
Self Studying the MIT Applied Math Curriculum - hsikka
https://www.harshsikka.me/self-studying-the-mit-applied-math-curriculum/
======
olooney
I did Strang's linear algebra course[0] (link goes to Youtube playlist of
lectures) several years after graduating and recommend it highly. I was
looking for refresher but I gained a deeper understanding of several important
concepts; in particular it's fair to say I barely understood, or perhaps even
misunderstood, SVD until Strang. If you're not sure if you need something like
that, I suggest doing something like testing yourself on this video[1] or the
MIT problem sets[2] it's easy to tell yourself that you "know" linear algebra
when it would be closer to the truth to say that you _used_ to know linear
algebra, but can't answer even basic questions today. After Strang, Golub's
book on Matrix Computations is also really incredible.[3]

[0]:
[https://www.youtube.com/watch?v=ZK3O402wf1c&list=PL49CF3715C...](https://www.youtube.com/watch?v=ZK3O402wf1c&list=PL49CF3715CB9EF31D)

[1]:
[https://www.youtube.com/watch?v=Cll03FUxjuk](https://www.youtube.com/watch?v=Cll03FUxjuk)

[2]: [https://ocw.mit.edu/courses/mathematics/18-650-statistics-
fo...](https://ocw.mit.edu/courses/mathematics/18-650-statistics-for-
applications-fall-2016/assignments/)

[3]: [https://www.amazon.com/Computations-Hopkins-Studies-
Mathemat...](https://www.amazon.com/Computations-Hopkins-Studies-Mathematical-
Sciences/dp/1421407949/ref=cm_cr_arp_d_product_top?ie=UTF8)

~~~
the_svd_doctor
Another great "intermediate" textbook (in my opinion) is Trefethen's Numerical
Linear Algebra [1]. Much more readable than Golub's I would say, which is more
like a reference than a textbook.

[1]: [https://www.amazon.com/Numerical-Linear-Algebra-Lloyd-
Trefet...](https://www.amazon.com/Numerical-Linear-Algebra-Lloyd-
Trefethen/dp/0898713617)

~~~
laserson
Strongly agree. This was actually the text book when I took numerical methods
at MIT. As far as math texts go, it is definitely a joy to read.

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throwawaymath
Speaking frankly, it's ambitious to complete even one of these courses in a
single summer. Most students would be taking one or two of these in 12 - 16
weeks while doing nothing else but being a student. Accomplishing the same in
about eight weeks from video lectures will be more difficult. I would strongly
urge the author to slow down, choose a single course they have the background
for and work from there.

I can't tell if the author has done a real analysis course before, but if they
haven't that's the one they should choose next. If they already have that
under their belt they should go to probability or complex variables. I don't
see the utility of re-doing calculus or linear algebra if the author is
already strong in both.

I appreciate the enthusiasm for math that's evident here, that's great! But if
you're learning math the "right" way - by actively engaging with the material
- there's only so much of it you can learn at once.

~~~
hsikka
Hey, author here, I really appreciate the feedback. Are there any specific
prerequisites to Real Analysis?

~~~
throwawaymath
Technically speaking, no. Real analysis is pretty self-contained. You
basically start out by constructing the reals from scratch and deducing
continuity as a consequence of the completeness of the real field. You use a
tiny bit of set theory to establish notation and define bounds, and then from
there you go into limits, derivatives, integrals and maybe the Lebesgue
measure. I wouldn't expect a real analysis course targeted at applied math
majors to do much other than that.

The reason real analysis is useful is because it's (loosely) a deeper calculus
course with proofs. Since probability theory becomes more proof-based (and
ventures into measures), real analysis is good preparation for it.

~~~
umanwizard
> You basically start out by constructing the reals from scratch

Not necessarily -- a lot of books just take the existence of a unique set with
certain properties as an axiom and call it R.

The main topics of basic real analysis IMO are differentiation, integration,
(uniform) continuity, compactness, convergence, etc.; how to construct the
reals from the rationals is a side point at most.

This could be personal bias as I just don't personally think that the exercise
of constructing the real numbers is very interesting.

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commandlinefan
I’ve been doing more machine learning for work, and I’ve realized a) how much
calculus is involved in it and b) how much calculus I had forgotten since I
took the courses as an undergraduate almost 30 years ago. I happen to still
have my old calculus textbook (I couldn’t sell it back to the bookstore back
then), so I dusted it off and started working through all the problems. I’m
about halfway through right now, and man I get strange looks from people when
I start working calculus problems. Good to know that I’m not alone, although
I’m about halfway through the equivalent of 18.01 (1/10th of the poster’s
coursework) after 6 months. It doesn’t look like there are that many sample
exercises in the online course, though - that may make it harder to really
absorb.

