breakthroughs

Yitang Zhang
Unlike other academic fields, mathematics is notorious for being a young thinker's game. G.H. Hardy, in his 1940 book The Mathematician's Apology, ticked off a list of great math whizzes from Isaac Newton to Carl Friedrich Gauss and noted that they all made their most important discoveries in their 20s. He concluded, gloomily (and famously):
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