# 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 sensor fusion.

Euler integral calculus.

Radon transform; Schapira’s inversion formula.

#### Homework:

Describe convolution of the indicator functions of an ellipse and a circle (consider different relations between the radius of the circles and the curvatures of the ellipse).