~~~
adamnemecek
Look into dual numbers and automatic differentiation, it's like Calculus on
steroids.

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dxbydt
It’s awesome to see more & more devs getting interested in math & stats as
machine learning, optimization & sequential testing (what you folks call a/b
test) becomes more mainstream.

The obvious benefits are, much of math literature is currently written for
consumption by math grad students & profs, so all of this will get a much
needed rewrite. I foresee a surge in material like Jeremy Kun’s popular “math
for programmers” book that so many of you loved on HN. Much of math lit is imo
unnecessarily dense & terse. Speaking as a phd student, I really appreciate
atleast 1-2 numerical examples with answers, not just plain theory. Maybe if
there was a Linear Algebra for programmers text with some code in python to
accompany the theorems, it would be a huge hit among applied ML researchers &
programmers. Graduate linear algebra is actually quite hard & very few phd
students pass the linalg quals in their first attempt ( Grad linear algebra,
not basic shit like matrix algebra, linear maps & decompositions). This
material should imo be rewritten ( “dumbed down” ) so regular folk can also
sample some of the magic.

In my previous work at an adtech corp, I built a whole bidding algorithm
straight from Strang’s linear algebra book (the chapter on Lagrange
optimizers). It’s still running, 5 years & counting. I once joked if they gave
a nickel out of every dollar that algorithm made to Prof Strang, he’d be a
multimillionaire.

~~~
jimmy_dean
The Coding The Matrix textbook is exactly the type of book you've described
that is tailored for programmers. I've read only the first few chapters and
found it a joy even if a little wordy in some parts.

[https://codingthematrix.com/](https://codingthematrix.com/)

~~~
dxbydt
it’s a good undergrad book. also on that page he says lights out game is a
“linear algebra problem”. It’s actually not. incidentally I worked on Lights
out for my thesis, and my advisor made a whole career out of lights out! That
game relates to Fibonacci polynomials, studied in Combinatorics -
[https://www.unf.edu/~wkloster/fib.html](https://www.unf.edu/~wkloster/fib.html)

~~~
sandGorgon
How would you compare coding the matrix to strang ?

I'm going to be starting out on one of these and coding-the-matrix is very
python centric (which is super helpful and practical) versus MATLAB in
Strang's book.

~~~
dxbydt
this is classic “mistaking the map for the territory”.

whether the code is in python or matlab matters little. the code is just a
supplement to the main material which is still math. trying to understand that
math by reading code is actively counterproductive and actually quite foolish.
my professor gave a symposium talk on this topic, so i will use one of his
examples -

At 0 minutes you have 0 dollars. suppose each minute you toss a coin. if it’s
heads you make a dollar. if it’s tails you pay a dollar out of your pocket.
What are the chances you’ll have $30 in an hour ?

This is the sort of classic problem that can be coded up in python or matlab
under 5 minutes. it’s just a for loop & an if statement. It won’t tell you
much about the actual probability because that number is so small your code
will have to run for a long time to get a meaningful answer. Whereas it’s
trivial if you just do the math by hand.

~~~
sandGorgon
it just happens that i dont want to download and install matlab on my computer
(which is company provided) and i'd rather work on python if there is an
equally good course.

not what you thought though.

------
usgroup
Good luck man. I did a similar kind of thing at one point. A lot of the
material is extremely boring and it’s really easy to skim it and pretend you
know it in principle. You’ll see... it’s one of those things. The unbridled
enthusiasm won’t let you acknowledge it but you know ifs true :) So IMO you’ve
got to find some way of doing some exams somewhere.

In the end the only way I could find enough motive to get through a whole
applied maths syllabus was to book myself for exams so that I could panic
about failing and study.

------
graycat
Suggestion: Set aside complex variables and do measure theory instead. For the
Fourier and Laplace transforms, cover those via measure theory. For measure
theory, H. Royden, _Real Analysis_ and the first half of W. Rudin, _Real and
Complex Analysis._

Then take a course in _graduate probability_ which is based on measure theory.
Good authors are Loeve, Neveu, Breiman, Chung, among others. So, learn about
the cases of convergence including almost sure convergence, a good version of
the central limit theorem, proved carefully, the weak and strong laws of large
numbers, ergodic theory, and martingale theory.

Learn stochastic differential equations.

Learn some potential theory.

Then do some work in optimization and optimization under uncertainty.

