Help this PFer with changing plans (suggerences needed)

[Redsummers]

This is not a thread about 'can I get into there or there' etc... It's just that I need some advice about my current situation. Which, after some searching in PF, I could not find any answers regarding a similar situation.

So, to get an idea, I am currently studying a Physics B.Sc. in Germany (taught in English). My grades are fine so far (A-ish? German grading is weird). And I will be graduating next year (i.e. June 2012). So, I would not be worried if I had to apply to a physics graduate program...

HOWEVER, this past year I have become really attracted to pure mathematics. By this I don't mean that I want to pursue a graduate degree in Theoretical physics; I want to pursue a graduate degree in pure mathematics.

Now the problem is, in Germany, the universities where you can pursue graduate studies in English for Mathematics are two: the BMS (Berlin) and Uni Bonn. Obviously, Uni Bonn is one of the top notch universities for Mathematics in Europe. They have the max Planck institute for mathematics there, etc... And the BMS is quite well known in Europe, so that I don't know whether I would really have a shot or not.

That said, since I want to broaden my chances, I have considered applying to the US. Now, my problem is that I don't really know how does a 'changing plan' like this would work in the US.

Can physics students apply to a mathematics department? Or they just get rejected right away? Do you think I should apply to an undergraduate math-degree, instead? (in that case, would I need to take the SAT? could I take the M-GRE to get into an undergrad math program?)

I have been doing some searching in different Math dept., but I don't see anything related being mentioned...

The branches I am most interested with, are Algebraic topology, category theory (specially the notions of Topoi) and some aspects of Algebraic Geometry. But that's definitely not all, of course. The familiarity with such branches comes from me reading books in the library.

Do all the departments in the United States have Algebraic Geometry and Algebraic Topology seminars? (I don't think that's the case, though...). Do you know of *reasonable* universities (i.e. where I would not get rejected right away) with such seminars?
I tried to find a list in google, but couldn't find any... Do you know if such a list exists? Or, say, a list with research interests arranged by universities?

If somebody has answers to this questions regarding Canada instead of the States, I would be pleased to hear them too. I have actually been considering Dalhousie university, but I think it is really well-known, so that my chances wouldn't be too high either. Well, that's my impression. Never been to Canada.

And for the UK... really, UK has a complex system. Because I think I would have to take the masters first (like in Germany), but they consider a master as the 4th undergrad year, and it's somehow of a transfer, etc. So I don't know if I should really consider UK..? (maybe it's easier than what I think!)



I would really appreciate your input on this matter. I have been googling too much, but I am sure that people here have such information already in their heads. Thanks!


P.S. For the ones wondering about background info.:
My math courses here in Germany consisted of Real Analysis I (sequences, differentiation), Real Analysis II (integration, integration in R^n, vector Analysis), Complex Analysis (Cauchy-Riem. eq., Liouville's thm., Cauchy's thm. cauchy's formula, Runge's thm., etc.), Measure theory (σ-Alg, etc.) together with Integration theory (Lebesgue's etc.) and its generalization to Functional Analysis (Hilbert spaces etc.), and finally theory of Distributions (Dual spaces, Schwartz' spaces, etc). Hmm... Also Linear Analysis (+ intro to Abstract Alg.), ODEs, and a small intro to PDEs. All courses are proof based, of course.
Hopefully my bachelor thesis will be about Complex Manifolds, showing some properties, theorems, extensions, and finally some applications to Physics (for a Physics degree, the Bachelor needs to be minimally related to physics lol.)

P.S.2 ops, the post was way longer than I expected it to be.. sorry!

 

[micromass]

So you want to do algebraic topology and algebraic geometry, but you never even had a course in abstract algebra or topology? How would you know you would like these fields?

The fields you mention are not easy fields and I think that you first need to work on the basics.
Take undergraduatecourses in

• Abstract algebra (including groups, rings, fields), the more you take the better.
• Topology
• Undergraduate algebraic geometry
• Lattice theory (handy for toposes) and sheaf theory
• Differential geometry (provides a lot of ideas for algebraic geometry)

You won't find undergraduate courses on category theory, but it is quite easy and you should read up on it independently.

I highly recommend that you do a few yours completing all these courses, and only then can you consider researching these things. If you wish to study these independently, I can recommend you good books on these topics

Note that you shouldn't expect to do something exciting soon. Algebraic topology and algebraic geometry are probably a bit out of reach for you now

 

[Redsummers]

Thanks for the input, Micromass!
Yeah, that makes total sense. Although, in my Linear Algebra class we went over definitions of rings, fields, etc; and its properties. We didn't mention galois theory though. Also, at the end of Real Analysis II, we still had three-four weeks left and our professor decided to give a brief introduction to topology, giving us definitions of metric spaces (even though this notion was later generalized in Hilbert Spaces), compactness, completeness, etc.

