What to study next? (Math)

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  • #1
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Just looking for a little advice on subfields of math to study up on. I'm asking since it's hard for me to know what some subfields are like looking at them from a "bottom-up" perspective.

I'm a huge fan of algebra/category theory, but I feel like some of the problems can be a bit uninspired. I've seen a little bit of algebraic geometry, but sometimes the focus on the geometry turns me off (especially when it gets around to some complex algebraic geometry). Likewise with algebraic topology, although I only know a little bit about the basics of the fundamental group, covering spaces, etc.

So I guess I'm wondering: is there anything out there that is very algebraic in nature, but that maybe has a view towards problems that lie outside just algebra? Maybe I should look into algebraic geometry/topology again?

Thanks.
 

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  • #2
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Nothing? Surely someone can give a little input from their perspective.
 
  • #3
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I don't have any firsthand experience, but I have similar interests and I've had algebraic topology and algebraic graph theory recommended to me. Maybe something along those lines might similarly appeal to you. I've also heard that, though it's not outside of algebra, Galois theory problems tend to be enormously interesting. There's a very popular independent study project in Galois theory going on right now in my school's grad program.
 
  • #4
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I'm a huge fan of algebra/category theory, but I feel like some of the problems can be a bit uninspired.

Could you be more specific? It might help in making suggestions.

I've seen a little bit of algebraic geometry, but sometimes the focus on the geometry turns me off (especially when it gets around to some complex algebraic geometry). Likewise with algebraic topology, although I only know a little bit about the basics of the fundamental group, covering spaces, etc.

How much geometry/topology have you seen/worked with? These are both such huge fields I would still be inclined to recommend taking a closer look at them.

So I guess I'm wondering: is there anything out there that is very algebraic in nature, but that maybe has a view towards problems that lie outside just algebra? Maybe I should look into algebraic geometry/topology again?

Thanks.

Like I said, I would recommend you look at geometry/toplogy again. But as mentioned above, Galois Theory would also be a great idea.

Another suggestion: module theory. If you haven't studied modules in your algebra class, you should definitely take some time and read over the basic theory.
 
  • #5
mathwonk
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maybe commutative algebra? It seems wise to go with what you like and feel good at, but eventually you run into the unity of all mathematics, and it helps to know enough other areas to see those interconnections. Is Shafarevich's Basic algebraic geometry too geometric? How about Mumford's red book of algebraic geometry? Or Fulton's algebraic curves? (available free on his web page I believe.) These all use the algebra to clarify the geometry, and to make it more precise.
 
  • #6
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I've had a graduate course in algebra, so I'm familiar with the basics of modules and Galois theory -- both of which I find interesting. It's hard for me to be more specific and say exactly what I like about algebra/category theory, but I guess it's the level of abstraction and the way that one reasons in those fields that attracts me.

I haven't seen a whole lot of algebraic geometry (I know enough to tell you what a sheaf is, but not a scheme), nor a lot of algebraic topology (just the basics of the fundamental group), so I'm definitely open to studying those fields further. (I guess, then, to help me decide which one: which of these would you say is more algebraic in nature?)

As for the algebraic geometry books mathwonk mentioned: Shafarevich does seem too geometric for my liking, but I like what I've read from Mumford's Red Book. I haven't read any of Fulton's Algebraic Curves, however.
 
  • #7
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You could try to read Hartshornes book on Algebraic geometry. You have had Graduate Algebra and don't want an overly geometric treatment. Another Option many people have found more palateble is to Read Ravi Vakil's lecture notes (very extensive on his blog) on Algebraic Geometry.
 

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