I'm finishing my undergraduate mathematics programme this summer. The graduate program at my university requires that you specify a specialization on entry. This is very difficult for me and would like some advise. I have no experience with the third set. (Courses I liked: Groups, fields and Galoistheory, algebraic topology)

The possible sets to choose from are :

Set 1 "Algebraic geometry & Number Theory":

Algebraic Geometry, Algebraic Number Theory, p-adic Numbers, Elliptic Curves, Diophantine Equations, Modular Forms, Analytic Number Theory, Riemann Surfaces.

Set 2 "Differential geometry & Topology":

Homotopy Theory, Homological Algebra, Sheaf Theory, Knot Theory, Quantum groups and Knot theory, Category Theory, Simplicial sets, K-theory and vector bundles, nalysis on Manifolds, Symplectic Geometry, Foliation Theory, Riemannian Geometry, Lie groups, Semisimple Lie Algebras, Differential Topology.

Set 3 "Logic":

Model Theory, Proof Theory, Computability Theory, Intuitionism, Category Theory, Topos Theory, Peano Arithmetic and Gödel Incompleteness, Set Theory, Type theory and $\lambda$-calculus.

Which one has the most algebraic 'flavour' to it

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    $\begingroup$ The one with "algebraic" in the title... anyway, if you don't know what these things are you should find out. They are all fascinating. I recommend reading the Princeton Companion to Mathematics. $\endgroup$ – Qiaochu Yuan Mar 5 '13 at 18:07
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    $\begingroup$ §-calculus? Logic must have made great strides since my day ... we had to make do with $\lambda$-calculus. $\endgroup$ – Henning Makholm Mar 5 '13 at 18:15
  • $\begingroup$ Thank you for pointing me to some literature. And the title argument is not a obvious one. And I assume its $\lambda$-calculus $\endgroup$ – bbnkttp Mar 5 '13 at 18:19
  • $\begingroup$ If you have finished undergraduate studies there, you know the faculty and what they are up to. You could ask them directly, or extrapolate from what they taught, or check their current research interests... $\endgroup$ – vonbrand Mar 5 '13 at 19:02
  • $\begingroup$ @QiaochuYuan: You should post that as an answer. $\endgroup$ – Jim Mar 6 '13 at 2:03

I'll differ from Qiaochu and PyRulez and suggest the second set. You said you liked algebraic topology, and many of the courses here are the ideal preparation to continue to study it. If you move in a certain direction, perhaps best represented by the courses in homotopy theory, homological algebra, simplicial sets, and K-theory, the subject takes on a very algebraic flavor. The same could be said for the courses in quantum groups and Lie algebras, though these would move you in a separate direction. Though I imagine they're teaching it to prepare you for the geometry courses, sheaf theory is also a key component of algebraic geometry, and you could conceivably use it to start travelling down that road. Finally, it's hard to see an algebraist disliking category theory.

The rest of the courses are solidly situated in geometry or manifold topology, which you might find less attractive. However, it's also worth pointing out that set 1 will force you to learn a decent bit of analysis, which you might also find less attractive.

(Disclaimer: I'm a homotopy theorist and thus pretty biased here. I also know next to nothing about the third set, which has been ignored by your responders but which you might end up liking as well.)

Another thing that might help you decide is that the two sets use different sorts of algebra. A number theorist will see a lot of discrete groups, rings, and fields, and an algebraic geometer will see a lot of rings. An algebraic topologist will work more often with modules, homological algebra, and diagrammatic/categorical reasoning; Lie groups are the primary sort of groups you'll see. You could decide which one you find more attractive -- skimming introductory books to some of the courses involve might help.

A third thing to take into consideration is the professors in your department. You're going to get chummy with the people working in the research area you pick, so you should hope you enjoy their company. For the sake of jobs post-grad school, it would also be nice if they're respected in their fields. They're also probably the best people to ask this question -- schedule a meeting with some of them, tell them about the math you've enjoyed so far, ask about what they do, and try to see if you'd be a good fit for their program.

Good luck!

  • $\begingroup$ Thanks for this constructive reaction! $\endgroup$ – bbnkttp Mar 6 '13 at 7:10

I would say set 1 since it has algebra in the title.

  • $\begingroup$ This is not a valid argument. subjects with algebra in the title tend to be a subfield of a subject, which is the more algebraic of the subfields. Algebraic geometry, algebraic number theory, algebraic topology. But Category theory does not have algebra in the title, so does that mean the other 3 subjects are more algebraic of nature? $\endgroup$ – bbnkttp Mar 6 '13 at 7:13

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