[HN Gopher] Self studying the MIT applied math curriculum (2019)
___________________________________________________________________
Self studying the MIT applied math curriculum (2019)
Author : jiggle123
Score : 191 points
Date : 2021-10-24 17:12 UTC (5 hours ago)
(HTM) web link (www.smallstepcap.com)
(TXT) w3m dump (www.smallstepcap.com)
| joconde wrote:
| Is this doable for someone with less basis in math? I almost
| stopped studying math after two years of undergrad, when I went
| to a more practically-focused school. It'd be nice to get back
| into shape, because I've started reading research articles and
| always feel like I'm behind on the theory side of ML.
| rsj_hn wrote:
| It depends very much on your own skills and drive.
|
| At some point, you need people to check your work, though, but
| you can search for online communities to do that. E.g. a post
| to mathoverflow asking about whether the following proof is
| correct, for example. Try to find a community of others online
| and work with them.
|
| Also remember that math is about _ideas_. It is not just an
| exercise in formal deduction. The field is dense with non-
| trivial, non-obvious, interesting insights, and understanding
| these insights so well that overtime they become obvious to you
| is what it means to learn math. Sites like 3Blue1Brown to a
| good job trying to explain these ideas, but they really put a
| lot of work into it. Most math texts do not, but you have a
| professor or classmate you can talk to -- so there is a gap
| between the text and what you need, but perhaps you can close
| the gap with online resources.
| jknz wrote:
| Mathoverflow is for level research questions.
|
| Homework type questions (anything before PhD student level)
| are typically asked on https://math.stackexchange.com/
| jimhefferon wrote:
| > At some point, you need people to check your work, though,
|
| Agree with that, 100%. Most of my students will, at some
| point in a course, respond to questions with things that are
| completely wrong. It's just how people learn. If there was
| not someone to correct their mistake then they would struggle
| a lot to find the error and fix it.
|
| One of the most rewarding things about working with students
| is helping them through tough spots. But it is real work, and
| takes the time and attention of someone who has training.
|
| > but you can search for online communities to do that. E.g.
| a post to mathoverflow asking about whether the following
| proof is correct, for example.
|
| I'll just observe that MO, while a great community, can not
| scale to helping lots of people check their homework. For
| instance, in a Linear class session I might have 20 people,
| each with 12 homework questions. MO is just not set up for
| that.
|
| Now, sometimes students are sure they got questions right,
| and usually their certainty is right, so there is no need to
| ask that one online. And sometimes people can find a relevant
| previous answer (although learners often struggle to find
| those, in my experience). Nonetheless, even after taking
| those out, there are still a lot of people with a lot of
| questions.
|
| > Try to find a community of others online and work with
| them.
|
| Yes, very good advice. I just wanted to observe that while it
| might work for OP (and I hope it does), it cannot work for
| lots of people. I don't know what the answer is, and maybe MO
| could help a lot, but it can't be the answer alone.
| rsj_hn wrote:
| Yes, I agree with all these points. I wasn't trying to
| suggest that access to knowledgeable teachers isn't
| necessarily, but giving second-best options to those for
| whom that's not an option.
| downrightmike wrote:
| It helps to have good resources
| https://openstax.org/subjects/math
| zsmi wrote:
| Not impossible but probably really hard.
|
| The issues will be the same with self studying anything.
| Without someone to critique your work, and point out
| deficiencies you don't even know are a thing, continuous
| improvement quickly becomes exponentially harder from the bad
| habits that are holding you back.
|
| Technique totally matters, even in STEM.
| Evgenii1 wrote:
| I self study machine learning here
| https://learnaifromscratch.github.io/ai.html it's an early and
| shitty draft and proof of concept that you can do self-directed
| learning for these topics while looking up the background you
| need to know, which for me is much more interesting than taking a
| generalized math curriculum of absolutely everything. The courses
| so far we haven't escaped the content of Wasserman's 'All of
| Statistics' book yet on classification or probabilistic graphs,
| so you could if you wanted watch the lectures and only do
| Wasserman's book.
