390 Quick Answers 15 April

I have finished over half of the papers.  You are all always welcome to come discuss with me.  Expect that you have much work to do when you get your feedback.  Do be sure that you have mathematical details.  Focus focus focus.  The more you focus the stronger you paper, the more you take on the more it will be a weak summary.   Oh and don't quote secondary sources.  _Do_ quote historical figures.

More exam topics?  

Now that playoffs are starting this week, perhaps you have heard of the Edmonton Eulers?

GREAT Day is in 9 Days.  Please plan to join us.  I will ask who is participating … oh, and remember that wonderful day we had after Diversity Summit where everyone talked about their experiences?   We will do that for GREAT Day.  Our session is at 4:30 in Welles 131, but there is so much good, and I love hearing about that which I miss. 

Yes, the mathematics is getting sophisticated.  Take what you can, and if something interests you that we talk about, it’s an opportunity for further study.  Remember 19th century is the end of undergraduate mathematics, so it’s serious.  20th century is yet more so.  We’ll regress quite a bit in chapter 10 on Friday.  Part of this growing sophistication leads to more and more sketchy descriptions.  This is _not_ that there is less, but that all of it requires so much to explain.  We do what we can.  Jeff and I both do what we can.  I will try to add some more.  (both more details and more topics). 

Speaking of Friday - does anyone look ahead enough to know what Chapter 10 is about?

Speaking of Chapter 10, Monday we will talk about organisations including the MAA.  On Friday I will be attending the local MAA meeting, the Seaway Section in Fredonia.  As a consequence, I will not be here (for class or office hours), and as a consequence, I will send you a link to a video in place of Friday's lecture.  I will send it before I leave.  I recommend watching it in the afternoon as usual.  The video will not include the quick answers, which will be posted here as usual.  Please process both.

Lecture Reactions


A bit more about Sylvester who gets sadly put in the Galois aftermath:  he is responsible for the words “matrix”, “graph” (the one with edges and vertices), “discriminant”, “nullity”, “canonical form”, “minor” for determinants, and “annihilators”.  Matrix equations, e.g. AX + XB = C, and f(A) for a matrix A. 

If FLT is true for an exponent, it is true for all multiples of that number.  Therefore usually one wants to prove it for primes (Euler proved n=4, since it is _very_ not true for n=2).  Originally n=7 was too hard, so they did n=14 instead.  Yes, Faltings and Wiles are alive, sorry for that not quite being history, but it felt responsible to catch you up on the story.  Lame’s proof relied on unique factorisation of x^n + y^n over the Gaussian integers (i.e. a+bi where a, b are integers).  This only worked for some primes, but not others.  Those for which it worked were then named “regular primes”.  

The sum of finitely many continuous functions is always continuous.  The sum of infinitely many may not converge, but even if it does converge, it may not be continuous (e.g. Fourier series).  The sum of _uniformly_ continuous functions is continuous if it converges. 



Reading Reactions

In the reading there is a reference to normal schools in Germany.  I remind you that Galois attended l’Ecole Normale.  SUNY Geneseo was established by New York State Legislature in 1867, and opened its doors in 1871 as the Geneseo Normal and Training School.  Since you were here for the sesquicentennial, I would hope that you had learned something of our school in this historic time.  Here’s a small bit of what I learned while preparing for our celebrations. 

Aside from other things named for him, ABELian groups are also named for Abel.  Oh, and I want to highlight that Abel thought he had found a way to _solve_ quintic equations until he was asked to find an example.

You may not know that the larger schools in the US are more focused on research than they are on teaching (and if on teaching only on training more researchers).  This is, by the way, why you are here and not there (or you were just lucky).  Berlin was an early example of this.  

Cauchy was in a high profile position to be reviewing papers.  As such he is partially responsible for both Abel and Galois’s tragedies.  
Ok, I need to talk about elliptic functions.  They are the key that connects to FLT out of number theory.  They are functions defined by elliptic integrals, which are similar to integrals used to compute arc-length of ellipses.  The connection to ellipses, in the end, is slight.  Jacobi also studied these.  He also contributed to partial differential equations, determinants, and differential geometry.  

Homogenous coordinates are used for considering the set of all lines through the origin.  [1,1,1] = [2,2,2] because they are on the same line.  This was studied by Plücker. 

Vectors are on their way; they are surprisingly late to the scene.  Matrices have a peculiar history - somehow they were used far before they were recognised.  I will try to tell that story in our closing days.  

I will talk about other ideas from Klein, but yes, he is associated with the Klein 4-group. 

Eisenstein is not Einstein (who we will discuss later).  One thing Eisenstien is known for is a way to identify irreducible polynomials (that cannot be factored over the rationals) … if there is a prime number that divides each of the cofficients after the highest degree, and this prime number squared does not divide the constant term.  E.g. 3x^4 + 15x^2 + 10.
  
Some transformations are continuous - translation and rotation.  Some transformations are not - reflection cannot be slowly done - points cannot slowly reflect over a line, as they can slowly translate or rotate over smaller distances and angles to get to the final results.  Lie was studying continuous transformations.  These are continuous groups, not discrete groups as you are most familiar with. 

Suzuki has a little secret buried at the end of chapter 9.  Failed French General Bourbaki was the namesake of a collective of anonymous mathematicians in roughly 1935-1970 called Nicholas Bourbaki.  It’s a fascinating story, you can read more here.