Objectives for Math 308 /
Analysis and Number Theory
The following is entirely optional, but describes some of my thoughts about teaching. As with all material in the course, I'm happy to discuss anything here in greater detail; I hope this will help start a dialogue about your undergraduate education (as well as my continuing education!). If anyone wants to convert this to a blog for a discussion, either email me direct or anonymously through email@example.com (password 11235813).
When I was in high school, I remember the Boston Globe ran an article where they asked numerous 'famous' people in the state: what 10 books should every high school student read? The answers were, for the most part, disappointing. The English professor had ten literary selections, ranging from some Shakespeare and Milton to stuff I can no longer remember. Most of the others had lists greatly biased towards the small part of the world they studied; very few had balanced lists that would help prepare the general student for the world (Governor Weld was one of the few who did).
I strongly urge you to view the following clip on YouTube: Did You Know (2009 version). Some of the more interesting statistics / quotes from this video:
Another great clip to listen to is the TED lecture of Malcolm Gladwell on spaghetti sauce. The main point of this lecture is
What does all this have to do with our class? The point is that it is hard to predict what will be useful to most of you in your career(s) after Williams (save for the few who go off to grad school to study number theory, where I can do a pretty good job predicting). It is imperative that you learn to be problem solvers. The content of a course matters; you need to learn the language, the basic facts, the key theorems, et cetera. But, at least as important, you need to learn how to use these on 'new' problems.
One of my biggest epiphanies as an educator was when I prepared my lecture notes for Math 209 (differential equations). This was the first class I'd ever taught as a college professor which I'd actually taken as an undergrad. I was looking through my old course notes, trying to decide what to include, when I noticed that we did the Bessel equation and function when I was a student. I was shocked; I had no memory of having done this, but I use the Bessel function in about a quarter of my number theory papers (in fact, I'm using it crucially in a paper with a SMALL student right now). What's the takeaway? You're going to forget much of the material you learn. That's to be expected. Hopefully you'll be able to re-learn it as needed / you'll know where to go to read more about it / you'll have some familiarity that such results exist. I had to relearn the Bessel function in grad school (and did, it's not that bad). What is more permanent is the techniques and methods. These are the things you'll use again and again. They can range from learning how to multiply by 1 or add 0 (two of the most difficult things to do well in mathematics!) to how to count something two ways to how to model the key features of a problem.
One of the goals of this course is to help you become problem solvers. The problems will come from analysis and number theory, but the methods and techniques, the mindset, should hopefully be applicable to a variety of problems. If you are taking an intro calculus class and Section 3.2 is on the Chain Rule, it's a safe bet that you should use the chain rule to solve problems from that section. The real world (or advanced academia) is not like that -- you frequently do not know what the 'right' way is to attack a problem (especially open problems that have stumped people for years). This is one of the reasons why I want you to create (and if possible solve!) your own homework problems. The act of creation is a huge part of research. Most math papers are mostly trivial (or, as I often say, trained monkey work). What does this mean? It means that over 95% of most papers is just straightforward manipulation of previous results (the further you go in math, the more things become straightforward). The most important parts of papers are usually the following two items: (1) asking the right question; (2) coming up with a novel way to attack a problem. Often once the question is asked and the method chosen, the paper writes itself; however, it is very hard at times to ask the right question, or to see a good way to attack it. (As an example, when I taught at Princeton years ago I wrote a handout for my students on how to prove the Fibonacci numbers satisfy Benford's law of digit bias. I decided to try and publish it, and did some research. I found a paper from the 1970s that was almost identical to mine -- basically same theorems and lemmas in the same order! This isn't too surprising, as once the question was asked, this truly was the most 'natural' way to go.)
Finally, there are remarkable connections between what seem at first very disparate branches of learning; if you are one of the first people to see such a connection, you have the potential of making a real breakthrough. I strongly urge you to tell me what you're interested in. I'll see if I can work it into the course (either in the lectures proper or in the additional comments). The more you explore, the more likely it is you can make one of these great connections. I've been fortunate enough to make some connections between nuclear physics and baseball, and between number theory and tax fraud. Because of this I've had a private tour of Petco Park (where the San Diego Padres play) and given a talk at the Boston headquarters of the IRS (with district attorneys, auditors, and secret service agents in the audience) and been interviewed by the Wall Street Journal about fraud in the recent Iranian elections.