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390 Quick Answers 14 April

Papers still pending.  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 Bailey 104, 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?



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.  We will see more from him later, also.   

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. 

Why is Galois' story not more well known?  It is relatively recent work in history (1996), and fights against long-standing traditions for how the story was told.  Again, largely growing out of the influential if unreliable _Men of Mathematics_ book by E.T. Bell. 




Reading Reactions

Italy and Germany are much younger countries than you might expect. 

Aside from other things named for him, ABELian groups are also named for Abel.  For those who haven't reached 330 yet, group theory is the main topic of 330.  For those who have - I hope you knew that.  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.  And, please don't lose track of the fact that Abel was Norwegian.  That is part of his story.  Galois knew of Abel and was scathing in his letters submitted with his papers "to those who have Abel's death on their conscience". 

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.  We have definitely transitioned to the time where this was where research was done.  This probably was true from the 1600s or so.

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 PlĂĽcker. 

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.  Linear Algebra has had a somewhat peculiar history - something like it was used before it was invented.  Because of that, it quietly slips by Suzuki for the most part. 

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, .