Also, I have been doing exercices form one of the books on Algebraic Geometry in the library (first two chapters lol, at least that's something). And then, again, now that you mentioned differential geometry, I will most likely learn about complex manifold next year, so that will come handy :)

But anyway, yeah, you're right. The problem is that such courses are not offered to physics students here. Needless to say that they would be in German anyway, and my German knowledge is still a bit shaky to attend a whole class of mathematics. So, do you think the most smart thing to do would be to apply to a graduate math program (because I will have to anyway –next year–) and once there, take those classes and see?

If I am lucky enough to get into BMS (Berlin would be cool...) or Bonn, then I would have to take the master and there I could take such courses. That would be surely perfect. But again, I need to broaden my chances. :/PS. You also mentioned category theory, yeah, I also started reading a book in the library two months ago. And once you get the general concept is pretty straight forward (since we have been using special cases in other math classes without even knowing), and *somehow* easy to follow –in comparision to other stuff. It's good to know that students normally read it by themselves, because I hardly found any university offering such course (either for undergrad or grad). Which made me think, more than once, that the subject was totally obsolete.

 

[micromass]

Thanks for the input, Micromass!
Yeah, that makes total sense. Although, in my Linear Algebra class we went over definitions of rings, fields, etc; and its properties. We didn't mention galois theory though. Also, at the end of Real Analysis II, we still had three-four weeks left and our professor decided to give a brief introduction to topology, giving us definitions of metric spaces (even though this notion was later generalized in Hilbert Spaces), compactness, completeness, etc.

Also, I have been doing exercices form one of the books on Algebraic Geometry in the library (first two chapters lol, at least that's something). And then, again, now that you mentioned differential geometry, I will most likely learn about complex manifold next year, so that will come handy :)

Well, yeah, you probably know what rings, fields, topologies and compactness are, but that doesn't suffice. You'll need to know the fine technical details very well, and that requires you to follow a course on them and making many exercises!

But anyway, yeah, you're right. The problem is that such courses are not offered to physics students here. Needless to say that they would be in German anyway, and my German knowledge is still a bit shaky to attend a whole class of mathematics. So, do you think the most smart thing to do would be to apply to a graduate math program (because I will have to anyway –next year–) and once there, take those classes and see?

I'm just afraid that no graduate math problem would even accept you at this point. To give a physics analogy, you're saying that you want to do research in quantum theory but you haven't studied mechanics yet. You will need to get the basics in order first.

Furthermore, if you want to do a PhD in Germany, you will need a masters degree first. You should check the schools to see what the entrance conditions are for a masters degree.

If I am lucky enough to get into BMS (Berlin would be cool...) or Bonn, then I would have to take the master and there I could take such courses. That would be surely perfect. But again, I need to broaden my chances. :/

Why not take some classes in German now? Taking a math class in German is quite easy. I don't know any german, but I can read math articles and math papers in German fairly easily. You will have so much more possibilities if you already know German!

PS. You also mentioned category theory, yeah, I also started reading a book in the library two months ago. And once you get the general concept is pretty straight forward (since we have been using special cases in other math classes without even knowing), and *somehow* easy to follow –in comparision to other stuff. It's good to know that students normally read it by themselves, because I hardly found any university offering such course (either for undergrad or grad). Which made me think, more than once, that the subject was totally obsolete.

No, universities don't usually offer category theory as a different subject (and certainly not topos theory!). The category theory stuff is often mixed in other subjects or students are required to learn it independently.
The subject is not obsolete, and certainly not in algebraic geometry and algebraic topology, where categories are used intensively. But the subject is perhaps too easy to spend an entire course on it.

 

[Redsummers]

I'm just afraid that no graduate math problem would even accept you at this point. To give a physics analogy, you're saying that you want to do research in quantum theory but you haven't studied mechanics yet. You will need to get the basics in order first.

Ouch! really?
But I have seen that in most (almost all) graduate programs –in US and Canada–, they offer Algebraic Geometry, topology etc. I.e. they do not hope the students to know it before starting the graduate studies. They even offer courses such as Complex Analysis, Functional analysis, etc. (which I have already taken)

I guess that in Europe is different, but that is why I was considering staying in Europe for a master (yeah, the BMS and uni Bonn that I mentioned would be to do a master). And as far as Bonn told me, I would be *eligible* for a masters. I am still waiting an answer from BMS.

