Geometric Analysis and Topology Seminar
Area-Contracting Maps Between Simple Shapes
Speaker: Larry Guth, Stanford
Location: Warren Weaver Hall 1013
Date: Friday, March 7, 2008, 2 p.m.
The \(k\)-dilation of a mapping measures how much the mapping stretches \(k\)-dimensional areas. The 1-dilation of a map is its Lipschitz constant, and the \(k\)-dilation for \(k > 1\) can be thought of as a generalization of the Lipschitz constant.
Here is a typical problem about the \(k\)-dilation. Fix a domain and a range, and a homotopy class of maps from the domain to the range. Then try to estimate the infimal \(k\)-dilation of the maps in this homotopy class. This problem is quite difficult even when the domain and the range are simple shapes. For example, we will discuss what happens when the domain and range are \(n\)-dimensional ellipses and the homotopy class is the class of degree 1 maps.
On the one hand, there are some rather complicated maps with much smaller \(k\)-dilation than 'obvious' maps. On the other hand, there are some non-trivial lower bounds on the \(k\)-dilation of any degree 1 map. The best known maps and the known lower bounds match for large \(k\), namely \(k > (n + 1)/2\).