Giving a mathematical talk

Dear NG,

I need a little guidance here. I have given 3 talks on seminars since
my graduation, and all three were awful in my opinion. It is very
unfortunate that I give little talks in seminar (although I attend into
a lot of them, I hardly give talks in the seminars I attend to). And
unfortunately, my university is not the ideal place where I can
"simulate" talks with other students (currently Im the only pure math
student here!).

I therefore need some help on how I can improve my talks. I noticed
that most of the time, I was not ready for the questions that pop up
from the audiences, if the audience has no idea of what Im talking then
I have no such problem but I wouldnt learn anything on improving myself
if the audiences were all clueless. There are things I say/write, and I
would not like to give the proofs, for various reasons (eg. lack of
time, not very sure of how the proof goes, etc.) and these usually
leads into a mess (sometimes they may be false statements, sometimes
they are vague ..etc.). So people get the idea that I do bad math, and
that leaves a bad aftertaste and reputation not to mention that it
could be extremely embarassing (of course, I might have done some bad
math but those are things that have to be improved on as well!).

My last talk especially was disastrous. I realize, I just pushed aside
things or questions that were arouse instead of trying to contemplate a
little and answer them if they an answer could be found. I dont tend to
"think" when giving talks and just go on with what I planned, and that
is fatal. A few people tried to come up with counterexample of some of
my statements (some of them were good counterexample, some of them were
wrong but I didnt tried to explain why they were wrong and I began
having doubt with what I did... but only after the talk I realized
these things). I thought maybe, I should be my own archrival and always
try to counter statements and theorems I make during the talk and try
to make 100% sure of everything before presenting it to the public. I
only learned that giving a mathematical talk to the public is very
different than writing a mathematical paper.

Many here probably have enough experience on giving me some tips on
this, I can at most try to give a dry run of my talk to "myself", there
will be no other person who would be hearing my talks (my advisor
perhaps, but he is not a person who will have free time to hear some
"dry run" talk from a third rate postgraduate :) ). I would appreciate
any tips and I apologize for my lack of experience.

Sincerely,
Jose Capco

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