|
| If you want to try the OCW linear route of taking everything for
| whatever reasons, you will have to get up to MIT student levels
| trying to unravel the algebra done in the early calculus courses
| and later where they just assume you possess this background. One
| way to do that is those problem solving books like this one which
| 'bridges the gap between highschool math and university'
| https://bookstore.ams.org/mcl-25 at least you then get worked out
| solutions. Another way is Poh-Shen Loh's Discrete Math course he
| opened up on YouTube which is done the same way he holds the CMU
| Putnam seminar, working through a bunch of combinatorics and
| algebra will more than prepare you to understand those continuous
| math OCW courses https://youtu.be/0K540qqyJJU
|
| Like everybody else there is of course the issue of: who is going
| to check my work. For me I went with the time tested tradition of
| hiring a tutor, a local grad student and paid them once a week to
| go on chat/zoom or meet at a coffee shop before the pandemic and
| spend a few minutes going over everything I'm doing wrong. In the
| early days however I used constructive logic ie: 'proof theory',
| to audit my own work: https://symbolaris.com/course/constlog-
| schedule.html and read a huge amount of Per-Martin Lof papers on
| the justifications of logical operators like implies,
| disjunction, conjunction, etc. Of all the math I've ever taken I
| would say that proof theory was the most useful for somebody by
| themselves who isn't sure of what they are doing (I'm still not
| 100% sure.. hence why I hire people now).
|
| If you want a great Calculus text that explains those nasty
| looking Euler's e nested statistics distributions try
| _Mathematical Modeling and Applied Calculus_ by Joel Kilty
| everything from partial derivatives, gradients, x^n, e^x, trig,
| integrals, limits is explained in terms of parameters to modeling
| functions, if you write software it will be easy to understand. I
| haven 't posted it yet but I tried going through Allan Gut's
| probability book using only that math modeling calc text and have
| not run into anything applied, as in concepts about limits or
| integrals, that wasn't already covered. Of course the concepts
| are much more abstract measuring a bunch of intervals and a
| different method of integration and I don't pretend I'll be
| making any advances in this area beyond applied usage but it can
| be done, jump in and pick up the background as you go as opposed
| to doing all the background at once, losing interest and giving
| up.
| ai_ia wrote:
| I have a pretty similar background. I have an undergrad in ChemE
| who fell in love with machine learning research. As I didn't had
| the appropriate background so I taught myself Computer science
| using mostly resources such as OCW and teachyourselfcs,
| videolectures etc.
|
| However, what stood out to me was how difficult it is to self
| study? Universities provide a setting which helps you learn
| difficult subjects over a longer period of time. Outside of that,
| no such avenues exist. It's not just reading up a book or making
| Anki flash cards(which is quite tedious to be honest) but the
| process of selecting, vetting what to read next and actually
| completing it. Or the question are we there yet, with no idea
| what "there" actually means.
|
| I decided to work on creating a platform where people can
| actually learn difficult subjects on their own. It's a bit
| different than normal video based platform, it actually uses text
| based conversation to facilitate learning. But I truly believe
| this is the way forward.
|
| I have written comic based guide about this problem:
|
| https://primerlabs.io/comics/introducing-primer-comics/
|
| We are creating self paced courses on Computer Science and also
| looking into creating self paced Mathematics/Physics courses as
| well.
|
| We have released two free courses for everyone to try out at
| https://primerlabs.io
| WhisperingShiba wrote:
| I would love to self study with more vigor. I studied
| Mechanical engineering, but I find Math to be so beautiful. I
| think I chose the right education for the work I find
| meaningful in the world, but I wish I had more time to learn
| math that isn't directly relevant to what I am doing.
|
| I'll certainly check out your site and save it for when the
| world is more peaceful.
| lordnacho wrote:
| Wow that's what I was gonna write. I did engineering too, and
| all of the topics listed in the course. But something about
| doing it to apply to other things is wrong, hard to say what.
| I remember in high school being quite fascinated by certain
| beautiful things in maths, and the applied stuff just isn't
| the same. I guess it just seems like tools. All the theorems
| are just kinda unsurprising, as far as I recall. Things like
| Stokes Theorem are kinda foreshadowed as you're doing your
| reading. Or various statements of continuity (DivGradCurl).
|
| Luckily he left a link for the pure maths course.