What are your thoughts about taking a master, in say, Belgium? (you're from Belgium, right?) Do you think they would consider a physics student? Hm, I may as well e-mail them.

Thanks again!

 

[micromass]

Ouch! really?
But I have seen that in most (almost all) graduate programs –in US and Canada–, they offer Algebraic Geometry, topology etc. I.e. they do not hope the students to know it before starting the graduate studies. They even offer courses such as Complex Analysis, Functional analysis, etc. (which I have already taken)

I guess that in Europe is different, but that is why I was considering staying in Europe for a master (yeah, the BMS and uni Bonn that I mentioned would be to do a master). And as far as Bonn told me, I would be *eligible* for a masters. I am still waiting an answer from BMS.

What are your thoughts about taking a master, in say, Belgium? (you're from Belgium, right?) Do you think they would consider a physics student? Hm, I may as well e-mail them.

Thanks again!

Ah, you are actually eligible for a masters in Europe? That's good because it means it's not impossible for you to do what you are planning to do. The thing is, if you planned on doing a PhD in analysis, then you'd have a nice background for it, you have almost all courses needed for that (except topology). What you're planning to do now will be very hard work for you (I'm not trying to discourage you, but that's how it is).

I really suggest that you start studying abstract algebra and topology as soon as possible. Study it independently if you must. You always have this forum to ask questions though...

If you are willing to work hard for it (starting now), then doing a masters in pure mathematics isn't impossible. You already have real analysis and such a things under your belt, which means that you're already mathematically mature.

You are likely eligible for a masters in Belgium, but you should email them. Brussels, Antwerps and Leuven have good research groups in algebraic geometry and topology. But you should know that you need background in algebra and topology before attempting a master, such a courses will not be offered in a masters program (but perhaps you could ask them to make exceptions). The good thing is that most courses in Belgium are offered in English (if you ask the lecturer).

Perhaps the US is a bit friendlier because they indeed offer courses in topology and algebraic geometry in graduate school, something which is considered known in Europe.

 

[physiker_192]

Some thought:

If admitted, you might be required to undertake an interview/oral exam, which I would assume it would cover all the contents of a Math BSc.

 

[Redsummers]

@micromass: Haha, yeah. You're right about the US/Europe University-duality.

But anyway, thank you really much for all your suggestions. I will just start studying Abstract Algebra and Topology this summer then. (The dilemma is, though, if I should rather study the GRE... maybe someone from the US can shed some information about it later).
Before doing this though, I will first have to go though my semester exams (in three weeks, yay).

@physiker_192: Hey! Ohh, I see. That's good to know. But you are surely talking about German universities, right?

 

[physiker_192]

@physiker_192: Hey! Ohh, I see. That's good to know. But you are surely talking about German universities, right?

Yes, at German universities.
Sometimes, when the admission committee is not sure about the qualifications of the applicant (e.g. foreign degree or different background), then they might require such a test

It may not be applicable if the MSc is in Applied Math,, but its a different story if its pure math.

And to be actually be able to follow up (and not lag behind) with the grad math courses, you should really have a similar background to someone with a BSc in Math (you should try doing some exercises from the Math BSc courses and see how you get along).Edit: You can ignore the rest of this reply, as I didn't read your first post completely earlier.
BTW, I am not sure whether you want to continue your research in pure Math, since as you see, sometimes the courses might seem interesting, but the actual research need not be.
So you can always go for a Physics MSc and attend additional math or any other courses of your choice.

 

[Angry Citizen]

Since you haven't graduated yet, have you considered a double major in physics and mathematics? I can't imagine it would extend your degree beyond an extra year, two at the most, and you'll probably wind up even more prepared for math grad school than other math grad school candidates. Just my two cents.

 

[Redsummers]

@ physiker_192: Yeah, that makes lot of sense. Maybe I should really consider applying for an undergraduate degree somewhere. I will see..

Since you haven't graduated yet, have you considered a double major in physics and mathematics? I can't imagine it would extend your degree beyond an extra year, two at the most, and you'll probably wind up even more prepared for math grad school than other math grad school candidates. Just my two cents.

That's a good point, Angry citzen. But again, if I was in the united states, that would probably be easier to do. But in Germany a) I am not 100% certain whether they transfer courses from one department to another and b) the classes would be in German.
Hmm.. maybe I should just check whether they allow students to double-major in physics and mathematics and consider studying in German language. Because, I guess I shouldn't worry much about my age... I will graduate next year and I will be 20, so maybe taking 1-2 more years to double major would be fine.


Thanks for the input!