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Some maths links

  1. May 24, 2005 #1

    matt grime

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    A gratuitous bit of self publicity, but at

    www.maths.bris.ac.uk/~maxmg/maths

    are some bits of maths people may find interesting.

    There are, if you look around, articles on representation theory, some introductory notes to groups, exercises of many kinds, some research notes if you look hard enough (try www.maths.bris.ac.uk/~maxmg/research.html), links to all of Grothendieck's published work.

    If there's anything anyone wants to add then please let me know. I'd be keen to collect lots of (hard!) exercises for people to use for whatever purpose they see fit - look at the pdf of exercises to see what kind of level of difficulty we're talking about.

    I'd also be keen to have anyone's contributions of a similar nature. Want to explain 3 different proofs of the fundamental theorem of algebra, feel free to write one. The aim is not encyclopedic but interesting. I may for instance write something about why Dynkin Diagrams are so pervasive in algebra, something understandable to someone who knows what a graph (vertex/edge sort) is and what a positive definite inner product is.
     
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  3. May 24, 2005 #2

    quasar987

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    How do we download? If I go to '"teaching", and in the 'download' section at the bottom, the files listed there are not "links".
     
  4. May 25, 2005 #3

    honestrosewater

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  5. May 25, 2005 #4

    matt grime

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    Will fix the problems.

    Edit: done. Any more? Working on a longer set of group theory notes without any assumptions.
     
    Last edited: May 25, 2005
  6. May 25, 2005 #5

    honestrosewater

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    From http://www.maths.bris.ac.uk/~maxmg/maths/introductory/frontpage.html I couldn't find a link back to the homepage. That would be nice- maybe I missed it.

    I read everything that I could find and understand (proofs, philosophical section, set theory, division by zero). I love the conversational style, and everything was great. I don't really have anything to add, though I would like to know if and when you think it is helpful to take a formal approach to proofs (vs. intuitive), i.e. does formality usually just get in the way? For instance, translating a problem to symbolic logic and negating or such has helped me a few times (e.g. with even n2 implies even n or here), but I don't know if it's worthwhile for more difficult problems.

    I'm looking forward to the new group theory notes- I tried to read them a while ago but didn't know enough to follow.
    Thanks for sharing :)
     
  7. May 25, 2005 #6

    matt grime

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    Ahh, there are no links "backwards" from that page - it was a self contained set of pages that i shoved somewhere where it wasn't intended to go.


    As for proofs and so on.

    When writing a proof it is important to be formal since it has to be universally understood.

    The down side to this is, in my opinion, that some of the things we understand well (and there are really very few of those) have become so formalized that they appear hard to follow to the new student, when really they aren't, and we can explain them to people in a way that makes more sense. If you like, we've forgotten where the proof came from. And these conversations came from attempting to demystify some things for my students.

    I think as long as you understand what you're doing when you translate between formal logic and the problem in hand you should think in whatever way helps you. I mean, for example, here at Bristol we (wrongly in my opinion) write things for the students in the :

    [tex] (\forall x)(\exists y)(P(x) \implies Q(y))[/tex]

    which is all well and universal I suppose but really does scare some and hides the wood behind a lot of trees, and doesn't acutally help you to ever prove P implies Q. That is, we should distinguish between the manipulation of strings of symbols and showing things satisfy those strings. Or, in yet another rendering, being able to show that if you have to demonstrate X, then doing Y is sufficient, but shouldn't be confused with actually showing Y.


    I think the most productive conversation I had with them was this:

    how do we show that 1/n^2 tends to zero?

    well, we know abstractly that we need to show

    [tex](\forall \epsilon > 0)(\exists N \in \mathbb{N})( \forall m \in \mathbb{N})(m>N \implies 1/m^2 <\epsilon)[/tex]

    which they didnt' understand, not unreasonably.

    However making them figure it out when I let e=1/100, or 1/64 they figured out that it was actually straight forward and all they had to do was let N be anything larger than 1/sqrt(e).
     
    Last edited: May 25, 2005
  8. May 27, 2005 #7

    honestrosewater

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    Thanks, that makes sense. I find that each informs the other, somewhat like the relationship between theory and experiment in the physical sciences. But it is easier for me to learn one at a time, and I've only seen one person mix them successfully.
     
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