Posted by
Mikhail Gorelkin on
URL: http://friam.383.s1.nabble.com/Fw-Getting-Math-Chops-Back-Up-tp520645.html
P.P.S. Years ago I read about toposes as varying topologies on sets! so it
would be natural to define varying arithmetic(s) for better modeling some
"singularities".
----- Original Message -----
From: "Mikhail Gorelkin" <
[hidden email]>
To: "The Friday Morning Applied Complexity Coffee Group" <Friam at redfish.com>
Sent: Sunday, October 09, 2005 9:45 PM
Subject: Re: [FRIAM] Getting Math Chops Back Up
> P.S. It seems Godel doesn't prohibit it :-)
>
> ----- Original Message -----
> From: "Mikhail Gorelkin" <gorelkin at hotmail.com>
> To: "The Friday Morning Applied Complexity Coffee Group"
<Friam at redfish.com>
> Sent: Sunday, October 09, 2005 9:37 PM
> Subject: Re: [FRIAM] Getting Math Chops Back Up
>
>
> > > "Oh what, you mean like 2+2=5 here?"
> >
> > Beauty of math is sometimes 2 + 2 = 5 (or something else like in real
> life!
> > Maybe even "everything else" like singularities in complex analysis) -
> > Lobachevsky, Puankare / Lyapunov (bifurcation theory), Lebegue (integral
> > Lebegue), and many others. Otherwise it would be boring :-)
> >
> >
> >
> > - Mikhail
> >
> > ----- Original Message -----
> > From: "Cheryl Fillekes" <cfillekes at mail.utexas.edu>
> > To: <Friam at redfish.com>
> > Sent: Sunday, October 09, 2005 12:43 PM
> > Subject: Re: [FRIAM] Getting Math Chops Back Up
> >
> >
> > >
> > > Interesting story --
> > >
> > > For some of the mechanics of working problems in grad school, I
> > > still would often go back to my original notes from Richard
> > > Rand's classes at Cornell. He is now known as an absolutely
> > > legendary educator now, but back then he was just another professor
> > > -- whose lectures just happened to be gripping, fascinating...
> > > almost unforgettable. I eventually got rid of my record collection,
> > > but I sure hung on to those course notes, even after ditching the
> > > course textbooks.
> > >
> > > When I started graduate school at Chicago in Geophysics, I'd
> > > come out of a full 3 years of engineering math from Rand,
> > > including a graduate level courses that were weeding-out
> > > courses for physics grad students (I got a B) as an engineering
> > > undergraduate -- so in my first year at Chicago, I decided to
> > > sign up for what looked like the equivalent graduate level
> > > courses in differential equations...for review.
> > >
> > > My grad advisor in our "get to know your faculty advisor and
> > > choose your courses" meeting thought this was overly ambitious
> > > and suggested I take "at least a course in *calculus.*"
> > >
> > > I pointed out that I'd done that in uh, high school. He countered
> > > (definitively revealing that he'd not even read my transcript)
> > > "well that was a long time ago." When I pointed out that I'd
> > > taken half a dozen engineering mathematics and physics courses
> > > that required calculus as a prerequisite, and used calculus
> > > almost continually, he stiffened and answered that "Mathematics is
> > > Different Here at Chicago." (!!!) So I was like, "Oh what, you
> > > mean like 2+2=5 here?"
> > >
> > > He suggested a compromise whereby I'd sign up for sophomore-level
> > > complex analysis first quarter and ODE's second. I figured this
> > > one wasn't worth fighting, and besides I could use the easy "A"
> > > if I didn't get too bored in the mean time.
> > >
> > > In practice, I was pleasantly surprised. Whereas my
> > > engineering math courses had focussed primarily on technique,
> > > the mechanics of solving specific problems, and I could do
> > > Schwartz-Christoffel Transforms in my sleep already --these
> > > courses at Chicago focussed almost exclusively on proving a
> > > variety of properties of functions in the complex plane, i.e.
> > > analytic functions vs piecewise continuous functions, contour
> > > integration and so forth. In other words, it was complex *analysis*
> > > based on Ahlfors' text, not Complex Functions based on, say,
> > > Church. What had previously seemed to be a chore with some
> > > incomprehensible beauty behind it, was now was something truly
> > > beautiful I was getting the tools to actually take apart and
> > > put back together, lectures from people with some real insight
> > > and understanding.
