Title: K-dilation, Steenrod squares, and h-principle for immersions

Abstract: The k-dilation of a mapping is a generalization ofthe Lipschitz constant that measures how much the mapping stretches k-dimensional areas. For example, the diagonal 3x3 matrix with entries 100, 100^{-1}, 100^{-1} has 2-dilation 1, because it does not increase the areas of any 2-dimensional surfaces.

We discuss the following question. Pick two n-dimensional ellipses E_1 and E_2. What is the infimal k-dilation of any degree 1 map from E_1 to E_2? I can answer this problem for some values of k, and I find the answers a little surprising. The linear map can have far from the smallest k-dilation. I constructed some near-optimal maps that are quite wiggly, and the construction uses the h-principle for immersions. I will also describe some lower bounds for the k-dilation that use Steenrod squares as a tool.