prelim/quals exam questions thread

Forum for the GRE subject test in mathematics.
frgf
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Joined: Wed Jan 30, 2013 2:24 am

prelim/quals exam questions thread

Postby frgf » Sun Mar 31, 2013 1:01 am

Hi everyone, with admissions seasons wrapping up I thought it would be a good time to start thinking about prelims. Most PhD programs have qualifying exams (sometimes called prelim exams) that test first year graduate material in algebra , topology, analysis etc.

I'm hoping to skip these first year core courses so I will be preparing over the next few months for the prelim exams, using past exams from umich, stanford, ucsd, and any others that I can find online. It would be nice to have some people to discuss these problems with.

My plan is something like this: we pick an exam, set a deadline ( a week? 2 weeks?) then try to do as many questions as possible. Then we discuss the solutions and any related material, and repeat. And feel free to ask about any related exercises or problems from textbooks.

PM or post here if you're interested.
edit: I've decided that perhaps it's not really necessary to set a exam and deadline for everyone to do. Just let everyone know what you've solved and what you're working on, that way others reading the thread will know what questions to work on if they want someone to discuss problems with.
Last edited by frgf on Sun Mar 31, 2013 7:47 pm, edited 4 times in total.

longtm1989
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Re: prelim study group

Postby longtm1989 » Sun Mar 31, 2013 1:06 am

I would love to join you.

tbil
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Re: prelim study group

Postby tbil » Sun Mar 31, 2013 2:37 am

I'd like to join, too!

Eowyn
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Joined: Sat Dec 08, 2012 2:19 am

Re: prelim study group

Postby Eowyn » Sun Mar 31, 2013 4:47 am

I'd also love to be a part of the group, but I still have a few weeks left of my last semester and so shall start this preparation from around May 15 onward (may be a couple of days earlier... not sure). I also wish to sit for the written comprehensive examinations in Courant, NYU, this August (or September)... although I'm not sure if I can manage it... since it may entail going quite a few days in advance, and I might not find accommodation during that time (on campus housing facilities start from August 21). But I'll try nonetheless.

Let's do all the discussion and posting of problems in this thread... what do u guys say?

frgf
Posts: 24
Joined: Wed Jan 30, 2013 2:24 am

Re: prelim study group

Postby frgf » Sun Mar 31, 2013 5:03 am

Let's do all the discussion and posting of problems in this thread


Sure, let's start with that and see how it goes. I've already started with Topology QR Exam (Jan 2012) from http://www.math.lsa.umich.edu/graduate/qualifiers/. I've done q1,q2,q4 and q5 (from the morning session).

If anyone ends up doing these questions, let me know , and I'll post a sketch of my solutions so we can compare.


Feel free to post questions from any other sources here. I've decided that perhaps it's not really necessary to set a exam and deadline for everyone to do. Just let everyone know what you've solved and what you're working on, that way others reading the thread will know what questions to work on if they want someone to discuss problems with.



e.g.


Michigan Topology QR Exam (Jan 2012) http://www.math.lsa.umich.edu/graduate/ ... RJan12.pdf

Done: q1,q4,q5, q2a), and maybe q2b)
working on the rest

For q2b), isn't this just a consequence of the fact that the only automorphism of Z_2*Z is the identity? I don't see what it has to do with covering spaces.
edit: oops, there is another automorphism of Z_2*Z that I forgot about. So we have to have to use the fact that it's a covering translation + the fact that the induced projection of a covering map is injective to deduce automorphism must be the identity.


Stanford Analysis Spring '10 http://math.stanford.edu/quals/real_10s.pdf
Working on q1-10. Some of this stuff is new to me so it might take a while.
Last edited by frgf on Sun Mar 31, 2013 7:36 am, edited 4 times in total.

math_applicant
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Joined: Sun Oct 14, 2012 12:15 pm

Re: prelim exam questions thread

Postby math_applicant » Sun Mar 31, 2013 5:53 am

I think this might help. Here is a link to Ohio State's Qualifying exams, they also have links to previous exams in many other universities in one place.

http://www.math.osu.edu/graduate/current/qual

Eowyn
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Joined: Sat Dec 08, 2012 2:19 am

Re: prelim exam questions thread

Postby Eowyn » Sun Mar 31, 2013 7:11 am

Tell me something guys: I found from the website of NYU Courant that people are allowed to sit for the written comprehensive examinations twice, without needing any kind of permission. If after two attempts, still they do not find the success they are looking for, then they are usually in trouble, unless the Dean or Director grants special permission.

