10 ms·
Self Studying the MIT Applied Math Curriculum
- mpoteat 7y ago7 non soft classes in a single summer is ambitious to say the least.
- hsikka 7y agoYeah, 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!
- Funes- 7y agoI admire your determination, really.
- deleted 7y ago[deleted]
- barry-cotter 7y agoWhat are you Mastering in? What’s your research on?
- hsikka 7y agoI'm studying CS and Biology, and I'm currently researching disentangling object representations using brain inspired auto encoders
- hooloovoo_zoo 7y agoIt's ambitious, but it reads like he's taken most of them already.
- deleted 7y ago[deleted]
- AnthonBerg 7y agoTrue. However there is also this perspective: https://sivers.org/kimo https://sivers.org/kimo
- leereeves 7y ago> In our three-hour lesson that morning, he taught me a full semester of Berklee’s harmony courses. I don't know anything about music theory, but if a course can be completed in three hours, it's a soft course.
- mikorym 7y agoI think that music theory lends itself to a pace that can be dramatically increased and at the same time learnt successfully. I don't think math works like this. I specifically decreased my pace when doing math courses that I hadn't done in undergraduate and I am glad that I took that approach. I now see any problem (given enough time) as more of a research question than just a box to tick.
- miguelmota 7y agoA good quote from Sivers' post: > He taught me “the standard pace is for chumps” — that the system is designed so anyone can keep up. If you’re more driven than most people, you can do way more than anyone expects. And this applies to all of life — not just school
- olooney 7y agoI 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=PL49CF3715CB9EF31D https://www.youtube.com/watch?v=ZK3O402wf1c&list=PL49CF3715C... [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-for-applications-fall-2016/assignments/ https://ocw.mit.edu/courses/mathematics/18-650-statistics-fo... [3]: https://www.amazon.com/Computations-Hopkins-Studies-Mathematical-Sciences/dp/1421407949/ref=cm_cr_arp_d_product_top?ie=UTF8 https://www.amazon.com/Computations-Hopkins-Studies-Mathemat...
- the_svd_doctor 7y agoAnother 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-Trefethen/dp/0898713617 https://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefet...
- deleted 7y ago[deleted]
- laserson 7y agoStrongly 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.
- colmvp 7y agoI like Strang and have worked through some of his books but when I was first learning Linear Algebra, I preferred learning from Dr. Aviv Censor's videos on YouTube[0] from when he was teaching at Technion. It's a few dozen videos where he spends a lot of time going through each component of an introductory course on Linear Algebra. [0]: https://www.youtube.com/watch?v=aefKXYYXT6I&list=PLW3u28VuDAHJNrf3JCgT0GG_rjFVz0-j9&index=1 https://www.youtube.com/watch?v=aefKXYYXT6I&list=PLW3u28VuDA...
- itsbenweeks 7y agoA lot of these earlier courses are available on edX, too. The platform won't grade you unless you pay, though.
- adamnemecek 7y agoWhile 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.
- hsikka 7y agoI've been meaning to invest some time into learning Julia. I'll definitely try my hand soon
- sn9 7y agoThe newest iteration of 18.06 [0] is using this textbook [1] that incorporates usage of Julia in a companion book. [0] https://github.com/mitmath/1806/blob/master/summaries.md https://github.com/mitmath/1806/blob/master/summaries.md [1] http://vmls-book.stanford.edu/ http://vmls-book.stanford.edu/
- sus_007 7y agoWow, I didn't know these resources existed. It's gonna help me a lot as I'm on a similar journey as the OP. Thank you very much, sn9
- bigred100 7y agoWhy is Julia better than MATLAB or Python? Most applied mathematicians I know are in MATLAB all the time.
- adamnemecek 7y agoI 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 7y ago
- usgroup 7y agoGood 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.
- skookumchuck 7y agoIt'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.
- mehh 7y agoHave to agree!
- afarrell 7y agoTo 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!
- throwawaymath 7y agoSpeaking 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 7y agoHey, author here, I really appreciate the feedback. Are there any specific prerequisites to Real Analysis?
- throwawaymath 7y agoTechnically 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.
- hsikka 7y agoFantastic, thank you for the tip! I'll probably prepend Real Analysis to the list!
- 7y ago
- commandlinefan 7y agoI’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.
- hsikka 7y agoAwesome, it's heartening to see someone else feel the same way! And great note about the problems, maybe one of the classic textbooks would be a good addition for problems
- adamnemecek 7y agoLook into dual numbers and automatic differentiation, it's like Calculus on steroids.
- mkl 7y agoCan 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"?
- moinnadeem 7y agoMIT student here, there's really no rhyme or reason. Generally, higher numbers mean more advanced courses, but that's about it.
- augustt 7y agoThe first digit is the general area: 18.0x are mostly intro classes, 18.1x are analysis classes, 18.7x is algebra, etc.
- amirhirsch 7y agoTwo 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 7y agoThanks!
- afarrell 7y agoThere is no strict system after the “.” however the 3-numeral course numbers tend to be advanced/grad classes. There are some smaller patterns within each major but nothing knits those together. 6.82x tend to be systems classes like networks or OS. But 17.828 isn’t like an intro to constitutional law and 6.42 isn’t Causes and Prevention of Software Project Failure.
- graycat 7y agoSuggestion: 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.
- hsikka 7y agoIncredible, thank you for the guidance! I'll look into these topics more carefully
- graycat 7y agoComplex valued functions of a real variable are common and important but well covered in real analysis. Functions of a complex variable I've never seen in practice. A good course in statistics, e.g., with sufficient statistics, needs graduate probability, and with a grad probability course can do elementary statistics easily in the footnotes. E.g., the proof I worked out for the Neyman-Pearson lemma is quite general but based on the Hahn decomposition based on the Radon-Nikodym theorem (Rudin gives von Neumann's cute proof) in measure theory. The grown up approach to conditional probability is based just on the Radon-Nikodym theorem of measure theory.
- verylongname 7y agoPersonally, 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.
- gumby 7y agoGreat 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?
- joflicu 7y agoOne 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.
- c0vfefe 7y agoI hope you get some good answers, but just consider that you'll never achieve satisfaction by having infinite goals (in any part of life). All the tips and self-discipline in the world might only take you from X to 10X, and somewhere down the line you'll have to learn to be okay with that. The alternative is to have the health of your life & the lives of those around you be consumed by ambition and its pursuit.
- moosey 7y agoIf 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.
- martincmartin 7y agoIs 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.
- dxbydt 7y agoIt’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 7y agoThe 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 7y agoit’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
- 7y ago
- sara7262 7y agoThey found world's biggest fish Even you can't believe how big fish is this That's unbelievable http://bit.ly/2FHl4pM http://bit.ly/2FHl4pM Even some goats are better than humans See that she did and they made her as mayor of city http://bit.ly/2UjTcS5 http://bit.ly/2UjTcS5 See the video a dog shopping and lots of other acts like humans http://bit.ly/2DCpo9I http://bit.ly/2DCpo9I Even you can't believe there is a hen in world's biggest animals See the video how they looks and what they do https://cutt.ly/1rOU7Q https://cutt.ly/1rOU7Q Fish walking with legs like humans See the video on link below http://bit.ly/2XRGyYD http://bit.ly/2XRGyYD
- mikorym 7y agoThis 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.
- sparcraft 7y agoI 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.