View AbstractAbstract: Multidimensional NMR (MDNMR) experiments are an important tool in physical chemistry, but can take a long time, in some cases weeks, to conduct. At first glance, the application looks ideal for compressed sensing because the object to be recovered is sparse and the under-sampled measurements are made in the 'Fourier' domain. Actually, MDNMR is not covered by the existing compressed sensing literature. First, the 'Fourier' domain is not the classical one, but involves the so-called hypercomplex Fourier transform. Second, random undersampling is not a really sensible option, because of the structure of the actual experiment. In this talk I will review this background and review recent work with Hatef Monajemi, Jeffrey Hoch and Adam Schuyler, where we find that the now traditional structures -- for example Gaussian phase transitions, which are thought to be universal -- don't accurately describe the sparsity-undersampling relation. We will derive an accurate description with we think novel and interesting structure. Based on joint work with Hatef Monajemi, Jeffrey Hoch and Adam Schuyler.