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What to study in mathematics after taking basic math courses?

  • #1

Main Question or Discussion Point

I've taken differential,integral,multivariate calculus, DEQs, Introductory Linear Algebra, and am currently taking complex analysis and vector calculus (an extension of vector calc from multivariate). As a math student, how do you decide what to study after that? Is it time to specialize?

I don't have a broad road map of mathematics but I do know that I like to think about space, mappings, surfaces. How do mathematicians go about finding what they like to research/learn (either as a hobby or for their research)?
 

Answers and Replies

  • #2
jgens
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Not even close to specializing time. You should try reading some general topology, abstract algebra and maybe some real/functional analysis books. If surfaces sound interesting to you, then look into some differential geometry. Shop around and see what you like.
 
  • #3
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Hi jgens, can you recommend any books for differential geometry for beginner?
 
  • #4
jgens
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Depends on your background and where you want to go with the subject. I think that Spivak and Lee are great books to get a feel for manifolds and the very basics of differential geometry. Do Carmo is another popular alternative.
 
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  • #5
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Is there any basic knowledge I need to know before studying differential geometry?
 
  • #6
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To OP, you are nowhere close to being able to specialize. Take two more years of general math, including more linear algebra, a couple semesters of analysis, some algebra, and a good topology course. You can always mix in stuff like intros to PDE, DG, graph theory, whatever here, but by no means should you specialize until you've tasted a sizable bit from each field.


Is there any basic knowledge I need to know before studying differential geometry?
What is your math background? Have you taken analysis, topology, algebra, a bit of algebraic topology, linear algebra, and how much of each? There are many places to start in Differential Geometry and Topology, but they all depend on your background. As mentioned already, if you have a mastery of the sophmore/junior courses associated with the above topics, you should probably be able to tackle Lee. However, it would be good to have a more 'intuitive' and elementary DG course first. DoCarmo is good. If you like physics, Nakahara is great.
 
  • #7
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You can usually see some on-going areas of research by taking some courses in number theory if you want to get a feel for some of the problems out there that is. You will likely need abstract algebra if you intend to go that route as well though. Many different things branch off abstract algebra, to name a few: algebraic geometry, algebraic number theory, algebraic topology.
 
  • #8
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What is your math background? Have you taken analysis, topology, algebra, a bit of algebraic topology, linear algebra, and how much of each? There are many places to start in Differential Geometry and Topology, but they all depend on your background. As mentioned already, if you have a mastery of the sophmore/junior courses associated with the above topics, you should probably be able to tackle Lee. However, it would be good to have a more 'intuitive' and elementary DG course first. DoCarmo is good. If you like physics, Nakahara is great.
Thanks for your advice. Actually I'm a engineer, the highest level of maths I have studied is engineering maths and numerical analysis. This is my first time heard about topology and manifold.
 

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