**Objectives for Math 308 /
406Analysis 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
mathephs@gmail.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:

- We are preparing students for jobs that don't exist yet using technologies that haven't been invented to solve problems we don't even know are problems yet.
- We are living in exponential times:
- 31 billion searches on google each month now; it was 2.7 billion in 2006.
- The first text message was in 1992; more are sent today than the population of the planet.
- Number of internet devices: 1,000 in 1984, 1,000,000 in 1992, 1,000,000,000 in 2008;
- Estimated that 4 exabytes (4 x 10
^{19}bytes) of unique information will be generated this year, which is more than has been generated in the past 5000 years! - Amount of new technical information doubles every two years (hey juniors: a lot of what you learned, if you're in a technical school, as a freshman is outdated!)

- Click here for another video in the series; it contains a lot of the same information, some interesting facts in the beginning for educations, but no cool (or uncool) soundtrack.

Another great clip to listen to is the TED lecture of Malcolm Gladwell on spaghetti sauce. The main point of this lecture is

- It is important to ask the right question; what you think is the right question frequently isn't. I won't do the lecture justice by summarizing it, so I'll just give the following tantalizing tidbit: this surprising question led to Prego making $600 million in 10 years on extra chunky spaghetti sauce.
- There are lots of great clips on TED; another one of my favorites is Dan Pink on Motivation. Some very interesting observations here on how to create an optimal environment for creativity to flourish.

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.