
Princeton University Math Major Course Guide - kercker
https://blogs.princeton.edu/mathclub/guide/courses/
======
mturmon
From the part on "Intro Classes" \--

"The introductory courses for math majors are MAT 215: Single Variable
Analysis, MAT 217: Linear Algebra, and MAT 218: Multivariable Analysis. Like
the great majority of math courses at Princeton, these three courses are
theoretical and proof-based. [...] These three are usually the first math
classes that math majors take at Princeton. However, the math department is
very flexible in allowing advanced freshmen to skip some or all of these
courses."

MAT 215 is classical analysis (epsilon-delta, differentiability, etc.). Its
own course page
([https://www.math.princeton.edu/undergraduate/course/mat215](https://www.math.princeton.edu/undergraduate/course/mat215))
advises regular students considering taking the course:

"Typically students have a 5 on the BC calculus exam together with a math SAT
score of at least 750."

The concept of being so advanced as a freshman that you _skip_ this class is
pretty amazing to me. I know those folks are out there, but skipping honors
analysis really brings it home.

I struggled through another Ivy's version of Honors Analysis for math majors
(out of baby Rudin) as a beginning grad student.

~~~
graycat
> "Typically students have a 5 on the BC calculus exam together with a math
> SAT score of at least 750."

Okay, I went to the Princeton site and looked at the course descriptions and
contents.

I noticed a surprising _theme_ : It looked like there was a big intention to
make the courses difficult. Gee, guys, a student is paying a lot of money to
go to Princeton. To get in, they had to do a lot of preparation and,
apparently, have a lot of aptitude. For such a student, the material listed is
not so difficult that the courses have to be difficult.

So, a question: Why the heck should a good student who wants to know some math
put themselves through such difficulties just to learn some material that is
not really difficult?

There's an alternative: Essentially every topic in the courses is in
beautifully polished textbooks. The students are being expected to work really
hard outside of class, anyway. So, just save the money, the stress, and the
difficulties, skip those Princeton courses, get 2-3 shelves of appropriate
books, and study independently. "Look, Ma, no expensive Princetion tuition,
expensive cost of living, etc.".

I did notice that in places the course materials were lectures and notes.
Gads: For that material, there is no excuse for other than beautifully
polished textbooks since so many of them exist.

Why? Why, what the heck is the goal of such a Princeton undergraduate math
major? Sure, the goal is to go to graduate school in math or something closely
related.

So, get the undergraduate material by independent study, save the
_botheration_ of Princeton, and go to graduate school.

With high irony, at least at one time, the Princeton math department's Web
site stated that graduate students are expected to prepare for the math Ph.D.
qualifying exams on their own, that graduate courses are introductions to
research and given by experts, and no courses are given for preparation for
the qualifying exams. Okay, so the attitude is that the students should learn
by independent study. Right.

Likely the main challenge of a Princeton math Ph.D. is the usual one -- the
research. But, now, there is a largely new approach, likely also permitted at
Princeton: Get a problem from the non-academic, real world, derive some math
likely at least somewhat new for a good solution, and let that research be the
Ph.D. dissertation. From those Princeton materials, with a lot of emphasis on
machine learning (ML), it looks like such a dissertation approach would be
acceptable.

My experience can serve as an example: In high school I took 1st and 2nd year
algebra, plane and solid geometry, and trigonometry but took no calculus. I
went to a college selected because I could live at home and walk to it! It was
not a good college! They wouldn't let me take calculus as a freshman and,
instead, forced me into some course beneath what I'd done in high school, so I
got a good calculus book and dug in. For my sophomore year, I went to a good
college and started on their sophomore calculus from the same text Harvard was
using. I did fine.

Lesson: It's possible to teach this stuff to yourself. Big buck tuition, big
challenges, etc. are not necessary.

Another Lesson: Don't need AP calculus. Instead, just get a good calculus book
popular for college freshmen and dig in. Indeed, the time I looked at the AP
calculus material, it seemed to be written by people who didn't understand
calculus well. Likely the used book collections are awash in good college
calculus books -- we've had very highly polished calculus texts for decades,
and the subject hasn't changed much.

Note: At some point, might want to learn vector analysis with Stokes theorem.
Okay. The high end approach is via Cartan's _exterior algebra_ , but really
that should be for a second pass. Besides, if you want vector analysis for
Maxwell's equations, potential theory, fluid flow, etc., then the exterior
