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So, I am a graduate student who is certain that he does not want to do analysis (I think...). What are the most exciting fields in mathematics right now? It seems to me that very generally, they are algebraic geometry and algebraic topology, with the latter being specifically concerned with homotopy theory (something I find very interesting). Perhaps this is not the correct place to post such a question and I'm welcome to redirection on that front.

Ultimately, I would like to do something highly abstract and categorical. I love the ideas of sheaves, homological algebra, and various categorical concepts. Does anyone have any good knowledge of where to begin? Also, do algebraic geometry and homotopy theory have a great deal of common ground?

Thanks, and sorry if this question belongs somewhere else.

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For the second-to-last paragraph, see en.wikipedia.org/wiki/A%C2%B9_homotopy_theory –  Akhil Mathew Dec 10 '10 at 0:35
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The "most exciting fields in mathematics right now" are entirely subjective. So IMO answers to that question aren't much more than an opinion poll. Of course algebraic geometry has something in common with homotopy-theory. Take a look at the work of Dugger, Isaksen, Shipley, Lurie, etc. –  Ryan Budney Dec 10 '10 at 0:46
    
Sure, I am pretty much polling people. –  Jon Beardsley Dec 10 '10 at 5:59
    
Also, unfortunately, to some of us these things are rarely "of course". –  Jon Beardsley Dec 10 '10 at 6:00
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LOL there's someone on your faculty, Romie Banerjee, whose listed research interests are "homotopy theory and algebraic geometry". This might be the perfect person to answer your last question! –  Aaron Mazel-Gee Dec 10 '10 at 18:51

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Wow, this is great! I would love to expound a bit on some of the things you have mentioned. While I agree with Ryan that what is exciting and hot is entirely subjective, I too find homotopy theory and algebraic geometry enthralling. I really only know things about homotopy theory though, so I can only talk on that.

There are loads of places to go abstract crazy under the general heading of homotopy theory. A lot of the model category theory is being put to use in many many places, like Algebraic Geometry! (This is the work of Dugger-Isaksen that Ryan mentioned above). There is a lot of beautiful abstract framework that people work with that falls under the umbrella of homotopy theory. Personally I am a bit more on the computational side, or rather old fashioned "how do we compute $\pi_n^S(\mathbb{S})$?" So I am interested in different computational aspects of the stable homotopy category of spectra. It really is a big field.

I just realized I did not address your question about where to begin. There are some amazing deep results that are really cool that you can get to in a finite amount of time. For homotopy theory I would work on trying to get to come classical results of Adams, like vector fields on spheres or Hopf invariant one. Both of these are addressed in Mosher and Tangora (now a dover book). It is a good book, but you should skip certain bits, like you don't need their construction of the steenrod operations. There is also the theory of formal group laws and how those relate to stable homotopy theory, that stuff is awesome. Being at JHU, I would begin by asking Jack Morava or Andrew Salch. They are both super nice guys that know a whole lot, but they are really smart and might be hard to keep up with. So maybe ask them what got them started in being interested in such things, or what they think is something that is really cool to work towards. I also think that Boardman and Wilson would be excellent people to ask, but I have had no interaction with them. They are also deep vats of information. I learned so much going through the first third of Boardmans CCSS paper, it was great!

As far as the interaction, it is large! As Mathew pointed out there is a whole "new" field called motivic homotopy theory that asks "what can homotopy theory tell us about schemes?" There are other interactions though, a lot of number theory and stackiness is interesting via the chromatic picture. A lot of people study the moduli stack of formal group (laws) in order to get at the stable homotopy groups of spheres. That moduli stack necessarily has a lot of AG in it.

I think the best thing to do would be to talk to people around about things that you are learning/want to learn. In fact, feel free to drop me a line at first intial last name at wayne.edu. Seriously drop me a line if you want to get into a bit more detail, it is unclear what will be beneficial for me to say without a bit more background.

PS: there are great people at JHU that know loads about homotopy theory, as well as its interactions with algebraic geometry. Morava pioneered the relationship with class field theory, and now it seems like Salch is taking it all the way to Langlands Land. It would be hard to be at a better place to study homotopy theory!

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Wow, this is awesome. Thankyou for all the info. Jack Morava has already sort of spurred this interest in me, he's a really great guy. I can tell I'm really at the first steps of something pretty deep, but I find all of this fascinating. Right now I'm just trying to get a better grasp on K-theory in general, as well as spectra, spectral sequence, classifying spaces, and some of these things that keep getting tossed around and sound like important building blocks. –  Jon Beardsley Dec 10 '10 at 5:58
    
I never really feel like I understand k-theory as much as i should. I have some advice about SS though, There are a lot of great easy computations you can go through (Mosher and Tangora) before reading Boardmans CCSS paper. Drop me a line ifn you want to chat or ask more specific things. –  Sean Tilson Dec 10 '10 at 6:22
    
Just rediscovered this thread. Haha! Good answer. @Sean –  Jon Beardsley Nov 9 '11 at 0:47
    
thanks beardman! –  Sean Tilson Nov 11 '11 at 7:24

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