Video of the lecture.

Began with 750 points randomly selected from two disjoint quarters of a torus in R³. Any linear transformation R³ ---> R² (obtained say from PCA or the Johnson-Lindenstrauss theorem) would lose geometric information. However, we saw that the geometric information seems to be retained when the points are mapped to the vertices of the clique complex K_ε for various values of ε.

Introduced the notion of homotopy between two maps f,g:X-->Y. Introduced the notion of homotopy equivalence between two topological spaces. Proved that the circle is homotopy equivalence to the projective plane minus the origin.