algebra material will likely not yet be popular. I suggest

Tom M. Apostol, _Mathematical Analysis: A Modern Approach to Advanced
Calculus_ , Addison-Wesley, Reading, Massachusetts, 1957.

It's great fun to read, especially after sitting through physics courses where
the professors struggle with this material and the math departments don't want
to teach it.

Get the book used. It's just the right compromise for, say, Maxwell's
equations. Leave the high end versions with differential geometry, exterior
algebra, emphasis on manifolds, measure theory, etc. for later.

So, in college, I got a math major. Okay, I found that after the first theorem
proving course, I could teach myself material that was about proving theorems.

Lesson: Take a course that tries to teach theorem proving; then you should be
able to continue on with independent study of the math that is all about
theorem proving.

So, right, my undergraduate math major had a theorem proving course from Baby
Rudin (W. Rudin, _Principles of Mathematical Analysis_ , hasn't changed much
in a very long time). I did well enough in the course, but at the end I
neglected it to finish my math honors paper. Later on my own I took a second
pass through Baby Rudin and learned it much better. That second pass was slow,
careful, where I _chewed_ on the proofs to try to _understand_ why they
worked, took the time to develop intuition, etc. Then I took a careful pass
through Fleming, _Functions of Several Variables_ and, in my career, studied a
wide variety of related topics. Then on my Ph.D. analysis qualifying exam,
those passes through Rudin and Fleming got me the best score in the
department.

Lesson: That analysis material at the level of Baby Rudin or a little more,
you can teach yourself plenty well enough.

The Princeton materials put a lot of emphasis on linear algebra. Okay. I'm not
sure that quite so much emphasis is crucial, but in the end I did have that
much emphasis. For linear algebra, I learned a lot, and nearly all of it I
taught myself from a stack of mostly quite good books. Then later I got pushed
into a _advanced_ course in linear algebra taught by a world expert guy. I
told the faculty that likely I didn't need the course. Yup, the course was
intended to _filter_ students, and that was sad because some of the _filtered_
students were quite good and should have done well. I was right: I didn't need
the course: The course was carefully graded, and on all the grading I totally
blew away all the other students, without so intending, effortlessly. I felt
sorry for the other students I made to look bad -- only near the end of the
course did I understand I was blowing away the other students.

Lesson: Linear algebra really is important, but you really can teach it to
yourself to a quite high level.

My Ph.D. dissertation was in stochastic optimal control. I had the main,
intuitive ideas on an airline flight. Then for the careful theory, I taught
that to myself from various sources and derived the rest myself. Really, I
wrote my dissertation essentially independently.

Lesson: it's possible to do that.

Overall Lesson: You can teach that math to yourself and do good research with
it, all with very few courses and without paying big bucks for tuition and
living in Princeton, putting up with rough _class notes_ , having people trap
you into some very stressful situations for no good reasons, etc.

Yes, at one point I did get accepted to graduate school by the Princeton math
department. I didn't go to Princeton. I also got accepted to Cornell, Brown,
and more.

Finally, why learn all that stuff to A+++ level, have memorized all the
difficult proofs, can work right away all the tough exercises from all the
relevant texts in the library?

Here's the main answer: There is no really good reason. Maybe for some student
there is a good reason, but IMHO mostly there is not. So, how do the
professors know the material so well? Usually from having taught a course in
it a few times! In the end, both in academics and outside, for any math you do
or apply, you will get paid for what you can create that is powerful and
valuable. Or, you won't get paid for carrying the library around between your
ears. The respect is for what you can create, not what you know. Yes, for good
creation, do need to know the prerequisites well enough, but you still can use
the books on your bookshelf and the ones in the library. Sure, the more stuff,
even just 100 challenging exercises, you know, the better, but it's rarely
necessary to have A++++ in Baby Rudin, etc. to do the needed research.

I will say, it does appear that learning the material well enough to have
built some good intuitive models, that let you make good guesses, is
worthwhile. Professors who are willing to pass out such models should be
listened to.

Yes, the Princeton quote above mentioned doing better than 750 on the SAT
Math. Okay, I confess, I did, both times I took it. But I don't believe that
such an SAT Math score is necessary. For me, likely the secrets were just that
I liked the material and wanted to learn it.

Finally, I fear that the Princeton Math Department is hurting a lot of good
students, maybe creating some sadistic professors, all for no good reason.