Now, these exams are held in late August or early September, and again in early January. I wish to be done with them by September, but that would possibly require me to reach New York quite a bit ahead of the starting date of the Fall Term, and that may cause me some trouble with the accommodation (since the on-campus housing facilities will start on August 21).

Any of you in this kind of situation? Any suggestions would be welcome...

mindreader
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Joined: Sat Oct 13, 2012 11:12 am

Re: prelim exam questions thread

Postby mindreader » Sun Mar 31, 2013 10:00 am

I am surprised that many of you are already looking forward to the prelims. Is this because you have taken a lot of graduate courses?

Legendre
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Re: prelim exam questions thread

Postby Legendre » Sun Mar 31, 2013 12:03 pm

I have begun studying for Analysis, Topology and Algebra in preparation of the year 1 exam as well.

But I think the syllabus might differ from school to school. No sure how good it would be to train for the exam of another school.

Anyway, I come from an Applied Math background so I am pretty much starting from scratch with regards to all these pure maths requirements. :oops: Can't wait until the year 2 quals - I am already semi-prepared for it from my extensive research experience. 8)

mohamedun
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Re: prelim exam questions thread

Postby mohamedun » Sun Mar 31, 2013 12:16 pm

I would love to join the preparation group too, any thoughts on the organization?

mindreader
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Joined: Sat Oct 13, 2012 11:12 am

Re: prelim exam questions thread

Postby mindreader » Sun Mar 31, 2013 1:40 pm

Can someone help me clear up the issue? Am I right in saying that the purpose of the preliminary exams is for the PhD candidate to skip the first year graduate courses?

I've never heard of such a thing before. I only know of the qualifying exams... If you fail the prelims, will they kick you out of the program?

Manyfolds
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Joined: Thu Jan 24, 2013 1:22 pm

Re: prelim exam questions thread

Postby Manyfolds » Sun Mar 31, 2013 1:54 pm

I'm in also. Eowyn, I'm planning to study for the Courant written exams as well and ideally take them in Sept (if I go to NYU).

Mindreader: The schools I am familiar fall into two categories. (Type 1) is that there is a basic exam taken in the first year at some point, and then qualifying exams taken after two years after which the student can do research. (Type 2) omits the first basic exam and just has a qualifying after year 2. Either way, if you take and pass the qualifying upon entry, you can skip to just doing research and you are effectively a third year student. You are typically given a few (~2) opportunities to pass the basic exams, but the sooner the better.

mindreader
Posts: 90
Joined: Sat Oct 13, 2012 11:12 am

Re: prelim exam questions thread

Postby mindreader » Sun Mar 31, 2013 1:58 pm

Manyfolds wrote:I'm in also. Eowyn, I'm planning to study for the Courant written exams as well and ideally take them in Sept (if I go to NYU).

Mindreader: The schools I am familiar fall into two categories. (Type 1) is that there is a basic exam taken in the first year at some point, and then qualifying exams taken after two years after which the student can do research. (Type 2) omits the first basic exam and just has a qualifying after year 2. Either way, if you take and pass the qualifying upon entry, you can skip to just doing research and you are effectively a third year student. You are typically given a few (~2) opportunities to pass the basic exams, but the sooner the better.


Hi Manyfolds, thanks for the response! Btw, have you gotten off the waitlist at Brown?

