One afternoon in January 2011, Hussein Mourtada leapt onto his desk and started dancing. He wasn’t alone: Some of the graduate students who shared his Paris office were there, too. But he didn’t care. The mathematician realized that he could finally confirm a sneaking suspicion he’d first had while writing his doctoral dissertation, which he’d finished a few months earlier. He’d been studying special points, called singularities, where curves cross themselves or come to sharp turns. Now he had u...