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I am interested in learning differential topology as Milnor, Guillemin, Pollack, Hirsch, Kosinski, etc.. did it. However, I am in University of Toronto as an undergraduate and none of colleges (or universities in Canadian English) near me do not have any professors working in that field. Is it the same trend in America? Or did a lot of things happen since 1960-70s (back then that area was flourished)?

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I'm not sure this is appropriate for the site (as I'm not sure it has an objective answer), but I'm not sure it isn't, so for the time being I will not vote to close. –  Alex Becker Sep 25 '12 at 4:23
    
By the way, I am asking these because I wanted to talk to professors working in that area (not self-studying, because I am doing that too), but that the only professor being close to that needs these tools for physics (eg Topological Quantum Field Theory for cobordism) –  chhan92 Sep 25 '12 at 4:27
    
The question is already answered partially, but to clarify further, I put those mathematicians because they refer to their work. For example, I meant more topological flavored differential topology such as transversality, intersection theory, degree theory, morse theory, surgery, cobordism, etc.. on smooth manifold settings (not topological manifold setting). Ryan Budney's answer is really nice, but I am narrowing down what I mean, so that it is more clear of what I mean. –  chhan92 Sep 26 '12 at 2:47
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up vote 6 down vote accepted

What you say isn't really true.

There's plenty of people at U.Toronto that understand that material. V. Kapovich is one. But plenty of people in symplectic geometry/topology and the dynamics group know that material as well. I'm at a Canadian university (U.Victoria) and I know that material quite well. A bunch of people at McMaster know that material well, also.

As a field, Differential Topology mostly began in the States, France and Russia. Germany, Switzerland, Denmark, the UK, Japan, etc, all these countries have plenty of people that know or developed that thread of ideas to some extent. But the bulk of that field is in the States. Not many people call themselves differential topologists anymore, mostly because as a field it's evolved into more specialized niches, all with their own names.

Also, "Differential Topology" as a term didn't really catch on in some countries, like Russia. You'll find some field specifiers are culture-dependent. More often than not differential topology is lumped-in with differential geometry by Russians. I apologize if I'm painting with a broad brush but your question is fairly broad.

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"Quote: Not many people call themselves differential topologists anymore, mostly because as a field it's evolved into more specialized niches, all with their own names." Thank you very much! I was going through the list of professors in my institution to check who is working inside, but finally you clarified this matter for me (as I think similar thing is true for University of Waterloo, which I also asked my friend about). Maybe I should try to talk to professors in symplectic geometry and complex manifolds first I guess. –  chhan92 Sep 25 '12 at 4:40
    
Thank you! I talked to a professor who taught Morse Theory before. Apparantely the kind of stuff I am referring to mostly belongs in Morse Theory and Cobordism (for cobordism, I mean in smooth manifold setting (not topological manifold)) (and any relevant ones such as Handlebody, etc). I guess Transversality belongs to basics of those topics above (and application to dynamical system), and intersection theory degree theory in Guillemin & Pollack belongs to different viewpoint to algebraic topology results (and applications to physics again). –  chhan92 Sep 30 '12 at 0:06
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