Fw: Getting Math Chops Back Up
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Fw: Getting Math Chops Back Up
 
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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 > > > 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 > > > > > > > > -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 > > > |
Fw: Getting Math Chops Back Up
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Hi gang --
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