Re: What to study?
- From: quasi <quasi@xxxxxxxx>
- Date: Sun, 17 Jul 2005 17:05:07 -0700
On Sat, 16 Jul 2005 09:47:11 -0500, Adam Born <nrvous6@xxxxxxxxxxxx>
wrote:
>I was under the impression that topology and geometry were similar in natural, and may even go hand-in-hand in some cases. This is just another example of my lack of knowledge.
>
>What should I study in regards to geometry? Do you have any books that you would recommend? Anything from the Dover Publishing series (cheap and easy to get) would be much appreciated, but I am open to any texts which are important, key-to-the-field, etc.
>
>Jim Spriggs wrote:
>> [OT] Should I be surprised that there is no geometry here? Given the
>> significance of hyperbolic geometry one would think that a general
>> course would be recommended with subsequent specializations being
>> optional.
>>
Adam,
Not to slight geometry, but what you want to do is maximize the gain
for the time invested.
Keep it simple -- don't try to do too many branches of math
simultaneously in your self study preparation. In my opinion, it's not
wide knowledge of different areas of math that you need at this point.
What you need is the foundations -- pay your dues!
>From what I've observed, the attempt to succeed as a math major when
only having had the standard calculus courses is typically blocked by
lack of proficiency with the foundations (sets, logic, proofs). Most
attempts fail because of this, so you should take that threat
seriously.
How to defend? I recommend a 3-pronged approach, focusing on
these 3 categories of skills:
-----------------------------------------------------------------------------------------------------------
(1) Language (definitions, terminology, notation)
-----------------------------------------------------------------------------------------------------------
Learn the language of abstract math and test yourself on the
definitions, terminology, notation.
-----------------------------------------------------------------------------------------------------------
(2) Proofs
-----------------------------------------------------------------------------------------------------------
Memorize selected proofs and test yourself that you can replay them.
Organize the proof in your head in outline form (in words), so that a
simple outline of the key ideas drives the rest of the proof. If you
conceptualize the right outline, the rest of the proof then "writes
itself".
Also, attempt selected exercises where the proofs are short and sweet
-- "baby proofs" (don't worry, baby proofs, if nurtured, grow up to be
big proofs). Do these baby proofs in your head (discovery mode), but
then actually write them down (presentation mode), and perhaps show
them to someone whose judgment you trust to get an idea whether (1)
the proof is correct and (2) the style is clear and appealing (to a
prospective reader). Always have an (imaginary) intended audience in
mind in presentation mode.
-----------------------------------------------------------------------------------------------------------
(3) Problem Solving (challenge problems)
-----------------------------------------------------------------------------------------------------------
Also play with challenge problems at various levels, including the
levels below calculus, since the solutions often involve great
problem-solving tricks and strategies which can then become part of
your arsenal of potential strategies for later problems. My preference
is for problems where the statement is simple but the solution
strategy, initially elusive, involves a creative, inspired idea.
Don't neglect problem solving. Many math majors complain that they can
reproduce proofs and do simple proofs if the statement to be proved is
close enough to proofs already studied, but have no idea how to
discover a proof for a statement that is not close enough.
My view is that by doing challenge problems, you sharpen your creative
skills, so that while you are exploring a problem, before you know
what the proof will look like, you already have some ideas as to how
it might go, and you can make a judgment as to which strategy should
be attempted first,
-----------------------------------------------------------------------------------------------------------
Ok, so those are the 3 aspects to focus on, in my opinion.
But back to my original point, don't spread yourself too thin by
trying to self-study too many things at once. Take it from me -- you
will accomplish very little and then rationalize the lack of progress
by claiming you got too busy with other things.
--- quasi
.
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