prelim/quals exam questions thread
prelim/quals exam questions thread
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.
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.

 Posts: 54
 Joined: Sat Mar 24, 2012 5:01 pm
Re: prelim study group
I would love to join you.
Re: prelim study group
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?
Let's do all the discussion and posting of problems in this thread... what do u guys say?
Re: prelim study group
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).Let's do all the discussion and posting of problems in this thread
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 q110. 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.

 Posts: 157
 Joined: Sun Oct 14, 2012 12:15 pm
Re: prelim exam questions thread
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
http://www.math.osu.edu/graduate/current/qual
Re: prelim exam questions thread
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 oncampus housing facilities will start on August 21).
Any of you in this kind of situation? Any suggestions would be welcome...
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 oncampus housing facilities will start on August 21).
Any of you in this kind of situation? Any suggestions would be welcome...

 Posts: 90
 Joined: Sat Oct 13, 2012 11:12 am
Re: prelim exam questions thread
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?
Re: prelim exam questions thread
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. Can't wait until the year 2 quals  I am already semiprepared for it from my extensive research experience.
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. Can't wait until the year 2 quals  I am already semiprepared for it from my extensive research experience.
Re: prelim exam questions thread
I would love to join the preparation group too, any thoughts on the organization?

 Posts: 90
 Joined: Sat Oct 13, 2012 11:12 am
Re: prelim exam questions thread
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?
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?
Re: prelim exam questions thread
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: 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.

 Posts: 90
 Joined: Sat Oct 13, 2012 11:12 am
Re: prelim exam questions thread
Hi Manyfolds, thanks for the response! Btw, have you gotten off the waitlist at Brown?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.
Re: prelim exam questions thread
Yeah, I had provided an old email. Am strongly considering it also.
Re: prelim exam questions thread
Thanks for the link!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
I've taken a few, but still have lots of gaps. I'm hoping to learn as I go.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 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.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.
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.
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.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?
Re: prelim exam questions thread
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.
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.
Re: prelim/quals exam questions thread
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 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

 Posts: 157
 Joined: Sun Oct 14, 2012 12:15 pm
Re: prelim/quals exam questions thread
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 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
Re: prelim/quals exam questions thread
I see. That makes sense. Thanks.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.

 Posts: 31
 Joined: Fri Jan 25, 2013 9:11 pm
Re: prelim/quals exam questions thread
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 ... 01725.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.
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 ... 01725.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.
Re: prelim/quals exam questions thread
I'm not going to uwisc but I'm happy to discuss those problems. What does the 'R' after Q7,8,9 denote?

 Posts: 31
 Joined: Fri Jan 25, 2013 9:11 pm
Re: prelim/quals exam questions thread
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.
Re: prelim/quals exam questions thread
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 = NN^2/2+N^3/3N^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.
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 = NN^2/2+N^3/3N^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.
Re: prelim/quals exam questions thread
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.
Re: prelim/quals exam questions thread
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?
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?
Re: prelim/quals exam questions thread
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 nonissue.
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 nonissue.