~~~
verylongname
Personally, I think complex analysis is important. I really like Serge Lang's
book "Complex Analysis." It is supposedly a graduate level text, but the first
half of it covers the basics at a level suitable for upper division undergrad
class. Ahlfor's book is also really really good, but a little harder to read
in my opinion.

I didn't like either Royden or Rudin. I really like Folland's book "Real
analysis: modern techniques and their applications." His Fourier analysis book
is also pretty good IMO.

~~~
graycat
Yes, I've got a copy of Alfors. And at one time I started a course from a
student of Alfors. And I have a book by Hille.

I just never could see much utility in complex variables: Complex valued
functions of real variables. Sure. Complex valued measures? Sure. Fourier
theory making a lot of use of complex numbers? Of course. A vector space where
the field is the complex numbers? Certainly. Hermitian and unitary matrices
with complex numbers? About have to like those due to the fundamental theorem
of algebra, that is, roots of polynomials and, thus, complex eigenvalues.
Functions of a complex _variable_? Never could see the utility.

------
mkl
Can someone please explain how MIT course numbers work? I get that "18" means
maths, but why do specific courses sometimes have two digits after the "." and
sometimes three, and if the level of study is encoded in it, why does it seem
to jump to starting with "6"?

~~~
amirhirsch
Two numbers like 18.XX means that it is intended for non-math majors so 18.02
is the basic multivariable calculus that every MIT alum needs to graduate,
18.022 is multivariable calculus with an emphasis on theory and 18.023 is with
an emphasis on application and usually taken by people considering math majors
(though also fulfill the core requirement) The class numbering schema is by
field:
[http://math.mit.edu/academics/classes.php](http://math.mit.edu/academics/classes.php)

18.1XX Analysis / Calculus

18.2XX Discrete

18.3XX Applied

18.4XX Computational

18.5XX Logic

18.6XX Probability and Statistics

18.7XX Algebra and Number Theory

18.8XX Just project lab

18.9XX Topology and Geometry

~~~
mkl
Thanks!

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martincmartin
Is there a place to find answers to textbook questions? I get that this is
discouraged, because it means paying students can cheat. But for people who
are self studying, there are many times you _think_ you understand, but you're
not sure and would like to see.

In particular, I'm hoping to self study the Structure and Interpretation of
Classical Mechanics, but can't find the answers anywhere.

~~~
yeahman
If you post a serious attempt to math stack exchange, they are usually very
helpful in telling you if you are right, or where you went wrong.

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mpoteat
7 non soft classes in a single summer is ambitious to say the least.

~~~
hsikka
Yeah, it is, its more of a loose goal, I'm doing research and working on
finishing 2 Master's alongside it, so we'll see what happens! I really just
want to learn and grow, so if it takes months or if it takes years, I'm in!

~~~
barry-cotter
What are you Mastering in? What’s your research on?

~~~
hsikka
I'm studying CS and Biology, and I'm currently researching disentangling
object representations using brain inspired auto encoders

------
mikorym
This is a cool thing to do. I followed a slightly similar path in that my
undergraduate degree did biochemistry up to 3rd year, but at the expense of
third year pure math courses. (I had a kind of applied PDEs and modelling
course but applied to biological problems like gene flow, disease modelling
and population modelling.)

I ended up doing honours in bioinformatics, but then took a year to do third
year pure math (and the honours course in lattice theory and topology). After
this I ended up doing what I really wanted to, which was an MSc in pure
mathematics.

I also did probably the first 1/3 of Strang's course (even though I had
already done linear algebra formally in my original course). I think online
math courses are a mixed bag and it could even be that it works better when
you have an idea already of _why_ you want to do it and the applications
thereof. Applications need not imply applied math—you can apply mathematics to
other branches of mathematics.

The more difficult thing for me is to independently find research topics and
subtopics. Mentorship in mathematics is very important.

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sparcraft
I studied a very similar curriculum over ten years ago during my Physics
undergrad. While it’s not strictly “useful” in a day to day sense, nor do I
remember much of the details of these courses or methods, I can tell you that
studying applied math has served me very well during my career as an actuary.

Studying applied math gives you the skills of being able to approach problems
in a logical, analytical, and above all PERSISTENT way (because let me tell
you will you ever have some late nights banging your head on the desk trying
to wrap your brain around these problems). These skills are crucial in any
field where you need to create mathematical representations of a system - or
break apart and understand someone else’a models.

Also - partial differential equations are just cool! They are like a
mathematical rubick’s cube. Solving them will legitimately make you smarter.

------
joflicu
One question for all the self-learners here (I have immense admiration). Do
these courses inform your day to day job in some way? Or is this for personal
knowledge (even more impressive)? How do you find the damn time? Need some
suggestions. My goals are infinite and my time is so short.