| vertak wrote:
| This is very well done. I've been hobby researching learning
| tools for the past couple years and this looks like one of the
| best. @ai_ia have you seen Andy Matuschak's Orbit project?
|
| https://withorbit.com/
|
| You both seem to be solving similar problems.
| ai_ia wrote:
| Thank you for your kind words.
|
| I am familiar with Andy's Orbit project given that I am a big
| fan of his writing.
|
| I even linked to his essay "Why Book's don't work"[1] in the
| introductory blog post[2] as well.
|
| [1]: https://andymatuschak.org/books/
|
| [2]: https://primerlabs.io/blog/introducing-primer/#the-
| problem-o...
| maddyboo wrote:
| Excellent intro comic, it got me really excited about your
| product! I would love to try it out once you release a more
| advanced course. I signed up, hopefully you will send e-mail
| updates as you release new courses.
| ai_ia wrote:
| > I signed up, hopefully you will send e-mail updates as you
| release new courses.
|
| Yes. That's why we have created guest-login for people to
| test out the product and if they want to get new course
| release updates, they can sign up.
|
| Thank you
| flavio_x wrote:
| I have backgroun in chem Engineering too! Thanks for making
| this resource availble for us all
| andi999 wrote:
| Personally I would add integration and measure theory (the sigma
| algebra and lebesgue stuff), but there seems no such module?
| bigdict wrote:
| This is covered towards the end of a typical undergrad pure
| math real analysis course sequence.
| tgflynn wrote:
| I think that would be more likely to be covered in a pure math
| program than applied math.
| nuerow wrote:
| > _I think that would be more likely to be covered in a pure
| math program than applied math._
|
| Integration is pretty basic and used extensively, and I'd say
| it makes no sense to cover contour integrals within the scope
| of complex numbers and differential equations but leave out
| integrals.
| tgflynn wrote:
| Integration is covered in any calculus course, the comment
| I was replying to do was about Lebesgue integration which
| is a much more advanced topic that as far as I know is only
| needed for integrating functions which are so pathological
| that they probably don't occur in the physical world.
| spekcular wrote:
| The applied math does teach integration: the Riemann
| integral. Knowing about Lebesgue integration is not really
| necessary for most of this kind of work.
| blparker wrote:
| I'm doing something similar, except for Stats. I've cobbled
| together a plan based on degree programs from Stanford, CMU, and
| Berkeley. It would seem easier to stay on track with directed
| course learning, but how do you stay on track with the self-
| directed learning?
| querez wrote:
| Could you share your plan, I'd be interested!
| [deleted]
| CamperBob2 wrote:
| The subhead "Why in hell would you do this?" certainly applies to
| the choice of colors and fonts. Holy cow, talk about reader-
| hostile.
| TrackerFF wrote:
| Just don't spread yourself out too thin, and know when to stop -
| it's super easy to burn out when taking on too much coursework,
| even if this extra part is self study.
|
| The author writes that he wants to pursue a Ph.D - in that case,
| I'd put all my effort into pursuing what would maximize my
| chances at getting into such program. Unfortunately, self study
| is quite difficult to prove, and does not hold much weight when
| applying for such positions.
| wespiser_2018 wrote:
| This guy was my TA in a GT for a course on Educational
| Technology. He totally blew off giving me any feedback until the
| last possible moment, super frustrating experience.
|
| I guess now I know where his time went?
| [deleted]
| jchristian- wrote:
| I feel bad for you.
|
| Exactly, his time went to a bullshit, self promoting endeavour
| while neglecting his duties as a TA. By the looks of his text,
| he looks like a self-centered POS. The text is basically him
| patting his back.
| paulpauper wrote:
| tbh, I would wat until hearing his side of the story before
| passing judgement.
| [deleted]
| mensetmanusman wrote:
| There are always three sides to every story
| (his/hers/truth).
|
| In the case of undergrads it can often be a case of
| everyone procrastinating equally and then initiating a time
| crunch crisis as they all ask the TA for emergency help
| simultaneously.
| toreply2021 wrote:
| wow so much flexing in the article about their degrees and all
| the things they "did" by themselves
|
| this had me curious into looking this person up - honestly
| looks like they are trying super hard to get followers by
| making up substance about their own work and experiences.
