Title: The (in)compatibility of 3 and 5 dimensional Heisenberg geometry with Lebesgue spaces
Abstract: The 3-dimensional (discrete) Heisenberg geometry is the shortest-path metric on the infinite graph whose vertex set is the integer grid $\Z^3$ and the neighbors of each integer vector $(a,b,c)$ are the four integer vectors $$(a+ 1,b,c), (a- 1,b,c), (a,b+ 1,c+ a), (a,b- 1,c- a).$$
Analogously, the $5$-dimensional (discrete) Heisenberg geometry is the shortest-path metric on the infinite graph whose vertex set is the integer grid $\Z^5$ and the neighbors of each integer vector $(a,b,c,d,e)$ are the eight integer vectors $(a\pm 1,b,c,d,e), (a,b\pm 1,c, d,e), (a,b ,c\pm 1,d, e\pm a), (a,b ,c,d\pm 1, e\pm b)$. The purpose of this talk is to describe the culmination of a multi-decade effort to understand the extent to which these metric spaces can be represented faithfully as subsets of an $L_p(\mu)$ space. It turns out that if $p>1$, then there is no qualitative difference between the answer to this question in dimensions $3$ and $5$. However, if $p=1$, then the behaviors of dimensions $3$ and $5$ diverge markedly. These results rely on several ideas and tools that were introduced over the years, and they relate deeply to a rich variety of mathematical disciplines. They answer major open questions in metric embeddings, Lipschitz factorization, dimension reduction, and semidefinite programming. We will describe key statements and ideas while highlighting the most recent step which introduces the notion of a foliated corona decomposition (joint work with Robert Young).
Despite the fact that our discussion is linked to a wide range of areas of mathematics, the talk is intended for a general audience of mathematicians and theoretical computer scientists. We will not rely on any prerequisites beyond an undergraduate degree in mathematics, and all of the relevant background will be introduced and explained.