~~~
moosey
If learning is part of a well-rounded set of activities that you are doing in
your life, then it will lead to several positive effects. One, learning new
things, by itself, with nothing else, reduces cognitive decline and increase
brain health. But optimal learning and brain health also derives from physical
activity and sleep, so it's a small part of the puzzle.

EDIT: Brain health also derives from a healthy social life, I forgot to add
that to the list involved. Having friends that are interested in learning the
same things as you, and discussing it with them, help you to develop self-
checking techniques.

Also, Alan Kay pointed out in a speech in the last few years that our economy
is doing really poorly right now producing researchers, and people who
understand research, and you really can't get there without a good education.

I'll go one further and suggest that 'Transfer', which is the application of
understanding of one topic to another, is one of the prime sources of what we
consider 'innovation' or 'paradigm shifts'.

I feel as if that should be enough proof that education is an end to itself,
but I need to be really clear: continuing education doesn't need to have a
goal, and I would strongly discourage you from adding one. I go to the gym all
the time without goals. I go for walks everyday, without goals. Each of these
items are basic maintenance. Learning new things _well_ is basic maintenance
for your brain. If you want to do continuous learning then you should install
Anki, get a notebook and learn Cornell notes, then go out and take a quick
class, lecture, or study material.

If you are learning something new, the fastest way to learn and retain the
material is to move to self-testing as quickly as possible. I can go over this
in more detail, but perhaps the first class I would recommend would be
"Learning How To Learn" (Coursera), which produced a serious attitude shift in
me about learning and what I am capable of as a human.

~~~
joflicu
Thank you for the great reply and apologies for dropping out. I would like to
know more about self-testing and how it helped you.

------
gumby
Great idea! I've used OCW to "re-take" a class I last had 25 years ago, plus a
couple I never took, but never through to do a whole program.

A suggestion: 18.01 and 18.02 are typically taken in the first two semesters
of freshman year (and many take 18.03 the first of sophomore year, or even the
semester before). As you're well past the high schoolers who push up at MIT, I
suggest 18.014 / 18.024 / 18.034 for more theory (those weren't available when
I was a student).

It's funny: when I was an undergrad Math was quote popular because it had so
few requirements and on your page it still looks that way (compare to course
6). I wonder if they same dynamic still applies?

------
adamnemecek
While at it pick up Julia, another MIT product.

esp for applied math it's invaluable compared with python or matlab.

Juno is a nice ide, it's gotten a lot better over the last two months.

~~~
bigred100
Why is Julia better than MATLAB or Python? Most applied mathematicians I know
are in MATLAB all the time.

~~~
adamnemecek
I mean julia is much younger. The main strengths are that it's open source,
it's a much nicer language, you can actually ship things in it (good luck
doing that in matlab), package manager, etc etc. Matlab is legit not great.

~~~
bigred100
Ah ok. The reason I like MATLAB is cause I want a “numerical experiment” and I
don’t want to have to learn a lot to get it to run

~~~
adamnemecek
Check julia. Its not unlike matlab.

------
itsbenweeks
A lot of these earlier courses are available on edX, too. The platform won't
grade you unless you pay, though.

------
skookumchuck
It's amazing and wonderful you can get an MIT education from watching youtube
videos of their lectures. I've used them to fill in gaps in my education.
Highly recommended, and it's to MIT's great credit that they're doing this.

~~~
afarrell
To really get an MIT education, you should team up with one or two other folks
and work through the labs and problem sets together. There’s no substitute for
the learning you get by explaining your understanding of a problem and hashing
out how to approach it with someone else — and by building things!

