Special Geometry/Dynamical Systems Seminar

The k-dilation of a mapping F measures how much the mapping stretches k-dimensional areas. If a map F has k-dilation L , then it maps each k-dimensional surface of k-volume V to an image of k-volume at most LV. If the k-dilation of a map is very small, it means that the map contracts all the k-dimensional areas. We discuss how the k-dilation of a map relates to the homotopy type of the map. For example, we prove that there are homotopically non-trivial maps from the unit 9-sphere to the unit 8-sphere with arbitrarily small 6-dilation. On the other hand, we prove that every map from the unit 9-sphere to the unit 8-sphere with 5-dilation less than 10^{-50} is homotopically trivial.