Choosing research areas in topology

Forum for the GRE subject test in mathematics.
grassman1
Posts: 3
Joined: Sat Dec 05, 2015 8:45 pm

Choosing research areas in topology

Postby grassman1 » Sat Dec 05, 2015 8:54 pm

It's about time to consider more seriously my research area, and as of now, my primary interest is in topology. Specifically, I'm interested in interactions between algebra and topology, (topology used in algebra, and vice versa), and have taken basic courses and several topics classes in algebraic topology. It seems that much of modern research in algebraic topology is heavily categorical and algebraic in contrast to using rather more topological techniques.
1) Would algebraic topology indeed be a proper area of research to start with given these interests? Are there instances where topology is used to do algebra? I've also been thinking a bit about geometric topology.
2) In general, is algebraic topology harder than geometric topology in terms of prerequisites to beginning research and does geometric topology generally use more analytic techniques, which I'm not as comfortable with as I'm more algebraic oriented?
3) In addition, it seems that these two fields are related, mostly through applications of techniques from algebraic topology to geometric topology. Would it thus be easier to begin with the former?
4) Which field in general has more applications to physics?

Thanks!

MathCat
Posts: 140
Joined: Thu Oct 23, 2014 12:17 am

Re: Choosing research areas in topology

Postby MathCat » Sun Dec 06, 2015 3:07 pm

I can't answer these questions, but have you looked into geometric group theory?

grassman1
Posts: 3
Joined: Sat Dec 05, 2015 8:45 pm

Re: Choosing research areas in topology

Postby grassman1 » Sun Dec 06, 2015 5:24 pm

MathCat wrote:I can't answer these questions, but have you looked into geometric group theory?


I have looked into it, but I would prefer to work in a field a bit more abstract. I'm primarily stuck between choosing geometric topology or algebraic topology, and was wondering which one in general was harder to begin research in.




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