390 Quick Answers 19 April

Thank you for your patience for today.  I will be back on Monday and tell you about it.  You will read some about the MAA for Monday.  It's a nice convergence.  I will try to do an extra thorough job with quick answers for Monday if you feel there is anything that had a poor treatment in the video (or here). 

I have read 3/4 of the papers.  I have completed all those submitted before the afternoon they were due.  It may be another week for those remaining.  As you might imagine traveling is busy.  I did also update the actual current average for the drafts I have now read.  That reflects both the recent reactions and the draft.  If you have draft comments back you have a new actual current average.  Please read your paper feedback.  As before, what if you need new sources. 

GREAT Day is on Wednesday.  Read the program.  Make GREAT Day plans.  Our talks are in session 3N, but only those presenting have obligations to us.  Learn something interesting - doesn't need to be mathematical (or historical).  Tell us about it on Friday.  I will say the same on Monday.

Someone bring some nice final topics for Monday.  More than one someone would be nice.  Someone asked about this - starting with really chapter 5, culture and region is something that loosely corresponds to modern countries.  So, Germany and France are different.  You really never get to say "they were from Europe" and have that be enough. 

Lecture Reactions

Why did Bourbaki want to be anonymous?  Because they wanted their work to be impersonal, to transcend people. 

According to a reference that I will discuss on Monday, Gauß had 15 PhD-level students. 

I feel bad for misstating Riemann’s statement about conformal mapping.  Differentiable injections are conformal, not all differentiable, which makes better sense to me as z^2 is surely not conformal.  

In the Riemann sphere there is only one point at infinity.  In Calc I, 1/0 = ± infinity, but because there is only one, this is much simplified.  

Remember that almost always the question in mathematics is about _all_.  A few billion examples is not very close to all.  

Complex analysis is … quite different from real analysis.  It is more visual, it is more intuitive, and the rules are simpler and nicer.  (Coming from someone who has taught both 371 and 324, and completed graduate quals in complex analysis.)  I do personally strongly believe that real analysis is a mistake.  It’s like studying a narrow strip instead of a coherent intricate picture.  

I think, despite so many contributions, that Riemann worked almost exclusively on the analytic side of mathematics.  I don’t know of contributions on the side of algebra.  Mathematics is getting big.  

The basic FTC you learn in Calc I.  The generalised FTC in arbitrarily many dimensions is the centrepiece of Vector Analysis.  Green's theorem is a special case of the generalised FTC.  So are the divergence theorem, and Stokes' theorem. 

I will talk about the history of linear algebra next time as a bonus.  There are a few extra stories like this that I enjoy sticking in when we have time as we near the end.  



Reading Reactions

As I will say in the video, I do not believe Jefferson's comments to Condorcet about Banneker reveal his true thoughts.  The point is that you are right to be suspicious. 

If I do it wrong in the video, "Adrain" is pronounced like what you do to solve a drought "add rain".

It’s hard to do mathematics when you’re more concerned with survival.  

Before the revolution education was mostly handled overseas, but one way to be independent is to be self-educating.  

Before the civil war it was common to say “the united states are”.  We don’t think about it much, but “united states” really meant that this was a collection of independent countries like the EU.  This is why we live with such a strange system of government now where senators have so much power, despite us not thinking of this place as 50 countries.  But, yes, the colonies all had their own currency.  

I hope you studied zero-divisors in algebra (330).  When working mod 6, 2*3=0, so they are zero divisors. 

A nilpotent (zero-power) is something that when raised to some power yields zero.  An idempotent (identity-power) is something that when raised to some power yields itself.  There are nice examples with matrices.