Diastoles
This is about slicing (or sweeping out) a manifold in such a way that all slices have controlled size. My favorite diastolic invariant is the Urysohn width, which I investigated for my PhD thesis at MIT under the supervision of Larry Guth. This width measures the approximate dimension of a space by answering the following question: how far should we zoom out so that the space looks approximately $d$-dimensional? Formally, the $d$-width of a metric space is $\le w$ if the space can be mapped to a $d$-dimensional simplicial complex with fibers all having diameter less than $w$. Together with Sasha Berdnikov, we answered several natural questions on the “robustness of the approximate dimension”. For instance, we constructed manifolds that locally look approximately 1-dimensional, but globally their approximate dimension coincides with their actual dimension.
