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Programs PhD in Math

  1. Apr 8, 2006 #1


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    I'm finishing my third year in the math program and the University of Toronto. After my fourth year, I'd like to do a PhD in mathematics. This is a rather open-ended question, but what are the best grad schools for this?
  2. jcsd
  3. Apr 8, 2006 #2


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    I'm sure people will ask you a few follow-up questions such as "Where are you willing to go?" and "Is there a specific field you are interested in?" etc etc.
  4. Apr 8, 2006 #3


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    Pure Mathematics? Applied?

    We need details.
  5. Apr 9, 2006 #4


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    Pure Mathematics. I'm willing to go to the States, stay in Canada, or possibly go to Europe - but I only know English. I don't have any specific field of interest, but of the courses I've taken so far, I've liked linear algebra, abstract algebra, real analysis, topology, mathematical logic, combinatorics, and number theory. On the other hand, differential equations, probability and statistics, and differential geometry weren't as interesting for me.
  6. Apr 9, 2006 #5
    What is your dissertation on?
    (On what mathematical topic?)
  7. Apr 9, 2006 #6
    A dissertation is usually something written in, not before, grad school...
  8. Apr 9, 2006 #7

    matt grime

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    Chicago, Northwestern, Harvard, Berkeley, UPenn, MIT are 'the best' in the US, with so many very good places just behind it isn't even worth beginning to list them. These places I pick out because they are large departments with varied interests where you will be able to pick up fantastic experience from many people. The ones I (and I stress it is my opinion) place just behind like Yale, UCR, UGA, have slightly smaller courses and won't offer quite as much to choose from (from what I can tell), though they have some great (truly great) people there. Or at least this was my experience of it: you need to have a more focussed idea of what you wanted to do when you arrived at these places.

    (nb I've chosen to omit Princeton from a classification since I don't konw much about the place, or more importantly I don't know what the intearction between the Department and IAS is like.)

    The UK is much much different. As it stands there is little if anything in the way of graduate lectures in the UK; you just get on with your work. That is changing to reflect the knock on effect in changes in degrees, but right now you need to look carefully at your options here (not least for the funding). Cambridge is about the only place with the facility for graduate lectures (when you are able to participate in part III courses). Other places are trying to catch up, but it is a difficult thing to do because British mathematics departments are very small places. A typical department (ie not one of Cambridge, Oxford, Warwick, Imperial Durham, or Bristol) might have only a dozen PhD students if that, so don't be sucked in by one of the British univeristy's webpage which states them to be 'the best place for post-graduate study according to a recent survey' (of postgraduates).

    (As a sample, Sheffield has 18 pure maths PhD students, exeter has 13 [in all areas of maths] any single London College might well be small, but they are all part of a larger organization and often have intercollegiate collaboration)
  9. Apr 9, 2006 #8


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    Thanks matt. What is IAS?
  10. Apr 9, 2006 #9


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    Also, what kind of questions should I be asking, what kind of things should I be looking out for when making my decision? I know this is a very vague and demanding question, but even small bits of guidance will help. Thanks.
  11. Apr 9, 2006 #10
    I believe IAS stands for the Institue for Advanced Study, at Princeton.
  12. Apr 10, 2006 #11
    I'm currently in the same boat, but I'm at Berkeley. I'm currently going along the lines of functional analysis (and that statement is vague because I don't really know specifically), but that could change. I'm definitely going pure though. I'm open for US, Canada and pretty much anything in Europe (I can already speak Swedish, German and Spanish) and I'm open to learn a language, if necessary (since that's one of my other interests anyway). Anyone have any insight into any other schools?
  13. Apr 10, 2006 #12

    matt grime

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    If you can stay, then Berkeley is a brilliant place to work (I was there a few weeks ago at MSRI: sure the campus maths building is crappy but the people....).

