I have linked before to the blog of my grandniece Emily, who is studying math in Budapest. Here is another of her posts on her classes. This one is the best description I have ever read of what it is actually like to study serious math. You can find it here. I would admit her to my doctoral program in a shot, if I ran one.
Let me say just a word about this subject, with reference to Michael Llenos' comment yesterday. There is an enormous difference between those of us who can, with hard work, master some bit of serious math, and people like Newton or Einstein or John von Neumann, who actually make the stuff up brilliantly. At one point in my career [mid-seventies] I studied Game Theory rather seriously, eventually teaching a graduate course in which I went through von Neumann's proof of the Fundamental Theorem of Game Theory [that every mixed-strategy two-person zero-sum game has at least one solution], complete with a full-scale proof of L. E. J. Brouwer's Fixed Point Theorem. That is hard stuff, but I did it more by brute force than by any real mathematical insight. By contrast, my old friend Robert Ackermann once told me that the great Russian-English mathematician Besicovitch could "see" objects in n-dimensional vector space. He would "look" at such an object [say, an n-simplex] and rotate it in his imagination until he could see that something was just true of it. Then he would go about the tedious business of actually producing a proof of what he could immediately intuit to be true.
Of all the things I cannot do [such as throw a curve ball or play the viola well or tap dance], there only two that I really would give a great deal for. One is some facility with foreign languages and the other is the sort of formal mathematical intuition that Besicovitch and von Neumann and the rest had. Lord, it must be wonderful to have that ability!
Meanwhile, I can read Emily's blog and enjoy vicariously her engagement with real math.