Self studying the MIT applied math curriculum (2019)
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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
I'll certainly check out your site and save it for when the world is more peaceful.
Luckily he left a link for the pure maths course.
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
You both seem to be solving similar problems.
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...
One major thing I found when learning at a university that I have trouble with when self studying is pacing.
Let's say the textbook covers some topics A, B, C, D, and E in that order. When self studying I start with A, and keep working on A until I'm confident and comfortable with it, then move on to B, and so on.
In school we'd spend a while on A, but then usually move on to B before I was fully comfortable and confident with A. Same for B to C, and so on.
This was fine, because while I didn't think I was ready to move on the instructors had a lot of experience with teaching this material and knew that we students knew enough A to understand B, and in fact that exercising our A to help us learn B would also strengthen our A.
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.
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.
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.
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.
Homework type questions (anything before PhD student level) are typically asked on https://math.stackexchange.com/
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.
Thanks for the insightful comment by the way.
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.
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.
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...
Honestly, I think that real analysis is the largest barrier for people trying to self study applied math. I'll mention here that I actually had a total of five semesters in real analysis all said in done. The first was an intro to real analysis course that used "Introduction to Real Analysis" by Bartle and Sherbert. The second was a more advanced real analysis course that was two semesters long using "Real Analysis and Foundations" by Krantz. When I entered my Ph.D, I was given an initial evaluation and the committee found my real analysis still too weak, which necessitated another two semesters using baby Rudin. Only at that point, did it really take and it's not been an issue since.
Anyway, mostly I wanted to echo that I agree with you. I think Rudin is the metric to say good enough, but I was not able to get through it on first go.
I ended up enrolling in the math 23 sequence at the Harvard Extension School and found it to be a lot more valuable.
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.
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.
I guess now I know where his time went?