    If you like functional analysis and can stomach somewhere (small than) the size of Berkeley without the back up of the City 6 stops down the BART then Penn State has some great people to work with (Roe, Higson and Baum, for instance).
  14. Apr 16, 2006 #13
    I'm of course considering Berkeley (being an undergraduate here as shown me how good the school is), but it's not easy to be accepted in general and they like to force their undergraduates to change schools to broaden their experiences (which I think makes sense) so they give non-Berkeley students priority. I also kind of feel like I want to change venues anyway, but I'm definitely not ignoring the possibility. I'll take a look at Penn State though. Thanks for the info.
  15. Apr 16, 2006 #14
    Go to Berkeley. Say hi when you drop by too :)
  16. Apr 16, 2006 #15
    Well I'm currently in Germany for this year, but I'll be back at Berkeley next year as a senior. You a student at Berkeley?
  17. Apr 16, 2006 #16
    Yeah :smile: (next year I will be a junior)

    Well I do not want to derail this thread, so to the original threadmaker, I would like to wish good luck in whichever place you decide to go to.
  18. May 9, 2006 #17
    I need help too.

    I am a student in China.I am a freshman.
    I find algebra and foundations of math (mainly set theory and logic for the moment) so fantastic to me.And I learn fast and have a strong willing to do math research after undergraduate study.
    I wish I could go to Berkeley for graduate study and I wish you Berkeley guys could help me.
    What text books are being used for the undergraduate students?What quality do I have to have to be admitted to the graduate school at Berkeley?Is it harder for a non-US student to be admitted?
    Any guidance will be appreciated.
  19. May 9, 2006 #18


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    If you are a successful student at berkeley you can probably succeed most anywhere. In general onbe should ask how many years of support will be available at the school to finish ones degree.

    In the old days places like Princeton expected you to finish in 2-3 years which is very fast for most people. Harvard may have allowed 4. Other places may allow more.

    The key thing is finding the right relationship with an advisor. And these people change. One of my friends studied with Curt McMullen at Berkeley in topology but McMullen is now at Harvard after receiving the Fields medal.

    I studied at Utah and found the ideal advisor for me in Herb Clemens, and Clemens is now in Ohio. In general, with the influx of superb foreign born mathematicians over the last decades, virtually all US math departments seem to be getting better and better faculty.

    The locale is relevant to some extent. A place like Berkeley is exciting but for some much too expensive and hectic. Last time i was there the grad students were striking for higher wages and the place was shut down, and there was a lot of strife.

    Wonderful as the mathematics is, my wife and I dclined to go to Columbia because raising kids in NYC seemed such a challenge. I suspect Michigan is a good choice, and Ann Arbor is a lovely little town.

    But there are lots of options. It is useful to have financial support, helpful faculty, and a student body that supports each other and interacts mathematically.

    And you might be surprized at what leads to success. Some of the strongest faculty our Univ has hired went to Yale, Berkeley, Princeton, MIT, but others have gone to North Carolina, Brandeis, Utah, Queens in Ontario,...

    I suggest looking at the webpages of the deprtments to see who is there and what their interests are.
  20. May 9, 2006 #19


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    another thing to do is try to get an idea of the prelim requirerments, how many prelims you must pass, how long you have to pass them, what they look like.

    some of this information is also available on websites under information dfor students. e.g. at UGA the website has a section called info for grad students, including prelim syllabi.

    here is an old one from UGA in algebra:
    Last edited: May 9, 2006
  21. May 9, 2006 #20


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    Study Guide for Algebra Exams

    Group Theory:
    quotient groups
    Lagrange's Theorem
    fundamental homomorphism theorems

