Postby **rsk** » Wed Oct 22, 2014 10:36 pm

I remember asking one of my professors that question my senior year when I was asking which grad level classes are most beneficial. He said the one thing you can't learn too much of is linear algebra. It shows up everywhere. I'm assuming you've had the basic undergrad course, but it never hurts to really learn the theory behind it (vector spaces, dual spaces, tensors, etc.), something that may be covered either in advanced undergrad classes or beginning graduate classes.

Two other areas I was told to focus on being good at are real analysis and abstract algebra, since almost every graduate program tests those two subjects in the qualifying exams. If you're interested in number theory, studying complex analysis would also be useful I think, as well as some differential geometry and topology.

As far as level (and this is what I am personally doing), if you haven't had a solid background in the subject, start with an advanced undergrad / beginning graduate level text. For the subjects I mentioned, those would be (for example):

1) Linear Algebra: Linear Algebra and its Applications, Lax

2) Abstract Algebra: Abstract Algebra, Dummit and Foote (this thing is a tome)

3) Real Analysis: This really depends on how much you know. If you're comfortable with Rudin's Principles, try his Real and Complex Analysis.

4) Complex Analysis: The Rudin book above, as well as Visual Complex Analysis (I love this one).

5) Differential Geometry: This is a tough subject to break into I find. If your multivariable analysis is shaky, it might not be a bad idea to read up on those chapters in Rudin before moving on. Alternatively, Spivak's Calculus on Manifolds covers background material really well. After this, you can try Lee's Introduction to Smooth Manifolds to get to the nitty gritty of the subject.

6) Topology: Munkre's text is the gold standard for beginners. For differential topology, Milnor's text is a classic.

Hope that helps.