And algebraic topology, and algebraic geometry, (..and I'm sure analysis and basically every other subset of mathematics). http://matt.might.net/articles/phd-sc

And algebraic topology, and algebraic geometry, (..and I'm sure analysis and basically every other subset of mathematics). http://matt.might.net/articles/phd-school-in-pictures/ http://matt.might.net/articles/phd-school-in-pictures/ You get far enough into any field (and it doesn't even have to be in the hard sciences,) be it law, medicine, linguistics, or something as seemingly trivial as making industrial bearings, and you'll need not only a graduate level education but an additional 5 or 10 years to get up to par with the rest of the field. The world is so catastrophically complicated at this point, it often overwhelms me.