Mathematical studies before graduate school

[Desordre]

Dear fellows,
my undergrad was not in maths, but I have quite a chance of getting into a master's program in pure maths, later this year. Long story, not particularly interesting.

My question is due to the fact that I am quite prepared to the examination (basic real analysis and linear algebra), but I don't think they do reflect an adequate undergraduate formation. So, I'd like to ask you: what do you think it is necessary to accomplish before graduate studies?

If you feel like recommending some bibliography, it would be very nice!
Desordre.

 

[micromass]

What do you want to do in grad school exactly?? Something analysis related?? Algebra?? Our answer will depend on this.

In any case, you're missing abstract algebra, which I feel is very necessary.

Do you have any familiarity with proofs?? (I don't know how rigorous the linear algebra/basis analysis was). If not, you're going to need that.

I can probably give a better answer if you give us some more information.

 

[Desordre]

Dear Micromass,
thanks for you answer. I have not a precise focus already (because, up to a certain point, it does not depend on me), but my master's will be in analysis for sure.

About abstract algebra, it is a hole in my background indeed. I know some group theory, but I need to develop this area. Proofs are something I am working right now. Tough self taught, my studies of analysis and linear algebra were rigorous, based on Spivak, Bartle, Rudin (analysis) and Strang, some Halmos (linear algebra). Nevertheless, I think I can improve my demonstrations, specially when I consider the formal situation of an exam.

My main concern are the disciplines I feel that lie on the threshold: differential geometry, PDEs, measure theory are undergraduate or graduate matters? And what about projective geometry, probability, further topics on algebra: are not those topics "cultural" from my point of view?


Ds.

 

[micromass]

OK, that's good. If you went through Spivak and Rudin, then I think you're familiar enough with proofs.

Since you want to do a masters in analysis, I think the following topics are necessary (roughly in order that you should do them):
1) Topology. Munkres is a good book.
2) Measure theory. "Principles of Real analysis" by Aliprantis and Burkinshaw is good.
3) Functional analysis. If you can handle it, then you should go through "Real Analysis" by Lang.
4) Complex Analysis. Hille's book is quite ok.

As for mathematical culture. I feel that one should be acquainted with a bit of abstract algebra. Basic stuff like groups, rings and modules should be known.

As for other topics, like probability, projective geometry and differential geometry, I feel that those aren't really necessary. It would be nice to know them, but it's better to focus on other things now.

 

[Desordre]

Dear micromass,
you brought a smile to my face. Of course I'd like to explore many branches of mathematics, and learn interesting stuff, but who has the time?!? These four topics are a reasonable goal, and are indeed an expansion from my current state of knowledge. I'm not acquainted with Hille's book: would you mind to give me the reference?
Thanks again!
Ds.

 

[micromass]

Dear micromass,
you brought a smile to my face. Of course I'd like to explore many branches of mathematics, and learn interesting stuff, but who has the time?!? These four topics are a reasonable goal, and are indeed an expansion from my current state of knowledge. I'm not acquainted with Hille's book: would you mind to give me the reference?
Thanks again!
Ds.

Oh, sorry. I mean this book: https://www.amazon.com/dp/0828402698/?tag=pfamazon01-20
There are many other good books on complex analysis though...