3$$ satisfy $$A+B+C+D$$=0.
Simplify
$$A\times B-B\times C+C \times D-D\times A$$

Solution: By the anti-symmetry $$-B\times C = C\times B$$
and $$-D\times A = A\times D$$ we can rewrite the expression as
$$… ## March 28 • Consider the $$k$$-form $$\omega$$ defined at $$x\in \Omega\subset \mathbb{R}^n$$, given by the sum$$ \omega(x)= \sum c_I(x) dx_I $$where the multi-index $$I=\{1\leq i_1\leq i_2 \leq, \ldots, \leq i_k \leq n\}$$. Each basis element $$dx_I:= dx_{i_1} \wedge • ## Exterior Differential forms \(\def\pd{\partial} \def\Real{\mathbb{R}}$$ • We define a differential $$k$$-form as$$\omega(x) = \sum_{I=\{i_1<i_2<\ldots<i_k\}} c_I(x) dx_{i_1}\wedge \cdots\wedge dx_{i_k},
an exterior form with coefficients depending on the positions.

(We will be using sometimes simplifying notation $$\xi_i = dx_i$$ or $$\eta_i=dy_i$$.)

• Pullback maps: Let