> > >
> > > ODEs and PDEs were even better in that regard, the ODEs course
> > > being based on Birkhoff and Gian-Carlo Rota's text, which is
> > > so beautifully written, it reads more like an exciting novel
> > > in places, *particularly* the proofs. I'd been through Green's
> > > functions at least 3 times in different courses, for example,
> > > and again, could blow through the problem sets -- but it was
> > > just symbol manipulation. It never even occured to me to even
> > > ask *why* Green's functions gave you the particular solution.
> > > It was just the technique you applied when you had a forcing
> > > function, and it worked.
> > >
> > > So one night, I'm studying for the midterm, and get sidetracked
> > > reading Gian-Carlo's one-page proof on Green's functions. He
> > > actually drew me in to the story, when I "should have been
> > > studying" in the only way I knew how back then: working problems
> > > (in this case correcting some of the mistakes in Birkhoff and Rota).
> > > I thought for sure I was going to blow the exam, but this proof was
> > > cool and so interesting and so clearly written -- that I was
> > > able to reproduce the proof on the exam the next day...and I
> > > was the only one in the class able to do that. So what I thought
> > > was "being sidetracked" -- actually taking an interest in the
> > > material for its own sake rather than chugging through that
> > > odious chore called math homework -- turned out to be a more
> > > effective study technique as well as a whole lot more fun.
> > >
> > > I had amost the same experience with a proof of the uniqueness
> > > and completeness of Fourier Series in PDEs, which Chicago taught
> > > from Weinberger's text.
> > >
> > > That first year at Chicago, math went from being a Beautiful BFJ
> > > to something even more beautiful and engaging -- like great art.
> > >
> > > When I was working through Guckenheimer and Holmes on my own
> > > (there wasn't a course at Chicago that used it) I used Hirsch and
> > > Smale as my ODEs reference rather than Birkhoff and Rota, because
> > > Hirsch and Smale uses the same notation and way of expressing
> > > things (Guckenheimer was Smale's student, after all).
> > >
> > > The nice thing about these classics is that you can go back to them
and
> > > re-read them like a good novel. They're incredibly enjoyable as well
> > > as merely useful.
> > >
> > > Some of these are really expensive these days, but I think most of
> > > them are on the bookshelf at SFI:
> > >
> > > Complex Analysis:
> > > Ahlfors
> > >
> >
>
http://www.amazon.com/exec/obidos/tg/detail/-/0070006571/qid=1128869678/sr=8> > -1/ref=pd_bbs_1/102-6245318-2684139?v=glance&s=books&n=507846
> > >
> > > ODEs:
> > > Birkhoff and Rota
> > >
> >
>
http://www.amazon.com/exec/obidos/tg/detail/-/0471860034/qid=1128869773/sr=1> > -1/ref=sr_1_1/102-6245318-2684139?v=glance&s=books
> > >
> > > Hirsch and Smale
> > >
> >
>
http://www.amazon.com/exec/obidos/tg/detail/-/0123495504/qid=1128871551/sr=1> > -3/ref=sr_1_3/102-6245318-2684139?v=glance&s=books
> > >
> > > PDEs:
> > > Weinberger
> > >
> >
>
http://www.amazon.com/exec/obidos/tg/detail/-/048668640X/qid=1128869893/sr=1> > -1/ref=sr_1_1/102-6245318-2684139?v=glance&s=books
> > >
> > > Nonlinear Dynamics:
> > > Guckenheimer and Holmes
> > >
> >
>
http://www.amazon.com/exec/obidos/tg/detail/-/0387908196/qid=1128869955/sr=2> > -1/ref=pd_bbs_b_2_1/102-6245318-2684139?v=glance&s=books
> > >
> > > Note that none of these really drag you into Courant and Hilbert
> > territory.
> > >
> > > Cheryl
> > >
> > >
> > > ============================================================
> > > FRIAM Applied Complexity Group listserv
> > > Meets Fridays 9:30a-11:30 at ad hoc locations
> > > Lecture schedule, archives, unsubscribe, etc.:
> > >
http://www.friam.org> > >
> >
> > ============================================================
> > FRIAM Applied Complexity Group listserv
> > Meets Fridays 9:30a-11:30 at ad hoc locations
> > Lecture schedule, archives, unsubscribe, etc.:
> >
http://www.friam.org> >
>