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 RSS).

Sard’s lemma

Whitney embedding theorem

Transversality. Thom’s weak transversality theorem.

Homework

  1. Consider the family of points on the plane given by
    $$
    (x_k,y_k)=(\cos(2\pi k/N),\sin(4\pi k/N)), k=0,\ldots, N
    $$
    (figure $latex\infty$). Take \(L\) random points \((X_l,Y_l)\) in the unit square, and define  functions
    $$
    g_l(x,y)=\exp(-k((x-X_l)^2+(y-Y_l)^2)), l=1,\ldots,L.
    $$
    Consider these functions as defining a mapping the \(N\) points into $latex\mathbb{R}^L$. Take a random projection of the resulting point cloud to $latex\mathbb{R}^2$; interpret the result.
    (Take \(N=200; L=16; k=4.0-16.0\) – tune the parameters!)
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