Manyfolds
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Joined: Thu Jan 24, 2013 1:22 pm

Re: prelim exam questions thread

Postby Manyfolds » Sun Mar 31, 2013 7:19 pm

Yeah, I had provided an old email. Am strongly considering it also.

frgf
Posts: 24
Joined: Wed Jan 30, 2013 2:24 am

Re: prelim exam questions thread

Postby frgf » Sun Mar 31, 2013 7:47 pm

math_applicant wrote:I think this might help. Here is a link to Ohio State's Qualifying exams, they also have links to previous exams in many other universities in one place.

http://www.math.osu.edu/graduate/current/qual


Thanks for the link!

mindreader wrote:I am surprised that many of you are already looking forward to the prelims. Is this because you have taken a lot of graduate courses?

I've taken a few, but still have lots of gaps. I'm hoping to learn as I go.


Legendre wrote:But I think the syllabus might differ from school to school. No sure how good it would be to train for the exam of another school.


I think the combined knowledge you gain from studying like this should be enough to pass most exams. Even if it doesn't, you end up learning a lot of maths in the process. But definitely check your syllabus to make sure you aren't missing any big topics.

mohamedun wrote:I would love to join the preparation group too, any thoughts on the organization?


See my edit to the first post. I will post what exams I'm working on, and you can work on them as well if you want to discuss the problems later. Or, if you want to do some other questions, post a link to which ones you're doing and I'll (and hopefully others) will try to work on them too.


mindreader wrote:Can someone help me clear up the issue? Am I right in saying that the purpose of the preliminary exams is for the PhD candidate to skip the first year graduate courses?

I've never heard of such a thing before. I only know of the qualifying exams... If you fail the prelims, will they kick you out of the program?

The terminology is non standard. For instance , at Berkeley, "prelims" means undergraduate material. If you look at their past exams you'll see what I mean. At some other unis, prelims cover first year graduate material. But sometimes this is called "qualifying exams". Most unis allow you to attempt the quals before first year starts, before taking the relevant courses, and if you pass, you don't have to do those courses. My intention is to study for the quals.

Legendre
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Re: prelim exam questions thread

Postby Legendre » Mon Apr 01, 2013 8:26 am

Do you guys think there are common topics for every PhD Math programme? E.g. Measure Theory, Lebesgue Integration, Rings & Modules, 2 Dimensional Manifolds?

I have very limited time (in a research internship and have another lined up all the way until Sept!) but I will try to participate on topics that are in my quals*.

*Year 1 qualifying exam - Analysis, Algebra, Topology. Then Year 2 qualifying exam focuses on specific research topics.

Legendre
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Re: prelim/quals exam questions thread

Postby Legendre » Wed Apr 10, 2013 12:34 pm

My department's exams (Analysis, Algebra, Topology) is quite similar to Harvard's syllabus: http://www.math.harvard.edu/quals/index.html

I suppose for USA PhD Math the syllabus for these 3 are roughly the same?

Edit: OSU's syallbus for Analysis and Algebra seems to be based on undergrad syllabus. e.g. no measure theory or lesbegue measure for Analysis. http://www.math.osu.edu/files/QualExamSyllabi.pdf

math_applicant
Posts: 157
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Re: prelim/quals exam questions thread

Postby math_applicant » Wed Apr 10, 2013 2:43 pm

Legendre wrote:My department's exams (Analysis, Algebra, Topology) is quite similar to Harvard's syllabus: http://www.math.harvard.edu/quals/index.html

I suppose for USA PhD Math the syllabus for these 3 are roughly the same?

Edit: OSU's syallbus for Analysis and Algebra seems to be based on undergrad syllabus. e.g. no measure theory or lesbegue measure for Analysis. http://www.math.osu.edu/files/QualExamSyllabi.pdf


I believe harvard's is meant to be passed after you have taken the first 2 years of graduate courses (or have taken similar ones at the undergrad level) so if you pass it, you can start doing research directly. At OSU, passing the exams just means you will start taking graduate courses right away (I believe you can skip these by taking individual exemption exams per required courses). I believe OSU's is similar to Berkeley's system.

Legendre
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Re: prelim/quals exam questions thread

Postby Legendre » Wed Apr 10, 2013 2:52 pm

math_applicant wrote:
I believe harvard's is meant to be passed after you have taken the first 2 years of graduate courses (or have taken similar ones at the undergrad level) so if you pass it, you can start doing research directly. At OSU, passing the exams just means you will start taking graduate courses right away (I believe you can skip these by taking individual exemption exams per required courses). I believe OSU's is similar to Berkeley's system.


