Reminder:  I will not be available in any fashion tomorrow after 4p.  

Well, perhaps you see that I made good progress on the papers.  Those who handed in on time should already have email feedback on your papers (tell me if you don't).  Here's an idea to think of with regards to revision:  if you do not make the suggested revisions (taken seriously) your grade will be a grade lower on the final than it was on the original.  In particular this is most true for those who earned a C on the draft.  In that case if you do not make significant improvements, you will earn a D on your final paper.  Sometime I will get to the 10% off papers, and sometime later I will get to the 20% off papers.  

I am interested in working with you all as much as possible on projects at this point, but there are some limitations as to how much is actually possible.  Try to see if I'm available at any time to help, and if I am I will try to help.  I am likely to be more available for papers after GREAT day than I am before.  As I sent to you a message on Monday, the GREAT day program is now on-line.  I strongly encourage you to check it out - to see some of all the excellent things that will be happening that day.  

Not only in our sessions, not only in mathematics (there are as many talks out of our sessions as there are talks in), but broadly.  That all being said, your classmates will appreciate having familiar people with some background to hear their talks - please join us when you can - at least please stop by sometime during the day.  


§9.4

One last look at Europe before we go back in time and across the ocean (with a slightly odd digression on Friday).  

There was l'Ecole Normale in Paris before in Germany (Galois was there when he was in higher ed.).  Yes, this is the origination of places like Geneseo which was formally a normal school - and which is now a more diverse place (we no longer do exclusively teacher training).  

The ways in which Weierstraß and the Berlin seminar set standards for modern mathematics mostly concern what level of precision is required in proving.  

It's my impression that Gauß detested teaching because he found it distracting and couldn't be bothered to explain his ideas to others.   I'm guessing this was different when working with PhD students, who were more like collaborators.  

Riemann taught a course on material that wasn't understood, so it wasn't understood.  Most could not appreciate its value.  

Riemann did not invent the ancient greek letter zeta.  He studied the series and first proved that the related series S 1/n^p, where p runs through primes only, converges.  Thus the original connection between the zeta-function and primes.  

Riemann was the first to precisely formulate Riemann sums the way he did.  There were others who did similar things and the definite integral goes back to the beginning of calculus and before.  

Plücker's homogeneous coordinates were a way to introduce coordinates for lines through the origin.  Roughly the coordinates can be viewed as a nonzero point on the line, but multiplying by a scalar doesn't change the line indicated.  They are surely still used for projective geometry.  The transformations given in the text are Plücker's transformations.  More representations of any idea - in this case geometry and transformations - allows for more connections to be noticed and more results to be proven.  (That is an umlaut in Plücker's name.)  

One result that I know of due to Eisenstein is the Eisenstein criteria, a simple test addressing situations in which polynomials can be factored over the rationals (concerning whether a prime divides the coefficients).  

The hyperbolic geometry story is coming - I keep trying to hold myself back from it.  It will appear at the beginning of chapter 11.  

That rationals are dense - there are rationals as close to any real number as we choose, but they are not continuous - there are gaps at all the irrationals.  

Gymnasium still means secondary school in Germany.  I do not know how it came to be used in its other context in US.

Elliptic functions first arose as integrals to find arc length of an ellipse.  Related functions cannot be presented in terms of other simpler functions.  They are important in complex analysis, analytic number theorey, and differential geometry.  

Yes, Felix Klein is for whom the Klein 4-group is named.  He saw it from a geometric perspective.  And for whom the Klein bottle is named.  

Jacobi is responsible for any and all mathematical things named "Jacobian".  

Dreyse rifle, chassepot, mitrailleuse.  

This the first time in history there is a Germany (or Italy for that matter).  The goal in creating Germany was to unite all German-speaking peoples.  While there were a vast number in Austria, there were also a vast number of non-German speaking peoples.  For that political tension reasons, Austria was chosen to be omitted.  Austria then was also much more vast than it is today.  It included most of present Hungary, Czech, Slovakia, Croatia, Slovenia, and some of northern Italy.  Hence while modern Austria is predominantly German speaking because Austria then could not be split, it was not included in the newly formed Germany.  

Concerns with symmetry are quite important in things like quantum mechanics.  In this context, things like Lie groups and Lie algebras are quite close to things like quantum groups and quantum algebras.  Lie groups are groups that also can be viewed as geometric objects (here geometric means differential geometry).  The unit circle viewed as the complex numbers of norm 1 is an example.  Lie's work was not encoded to be secretive, it was merely mathematics, which looks like it is in code to someone who doesn't understand it.  

In my impression, mathematics is more often justified in terms of the extent to which it advances knowledge rather than the extent to which it answers daily life questions.  Ultimately mathematics is not a science - we do not base our truth on experiments and we do not base our value on practical uses.  In many ways it is much closer to humanities and the arts in this regard.  

Polytechnical schools are engineering schools - like RPI which will appear in chapter 10.  

I will say nothing more about Joachimsthal.  

in German "and" is "und", therefore "unapplied" and "und applied" are quite close.  

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Past: group theory is the study of groups, which I have explained.   Remember a diet is a a meting of a governing body.  

Last time:  yes, Abel is the first to prove the unsolvability of the quintic (5th degree polynomial).  (why?  Too many permutations of the roots)

Future (to which we won't directly get):  

Andrew Wiles proved Fermat's Last Theorem, Perelman (sp?) proved the Poincare conjecture.  There are two living mathematicians worth knowing about.  I could tell you many more, but they would also be too numerous to mention.  A good idea here - ask any faculty member about the best known modern mathematics they know.