My advice would be to get a book on any topic which interests you, read through it, and do a significant number of the exercises. You might try Epp's Discrete Mathematics, Hefferon's Linear Algebra, Colley's Vector Calculus, Dudley's Elementary Number Theory, Spivak or Apostol's Calculus (these go far beyond ordinary freshman calculus), Pinter's A Book of Abstract Algebra, among many others. Some of these books are expensive to buy new, but just buy older editions.
Resources like Khan Academy and 3blue1brown are also fantastic, and I have shared some of these with my students. I'd recommend using these as a supplement; if you rely on them solely then you'll develop vague intuition but not much else.
Also, with the pandemic, there are a huge number of traditional university courses that have moved online, and you could probably enroll in one for not too much money. Check the RateMyProfessor reviews -- you want a mix of positive reviews claiming the prof inspired them, and negative reviews complaining that work was expected. I have a RateMyProfessor review which complains bitterly that "homework is graded for accuracy and not completion". :)
In the (simple!) maths I took, my misunderstandings often led me to do exercises incorrectly, but to think I'd done them right, or to have half a proof and no idea what insight I was missing. Even when solutions were provided (which was rare!), the path connecting the dots could be foggy. "What made them think to introduce this lemma?" kind of questions.
I often needed discussion with others to help figure out my misconceptions and correct them -- so... big ask, but do you have any further suggestions on how to replace talking with a TA, or with a smart group of upper-year friends?
Since the parent mentioned my text, Linear Algebra, I'll note that it is Free, and comes with worked answers to all exercises, including proofs. And since other posters mention videos, I'll also note that there are videos (and the slides are available), along with a lab manual using Sage. https://hefferon.net/linearalgebra/index.html
This is difficult. One way to do this is to participate on Math Stack Exchange:
https://math.stackexchange.com/
I'd recommend not only asking, but also answering others' questions. It's not the same as having a group of people that you regularly talk to, but it's something.
You could also hire a graduate student tutor. For example, here is a list of math grad students at the University of Wisconsin willing to do this:
https://www.math.wisc.edu/tutors
An hour every two weeks would go a long way.
This discord server in particular is good if you want help, although you might not recieve instant help, you will eventually get help if you are patient. There are also other channels in the server just for the purpose of discussion, i.e discussing general topics in math.
This might not give you what you are looking for, but it's a possible alternative I guess.
First, the amount I learnt on a day-to-day basis from self-studying in my free time pales in comparison to the amount I learnt from a structured course ending with an exam, in which you have to commit to studying 2~4 hours of math every single day.
Second, classroom lectures mainly act as a structured table-of-contents for working through various textbooks. The problem sheets they distribute in the tutorials are very helpful, as is the discussion of problems that some students might have got stuck in.
Graduate-level math textbooks are fantastic, and the authors have poured 10+ years of their lives writing them. No online course with cute interactive video can substitute them. Having said that, there are some video lectures that can act as good motivation for studying a subject. For instance, Wildberger's lectures on algebraic topology (find them on YouTube) are a great starting point for the subject.
You first have to decide on what you want to learn, by looking through various fields, and narrow your scope. Mathematics is a huge discipline, and you have to decide what you like, and work through the prerequisites in a disciplined manner.
I cannot emphasize the importance of working through exercises enough. All mathematics textbooks have exercises at the end of each section, a subset of which you must work through.
Having said all this, you might find the following site interesting: it contains a lot of unpolished notes from studying various math textbooks, but it requires a lot more work to be useful to a larger audience. Look through it, and pick something you like. On request, I might find the time to add some material.
I am just curious about your process of actually joining the masters program. They usually require 2-3 letters of recommendation etc. How did you handle such requirements?
> Second, classroom lectures mainly act as a structured table-of-contents for working through various textbooks.
On a side note, actually studying math as an undergrad, I can't stress enough how much I agree with this, especially in a remote setup.
1. Spivak's Calculus.
2. Johnstone's Notes on Set Theory and Logic.
3. Gamelin's Complex Analysis.
4. Hoffman & Kunz' Linear Algebra.
5. Dummit & Foote's Abstract Algebra; just the group theory.
6. Munkres' Topology; just the general topology.
Once I was happy with my preparation, I strived for a deeper understanding of Group Theory. I bumbled through Herstein, but didn't understand it very well. Then, I stumbled upon Artin's book, and worked through it using the outline provided on the MIT OCW course page, and I could confidently solve most of the exercises.
For category theory, the top resources that I would recommend are:
1. Mileweski's Category Theory for Programmers, the video lecture series. As is always the case with video lectures, this one can help motivate a Haskell programmer uninitiated in category theory.
2. Goldblatt's Topoi. It's fairly dated, but teaches category theory well, via its application to topoi.
3. MacLane's CatWork. I'm not especially fond of this one, but it's necessary to work through it.
The most important thing to understand when learning category theory is that it cannot be learnt in a vacuum: the subject is entirely vacuous, and you need to use it in other mathematical disciplines to give it meaning.
Good luck.
Merci!
In mid-2017, I moved to California and spent ~2 hours a day studying by myself, while maintaining a day job writing Haskell and Coq. Then, in late-2019, I quit my job and moved to India to study mathematics full-time (read: 4~6 hours every single day). I started meeting a professor at the local university once a week to discuss my solutions to problems in Miles Reid. We worked through Reid together. I also audited a course in algebraic topology at the university, simultaneously.
