Jordan Ellenberg, Noam Elkies, and Brian Hunt.
They all have this in common - they’re American, they’re close to my age, and they were the most brilliant kids on the national level when I was at a high school math tournament or two. 
No mention yet of John Conway? I’m a big fan.
Not as high some others have mentioned, but I’m also a fan of Ron Graham (both he and Conway are good popularizers and do top work).
I was tempted to mention him. I’m not qualified to evaluate his (or anyone else’s) mathematical significance, but I do keep seeing references to all sorts of things he’s done.
Saharon Shelah has done seminal work in Logic, and has authored over 1000 papers.
shelah.logic.at
Other than that vampire on Sesame Street? I got nothin’.
Who sadly passed away recently, so he can’t be considered for the list of ‘top five living mathematicians’ in any event :dubious:
But does being undead make him a candidate for honorable mention?
So Shelah is now pushing Paul Erdős for having the most co-authors on papers. He has 220, while Erdős had 511. And Shelah has some time yet to catch up with Erdős, since he’s a mere lad of 67, while Erdős lived to be 83. Soon we’ll be able talk about Shelah numbers in addition to Erdős numbers.
I have to second Serre (for those who don’t know, his thesis “Homologie singulière des espaces fibrés” was published when he was 25 and completely revolutionized algebraic topology.)
Is it known for sure that Grothendieck is still alive?
All anyone can say is that since there has been no announcement that he has died, then he must still be alive:
http://boxen.math.washington.edu/home/wstein/www/sga/circle/pragacz.pdf
If a mathematician opens up a whole new field of study that is still popular decades later, then it’s pretty clear that he (or she, I guess) did something important.
Milnor, Serre and Atiyah are living legends, although I don’t know how prolific they are nowadays.
this Milnor guy must be pretty important
can someone tell me more about him
Well, here’s the Wikipedia entry:
One interesting fact not mentioned there is that if the movie A Beautiful Mind had been more accurate, Milnor would have been a character in it, since he and John Nash were fellow grad students and shared an apartment one summer.
He’s a brilliant topologist who’s done a lot of work on knot theory, singularities, K-theory, and surgery theory. The two most readily identifiable (which is not the same as significant or deep) results of his are the explicit construction an exotic S^7 and the construction of the higher K-groups, along with their calculation in certain cases. The former is a manifold that homeomorphic but not diffeomorphic to the 7-sphere S^7; it was the first manifold for which that was known to hold. The latter is harder to describe, but it’s some high-powered machinery that describes things like equivalence classes of vector bundles modulo stabilization; that’s useful and interesting on its own, but it turns out that a bunch of surgery obstructions live in K-theory, and you actually get a full cohomology theory out of it rather than just the ground level K_0. (Wikipedia has more details and background material on all the stuff I mentioned above.)
would you consider him the top mathematician in america…
That question is unanswerable, just as the question in the OP is unanswerable. Naming the five best living mathematicians in the world is as hard as naming the five best living novelists in the world. I pick that as an example because I believe that there are approximately as many mathematicians in the world as there are novelists in the world. Let me define a mathematician as someone who either (a) has at least a master’s degree in math or (b) has published a paper in a refereed mathematical journal or © works at a job where their title is mathematician or mathematics professor. Let me define a novelist as someone who has published a fictional work of at least 40,000 words in printed form that has sold at least one copy. Then I suspect that there are several hundred thousand mathematicians in the world and several hundred thousand novelists in the world.
Now, if you were to ask someone with some thorough knowledge of current mathematics who the best living mathematicians were, their answer wouldn’t be random. They would be unlikely to name any of the 90% of mathematicians whose work is very minor. They would be likely to name some of the twenty or so mathematicians who are usually accepted as being top mathematicians. But it’s quite unlikely that the list of the five best living mathematicians that one knowledgeable person comes up with is the same as the list that another person comes up with.
The same is true of novelists. If you were to ask someone with some thorough knowledge of current fiction who the best living novelists were, their answer wouldn’t be random. They would be unlikely to name any of the 90% of novelists whose work is very minor. They would be likely to name some of the twenty or so novelists who are usually accepted as being top novelists. But it’s quite unlikely that the list of the five best living novelists that one knowledgeable person comes up with is the same as the list that another person comes up with.
This is partly because it’s impossible for anyone to know the entire field of current mathematics, just as it’s impossible for anyone to know the entire field of current fiction. This is also because there is a certain amount of personal taste in what constitutes top-notch mathematics, just as there is a certain amount of personal taste in what constitutes top-notch literature. In any case, no, it’s not possible to say who the top mathematician in the U.S. is with any certainty.
No. He’s 82. As for the top American mathematician of his generation, the question isn’t readily answerable. He’s definitely in the top tier, but sorting that that tier is more a matter of personal taste than any mathematical judgment.
i also here that this french guy serre, is legendary as well
It’s actually a good deal harder to pick the five best living mathematicians than the five best living novelists.
After all, anyone can read the work of those novelists that has been translated into their language, and once one has done that, one can compare and contrast the work of different novelists.
But in order to do the same with different mathematicians working in different fields, one has to have a sufficient understanding of each field to be able to appreciate the quality of each mathematician’s contributions to his/her field. IOW, you’d need a PhD-level grasp, or close to it, of all the major areas of mathematics.