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What exactly is compressed sending and did Terrence Tao get into it?

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Emmanuel Candes and Justin Romberg had found empirically, using numerical optimization software, that certain solutions of nonconvex optimization problems (generally computationally intractable) coincide exactly or near-exactly with solutions of convex optimization problems (generally computationally tractable). Candes brought this up to Tao, who worked out a formal proof. It's actually pretty simple work, coming down to the triangle inequality and a basic construction from the field of harmonic analysis (Tao's area of expertise). David Donoho also got similar results at the same time. I don't see any reason to think that Tao knows anything in particular about computers or data science.
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It's more math then anything else. I think it originated in the optimization community. Basically you try to recover sparse signals (i.e. signals that have only few non zero entries) from as few linear measurements as possible. That is an NP hard problem, but luckily there are relaxations of the problem that under certain conditions find the solution. A lot of research was focused on how many measurements you need for a given sparsity, designing fast algorithms and proving, under which conditions on the sensing matrix these algorithms find the correct solution. It had found a ton of applications everywhere, e.g. wireless communication, but the hard part of it are the mathematical results. Terrence Tao write one of the first papers on CS in 2007 i think.
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