    group actions with applications to the structure of groups such as the Sylow Theorems
    group constructions
    such as:
    [free groups
    generators and relations]
    direct [and\0 \0s\0e\0m\0i\0-\0d\0i\0r\0e\0c\0t\0] \0p\0r\0o\0d\0u\0c\0t\0s\0
    \0s\0t\0r\0u\0c\0t\0u\0r\0e\0s\0 \0o\0f\0 \0s\0p\0e\0c\0i\0a\0l\0 \0t\0y\0p\0e\0s\0 \0o\0f\0 \0g\0r\0o\0u\0p\0s\0
    \0 \0s\0u\0c\0h\0 \0a\0s\0:\0
    \0[s\0o\0l\0v\0a\0b\0l\0e\0 \0g\0r\0\0\0o\0\0\0u\0\0\0p\0\0\0s\0]\0\0\0
    \0\0\0d\0\0\0i\0\0\0h\0\0\0e\0\0\0d\0\0\0r\0\0\0a\0\0\0l\0\0\0,\0\0\0 \0\0\0s\0\0\0y\0\0\0m\0\0\0m\0\0\0e\0\0\0t\0\0\0r\0\0\0i\0\0\0c\0\0\0 \0\0\0a\0\0\0n\0\0\0d\0\0\0 \0\0\0a\0\0\0l\0\0\0t\0\0\0e\0\0\0r\0\0\0n\0\0\0a\0\0\0t\0\0\0i\0\0\0n\0\0\0g\0\0\0 \0\0\0g\0\0\0r\0\0\0o\0\0\0u\0\0\0p\0\0\0s\0\0,\0 \0c\0y\0c\0l\0e\0 \0d\0e\0c\0o\0m\0p\0o\0s\0i\0t\0i\0o\0n\0s\0\0\0
    \0\0\0T\0\0\0h\0\0\0e\0\0\0 \0\0\0s\0\0\0i\0\0\0m\0\0\0p\0\0\0l\0\0\0i\0\0\0c\0\0\0i\0\0\0t\0\0\0y\0\0\0 \0\0\0o\0\0\0f\0\0\0 \0\0\0 \0\0\0A\0\0\0n\0\0\0,\0\0\0 \0\0\0f\0\0\0o\0\0\0r\0\0\0 \0\0\0n\0\0\0 \0\0\0> \0\0\04\0\0\0\0

    \0L\0i\0n\0e\0a\0r\0 \0A\0l\0g\0e\0b\0r\0a\0:\0
    determinants, \0
    eigenvalues and eigenvectors
    Cayley-Hamilton Theorem
    canonical forms for matrices
    l\0i\0n\0e\0a\0r\0 \0g\0r\0o\0u\0p\0s\0 \0(\0G\0L\0n\0 \0,\0 \0S\0L\0n\0,\0 \0O\0n\0,\0 \0U\0n\0\0)\0
    dual spaces: definition, dual bases, pull back, double duals.
    finite-dimensional spectral theorem



    Zorn's Lemma and its uses in various existence theorems such as that of a basis for a vector space [or the algebraic closure of a field], and existence of maximal ideals.

    T\0h\0e\0o\0r\0y\0 \0o\0f\0 \0R\0i\0n\0g\0s\0 \0a\0n\0d\0 \0M\0odules
    basic properties of ideals and quotient rings
    fundamental homomorphism theorems for rings and modules
    characterizations and properties of special domains
    such as:

    classification of finitely generated modules over Eucl dom
    applications to the structure of
    finitely generated abelian groups and
    canonical forms of matrices
    [Noetherian rings and modules]
    [tensor products of vector spaces]

    Field Theory:

    algebraic [and transcendental] extensions of fields
    fundamental theorem of Galois theory
    properties of finite fields
    separable [and inseparable] extensions
    computations of Galois groups of polynomials
    of small degree and cyclotomic polynomials
    [elementary symmetric functions]
    [solvability of polynomials by radicals]


    [1] Thomas W. Hungerford, Algebra, Springer, New York, 1974.
    [2] Kenneth Hoffman and Ray Kunze, Linear Algebra, Prentice-Hall, 1961.
    [3] Nathan Jacobson, Basic Algebra 1, W.H. Freeman, San Francisco, 1974.
    [4] Nathan Jacobson, Basic Algebra 2, W. H. Freeman, San Francisco, 1980.
    [5] Serge Lang, Algebra, Addison Wesley, Reading Mass., 1970
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