~~~
jvvw
I studied mathematics at university (though in the UK at Oxford so a different
system) and studied a good proportion of the first year course independently
from text books before I go there. But I still learned so much more in my
first year by being taught by the experts, having my thinking dissected and
generally being put through my paces. I don't think I could have become as
good a mathematician as I did just by studying independently.

Obviously, the tutorial system at Oxford was a great benefit though and I
don't know how that compares to teaching at Princeton. (This was also long
enough ago that I didn't have to pay for tuition so it was a no-brainer
decision to go there rather than study independently!)

------
Matetricks
For those interested in the "Probability and Statistics" portion of the guide,
Princeton recently created a certificate program in statistics and machine
learning that has some more updated information on courses:
[http://sml.princeton.edu/](http://sml.princeton.edu/)

I'm pursuing the certificate right now and the courses have been great so far.
Princeton's known for having a rather theory-heavy approach in their
quantitative classes but I've found a good balance with applications in some
of the classes (COS 424, COS 402).

------
azhenley
Every school should have a guide like this for every major.

~~~
surrey-fringe
Yes, and a curriculum, and textbooks that contain everything you need to know
for finals!

~~~
ianai
Pretty sure that's required by accreditation

~~~
RCortex
Oh you, not every top universities has to play by those rules.

------
byh
The intro courses have changed slightly. Now, there is the 215-217 track
(analysis + linear algebra) and the 216-218 track (basically honors version).
Most math majors take the latter.

The algebra introductory courses have different numbers, and are now 345-346
instead of 322-323. There is now an advanced graph theory class (477) that
follows the introductory one. Also, there is a theoretical machine learning
class (COS 511) and the algorithms/complexity graduate sequence (COS 522-523)
may be relevant as well.

I will probably update these changes on the website sometime in the near
future...

~~~
Matetricks
They've added a few more ML courses—the go to class for undergrads is now ORF
350 (Analysis of Big Data) with Han Liu. ELE 535 (Pattern Recognition and
Machine Learning) is also a new course that was added last year.

------
sn9
I've always liked Tim Gowers' welcome to incoming students to the Cambridge
Mathematical Tripos program. Lots of great tips on learning/studying math,
asking intelligent questions, etc.

[https://gowers.wordpress.com/2011/09/23/welcome-to-the-
cambr...](https://gowers.wordpress.com/2011/09/23/welcome-to-the-cambridge-
mathematical-tripos/)

~~~
plinkplonk
This is a great link. Thank You.

a dumb question if I may - What is an "example sheet"?

~~~
soVeryTired
Same thing as a problem sheet.

~~~
plinkplonk
Thanks!

------
OJFord

        > Lastly, don’t forget the ever-present Rule of 12!
        > (That’s “twelve,” not “twelve factorial.”)
    

[https://blogs.princeton.edu/mathclub/guide/other-
fields/cs/](https://blogs.princeton.edu/mathclub/guide/other-fields/cs/)

------
princetontiger
The math department has incredible professors.

~~~
tigertiger12
I agree, although I always struggled to overcome their strong accents. Every
professor and TA that I studied under had a thick accent, either Chinese or
Russian.

~~~
madcaptenor
One of the practical skills I learned in grad school was understanding foreign
accents.

------
imranq
What's up with all these major guides

~~~
dsacco
People realize that math and statistics are a very solid foundation for
machine learning, and machine learning is in vogue. Instead of going back to
school to get a graduate degree, they take the autodidact route.

~~~
tunesmith
Any good guides on auto-didacting math? That seems about impossible to me,
it's a subject where you'd really need study groups and office hours.

~~~
whitepoplar
This book is _fantastic_ and should get you pretty far:
[https://www.amazon.com/Mathematics-Content-Methods-
Meaning-V...](https://www.amazon.com/Mathematics-Content-Methods-Meaning-
Volumes/dp/0486409163)

~~~
tunesmith
btw I bought this on your recommendation. It's... big! :) I haven't started
reading it yet, I'm in the middle of Gödel Escher Bach right now.