|
| (Currrently at GT for part-time program and for some of my
| classes, the TAs are receptive and very helpful! I would hate
| to have someone who waited until the end, especially when
| everyone is figuring out how to schedule their coursework on
| top of their job)
| adnmcq999 wrote:
| This is a random flex if there ever was one
| [deleted]
| kxyvr wrote:
| If anyone wants to attempt this, that's awesome. It's a lot of
| hard work, but I think it opens a lot of opportunities for
| personally and professionally. I've a Ph.D in Applied Mathematics
| from a traditional program, so I wanted to chime in based on some
| of the comments I'm seeing. As a note, this is an opinion and
| others may feel differently and strongly at that.
|
| To me, applied mathematics is the art of transforming something
| into a linear system, which is something that we can tangibly
| solve on a computer. There are lot's of ways to do this such as
| Taylor series and Galerkin methods, so a lot of the field is
| understanding how, when, and why each method can be used. This is
| coupled with mastery over linear solvers, which includes direct
| methods, iterative methods, preconditioners, etc.
|
| I wanted to write this comment, though, to focus on certain areas
| that may end up blocking what I view as appropriate progression
| in the field. These are things that I believe are necessary to
| understand advanced topics, but don't necessarily fall under
| applied mathematics. First, you really do need mastery of real
| analysis. It's necessary because it covers formally topics such
| as differentiation, integration, and series, which are required
| to understand theorems and algorithms. In my opinion, calculus
| books are not sufficient. Rudin's Principle's of Mathematical
| Analysis is the most concise, well written book that contains
| enough. Second, enough functional analysis to understand Hilbert
| Spaces is required. This prerequisite to this is the real
| analysis above. The issue here is that algorithms for things like
| differential equations require function spaces to do properly.
| Certainly, you can go really deep in this regard, but Hilbert
| Spaces are generally enough for practical algorithms. This also
| affects optimization theory, which impacts machine learning.
| Technically, you can do optimization theory with only real
| analysis, but the theory is cleaner in Hilbert Space. Questions
| that need to be answered are things like does the infimum exist
| and can it be obtained? Working with a general inner product is
| also a valuable tool for parallelization as well as a modeling
| tool. Third, some integration or measure theory is required. It
| depends on what you're doing, so I don't think mastery is
| strictly necessary, but spaces like L2 don't make a lot of sense
| unless you know what a Lebesgue integral is. Even if you want to
| just work with spaces that are Riemann integrable, measure theory
| helps understand when this is possible and the ramifications of
| it. And, to be clear, this is important outside of differential
| equations. If you want to understand optimization theory in a
| Hilbert Space, the inner products used will require some
| understand of measure theory.
|
| Anyway, these are some random thoughts and ideas about the field.
| I do believe strongly that any amount of study is beneficial as
| most engineering fields benefit from applied mathematics.
| bigdict wrote:
| What would you recommend on Hilbert spaces to someone who has
| worked through Baby Rudin?
|
| Thanks for the insightful comment by the way.
| kxyvr wrote:
| Laugh. I was hoping no one would ask since I don't have a
| great answer! We used "An introduction to Hilbert space" by
| Young. It's fine? There's stuff in there like Sturm-Liouville
| systems that I don't particularly use or care for. I also
| never use it as a reference.
|
| Here's a hodgepodge of other books related to functional
| analysis that I like more, but don't directly answer your
| question. I got a lot of benefit out of "Convex Functional
| Analysis" by Kurdila and Zabarankin. They sort of have a high
| level overview of different functional analysis topics,
| including Hilbert Spaces, but with the ultimate goal of
| proving what they call the Generalized Weierstrass Theorem.
| Essentially, when does a function has an inf and when is it
| attained. Even if you don't care about optimization theory, I
| very much appreciated their survey of topics to get there.
|
| I occasionally also use "Introductory Functional Analysis
| with Applications" by Kreyszig as a reference. I think this
| was the first time I saw cleanly the difference between an
| adjoint and the Hilbert-adjoint of an operator, which was
| constantly confusing to me prior to that point.