In mid-2019, I moved to Paris to continue studying part-time (1~2 hours a day) while working on a Coq project. After some shuffles, I found a professor I really liked, and we started working together. He wrote me the primary recommendation for the Masters program.
I hope the elaboration was more useful than two numbers.
I'm a firm believer that you don't truly understand material unless you also understand a proof for it. Ane while those resources are AMAZING for simple intuition building, they aren't sufficient.
Go pick up a book about proofs to develop mathematical maturity and then delve into some undergrad books for analysis (Abbott) or linear algera (Axler)
However, I don’t think you that means you should skip watching 3B1B videos at all. In fact, Grant repeats quite often in his videos that they’re not a substitute for doing problems.
I’d highly recommend watching the videos side by side with reading through Axler. They will help develop the right geometric intuition (which Axler does a decent job of, but books can only do so much as a medium). I believe the 3B1B videos can also be quite motivating, which in my experience is the most important factor when self studying.
I'd say the same is true for proofs. Simply reading a proof isn't the best if you want to understand/retain it. If you can "re-invent" the proof step by step (or in some contexts you might be able to translate steps between geometric and algebraic representations), then you have probably internalized it sufficiently.
I was wondering how you feel about Professor Leonard? He's pushing recordings of his college classes online on YouTube. It's a veritable trove of useful lectures. [1]
I would advise following a more traditional path, with a good book, but I cannot think of any good reason to not take a break from the book to watch some related 3B1b video from time to time.
Here's another request: Terrence Tao does a recording of a measure theory course.
Personally I find the Khan Academy, 3B1brown stuff to be a waste of time. It's too elementary and cutesy. That is not anything like what I've experienced real mathematics to be like. It is a continuous painful struggle.
[1] https://www.youtube.com/watch?v=ab41LEw9oiI&list=PLun8-Z_lTk...
Sorry, but what?
You'd be hard pressed to find any single person, in the past 10 years, that has ushered more people into STEM by providing a solid fundamental education, than Salman Khan.
I understand that you're wanting more rigorous math, on a higher-level, but still - to blow off KA, 3B1, etc. as a waste of time sounds more ignorant than anything. Of the millions and millions that have seen these those guys, I'm sure a certain % have gotten their "Aha!" moments there, which have helped them tackle on much more comprehensive topics / fields.
edit: Just to clarify, I do not think these vids are any replacement for a solid book, just in the way I do not think lectures or tutorials are a replacement. But they can be fantastic aids for your learning.
Interactive websites are IMO vastly more efficient at conveying information, compared to static pages (like books, for example).
I mostly fudge together what I need from that and Khan Academy when I’ve forgotten something fundamental.
I don't maintain a collection of these, but I can probably recommend
* Stephen Abbott, Understanding Analysis, for mathematical analysis and a little bit of real analysis
* David Williams, Probability with Martingales, for advanced probability
* Dym and McKean, Fourier Series and Integrals
Alternatively if you have a more cursory interest, you can probably check out Metacademy [1] which provides roadmaps to learning a variety of math (and ML) concepts.
If you like learning math by programming https://greenteapress.com/wp/think-complexity-2e/ also free.
Mit Open Courseware also free.
Perhaps a hands-on approach such as solving the exercises while working through this book will prove beneficial and is complimentary to watching, say, KA or 3B1B.
[1]: https://www.amazon.com/How-Prove-Structured-Daniel-Velleman/...
Each section has a 1) written explanation, 2) video explanation with an exercise walkthrough and a 3) set of exercises. Each exercise also has a well-written walkthrough in case you can't solve them.
I've been working through them part-time for 4 months now, and it's honestly the first time that maths has clicked for me (I've found that most books tend to skip concepts which leave me lost).
[1] https://www.udemy.com/user/kristaking/ [2] https://www.kristakingmath.com/
Mathematical Techniques: An Introduction for the Engineering, Physical, and Mathematical Sciences by Jordan and Smith - This book is really accessible.
Mathematical Methods for Physics and Engineering: A Comprehensive Guide by Riley, Hobson, Bence - A more advanced coverage of topics than the one above.
Mathematics for Physicists: Introductory Concepts and Methods by Altland and Delft - The coverage is excellent but somewhat challenging to read.
On Coursera[0] there are a lot of courses on math[1], including on discrete math[2].
[0] https://en.wikipedia.org/wiki/Coursera
[1] https://www.coursera.org/courses?query=mathematics
[2] https://www.coursera.org/courses?query=discrete%20mathematic...
Maths is a very written-oriented domain, but still if you are more a video-oriented learner you can probably find a lot on Youtube, and a lot of great researchers have videos there.
It’s not perfect, but it is pretty good — the single largest change I would make is “more practice examples for any particular topic”.
Besides that good math books are usually very cheap. I would highly recommend those over a website.
https://www.youtube.com/channel/UCafIamqsRUHoRT4496jgvMQ
And I'm also working on a website that has a fair bit of the foundations:
Admittedly, we don't have a huge amount of the higher stuff yet. But we are making our way through steadily :) Hopefully it's of some help to you
"Enrol Now" should be "Enroll Now" :)
