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How many linear measurements (equations) do you need to recover a high-dimensional signal (unknowns)? If you know a basis in which the signal is sparse, and your measurements are not too aligned with this basis, then far fewer than you might expect. Moreover, you can recast your underdetermined problem as a convex program and solve it efficiently. I will talk about when and why this works, mentioning some now-classic applications and a few exciting possibilities in biology.