|
| The last one I like is "Nonlinear Funtional Analysis and its
| Applications I: Fixed-Point Theorems" by Zeidler. He wrote, I
| think, five volumes, but this is the only one that I use.
| Anyway, he presents differentiation, Taylor theorem, and the
| implicit function theorem very well in function spaces. The
| first four chapters are great as a reference.
|
| Since I'm listing off obscure books, for integration, I like
| "A Concise Introduction to the Theory of Integration" by
| Stroock. I actually don't like his newer book, "Essentials of
| Integration Theory for Analysis" as much as the older book.
| Anyway, I find it very dense, but well written. Essentially,
| I like the first five chapters, which culminates with the
| divergence theorem, which ultimately gives a precise
| description of integration by parts in more than one
| dimension. He also answers precisely the question about the
| difference between Reimann and Lebesgue integrals.
| bigdict wrote:
| Thank you! What is that extra power that comes from
| considering Hilbert spaces as opposed to staying in good
| ol' R^n?
| kxyvr wrote:
| Primarily, it gives us the power to work with functions
| that return other functions. In many ways, it's kind of
| like going from array processing in Fortran to functional
| programming. Outside of the esoteric, there's a choice
| that has to be made while modeling as to when to
| discretize the system into a linear system. The ability
| to use a function space means that we can manipulate the
| formulation into another form before discretizing, which
| impacts the convergence theory as well as the practical
| performance.
|
| The reason that I mention Hilbert spaces is that they
| have more structure than a general function space, which
| makes working with them easier, but still general enough
| to be useful. Essentially, we get an inner product as
| well as the ability to enumerate an orthonormal basis,
| which makes it feel more like working with linear
| algebra.
|
| Even in R^n, I believe strongly that it's important to
| code with these abstractions. First, it makes
| parallelizing the algorithms easier. If we treat a vector
| as simply an array of numbers and use the dot product,
| then the code requires a more significant rewrite when
| moving to multi-computer parallelism. If we treat them as
| a generic vector object and have an interface that works
| with inner products, addition, scaling, etc, then the
| same code can work either in serial or in parallel given
| two different implementations of the vectors and their
| operations.
|
| In addition to the abstractions, the choice of inner
| product is important. If we have a linear operator in a
| Hilbert space and discretize it, we generally have to
| discretize three things: the operator, it's Hilbert
| adjoint, and the inner product. If we implement this
| blindly, they're not consistent after discretization.
| Meaning, the property that we should get is that <Ax,y> =
| <x,adj(A)y>, but this probably isn't true if not done
| carefully. Generally, we can freely discretize two out of
| three of those operations and then the third one needs to
| be adapted for consistency. Maybe you want to choose the
| discretization for the operator and its adjoint, but this
| probably requires a non dot product for an inner product.
| Alternatively, maybe you're optimizing some problem and
| realize that some of your variables are out of scale, so
| it's converging slowly. You can certainly just rescale
| your variables with a diagonal scaling. However, you can
| also change your inner product, which changes the
| gradient, which also rescales the problem in a different
| way.
|
| Mostly, that's to say that an inner product should be a
| choice that one freely and intentionally makes. Learning
| to work in Hilbert spaces forces us to become comfortable
| with this approach.
| RheingoldRiver wrote:
| As far as books go, we used "Real Analysis" by Carothers for
| analysis in undergrad at Caltech and it's one of only two math
| textbooks (the other being Dummit and Foote for algebra) that I
| go out of my way to recommend, particularly for self study.
| It's probably not a sufficient book if you're pursuing a Ph.D.
| or anything, but I would definitely not start out with Rudin
| (too little hand-holding, you'll die), and Carothers was
| amazing.
| rsj_hn wrote:
| I recommend anything by George F. Simmons, as well as
| anything by Russians. Seriously, if it's a Russian sounding
| author's name, then odds are good that it's well written.
|
| They had amazing pedagogy and the EMS series (a joint
| publication by Springer and the old Soviet publisher VINITI)
| is first rate.
|
| https://www.amazon.com/Encyclopaedia-of-Mathematical-
| Science...
___________________________________________________________________
(page generated 2021-10-24 23:01 UTC)