I see. That makes sense. Thanks.

IfhhoteCannot
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Joined: Fri Jan 25, 2013 9:11 pm

Re: prelim/quals exam questions thread

Postby IfhhoteCannot » Fri Apr 26, 2013 1:41 am

Hi all! I'm not sure if anyone here is going to Wisconsin, but I've heard their quals are really tough, and from my first glance at the previous tests, this is an accurate statement. I'm starting to study for real analysis, and doing the most recently posted test.

Here's a link to all their previous quals
http://www.math.wisc.edu/graduate/quals/

And here's a direct link to the test I'm looking at now (from January 2012)
http://math.library.wisc.edu/reserves/e ... 01-725.pdf

I would love to start working on this with anyone who is interested. I'm sure it would be good practice for any school; I just figured I'd get the ball rolling with a test we can all talk about.

frgf
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Re: prelim/quals exam questions thread

Postby frgf » Fri Apr 26, 2013 5:01 am

I'm not going to uwisc but I'm happy to discuss those problems. What does the 'R' after Q7,8,9 denote?

IfhhoteCannot
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Re: prelim/quals exam questions thread

Postby IfhhoteCannot » Fri Apr 26, 2013 4:55 pm

frgf wrote:I'm not going to uwisc but I'm happy to discuss those problems. What does the 'R' after Q7,8,9 denote?


I have no clue.

frgf
Posts: 24
Joined: Wed Jan 30, 2013 2:24 am

Re: prelim/quals exam questions thread

Postby frgf » Wed May 01, 2013 7:52 am

Does anyone know how to do q3 of uwisc analysis January 2012? The question basically asks for the existence of a matrix logarithm of B, where B is any real invertible matrix.


My solution is mostly algebraic, so I was wondering if anyone had another, more analytic, solution?

Here's my solution.

1. If B is diagonal with no zeros on the diagonal, then let log B be the matrx obtained by taking logarithms of the diagonal elements of B. Then we are done.
2. if B is invertible and diagonalizable, then the matrix logarithm clearly exists: if B=ADA^{-1} where D is diagonal, then let log B = A log D A^{-1}, which is well defined by 1.
3. If B=1+N for some nilpotent matrix, then set log B = N-N^2/2+N^3/3-N^4/4+...... which is convergent because it is a finite sum.
4. For an arbitrary invertible matrix B, the jordan decomposition allows us to write B=C(1+N) where C is diagonalizable, and N is nilpotent, and CN=NC. Thus log B= log C + log (1+N), which is well defined by 2. and 3.

Rikimaru
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Re: prelim/quals exam questions thread

Postby Rikimaru » Fri May 03, 2013 1:15 pm

Just do your steps 1 and 2, then use the fact that the map A ->e^A is continuous in the space of matrices and the density of diagonalizable matrices.

frgf
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Joined: Wed Jan 30, 2013 2:24 am

Re: prelim/quals exam questions thread

Postby frgf » Fri May 03, 2013 8:37 pm

I thought it might be something like that , but when I went to write it down properly I couldn't get it to work.

Since the diagonal matrices are dense, we have a sequence B_n \to B such that each B_n is diagonalizable. We can take the log of each B_n by step 2. But there's no guarantee that log B_n has a convergent subsequence. So we can't deduce that log B exists. Can we?

Rikimaru
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Re: prelim/quals exam questions thread

Postby Rikimaru » Thu May 09, 2013 2:16 pm

Well...

Assuming your matrices are m x m, log B_n is composed of m sequences of complex numbers of the form log a_n (m). Where a_n (m) -->0 lambda_m as n goes to infinity where lamba_m is the eigenvalue of B. Since B and B_n is invertible, the eigenvalues can't be 0, so that's not a problem. For negative values, we obtain imaginary eigenvalues, so that's okay too. For positive numbers, this is a non-issue.




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