Travis McVoy
Resources
Self-study
Ideally, one studies under the guidance of some kind of mentor. It is not always the case that a mentor is available, though. In such a case, there are websites and textbooks one can use.
Note: I generally don't recommend YouTube as a source of self study. There are exceptions, but if you can learn to read textbooks, it's much, much faster and you severely decrease risk the chance of learning something that is incorrect (textbooks are reviewed by experts whereas any doofus with an opinion can post on YouTube).
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MIT OpenCourseWare
MIT OCW is a website that provides materials from MIT courses. Some courses have everything one could want (syllabus, readings, calendar, lecture vides, problem sets and solutions, tests etc.) while some just have minimal resources. Either way, MIT OCW is always the first place I look when finding new resources.
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The Art of Problem Solving
AoPS is a team of former math olympiads that provide training in competitive problem solving. Even if you have no interest in academic olympiads, you should consider AoPS. Their textbooks are amazingly well written in that they take a "learning by doing" approach. That is, they walk you through the steps one would take to discover theorems on their own, rather just stating and proving them. Moreover, AoPS has other resources like a list of recommended textbooks by subject, an entire Wiki dedicated to documenting concepts from various fields, and help with LaTeX.
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Course Listings
While I certainly don't go to MIT, Stanford, Berkeley or other highly selective schools, that doesn't mean I can't use their resources if they're available to me, and the same is true for anyone else. If you're interested in some subject, say probability or machine learning, there's a chance that your school (or a school you admire) might have published materials on that subject. Even if the course itself isn't available, some schools (like MIT) list out required textbooks for courses in their current semester's offerings. If all you know is the book, that might be enough to get you started. Also, schools sometime list who's teaching the course. Occasionally that person might have a website with resources on it. In general, if you want to learn about something, just assume the resources are out there and search relentlessly for them.
Other Helpful Websites
Pythontutor.com
Pythontutor provides a visualization of the execution of your code. It's a great way for beginners to start to develop an intuition for what the machine is actually doing when it enters and exits a loop or what have you.
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Desmos.com
Everyone studying any STEM subject should familiarize themselves with Desmos. Desmos is a cross between a graphing calculator and a programming language. It's great for dynamic visualizations and computations that aren't easy to do on your phone (binomial coefficients, several values of a polynomial, and more).
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GeoGebra.com
In the rare event that someone needs to work on some geometry, GeoGebra is the place to go (if you're not willing to code). Also, it can graph in 3 dimensions, which Desmos can't really do (yet).
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Putnam Archive
Admittedly, I don't think the Putnam Archive is useful generally. If you're interested in the Putnam, though, it's definitely a good place to find solutions for past exams (there's also books for older exams). Additionally, many schools publish problems from their preparation sessions (U Toronto and Northwestern in particular).
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Paul's Online Math Notes
Paul is a professor at Lamar University and has a set of notes that are useful when you need to quickly look up a formula or theorem from calculus but don't have textbook handy.
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Other
I imagine there's more that I'm forgetting. I'll try to update this page as I think of them.
Textbook Recommendations and Advice
I don't have the time as I'm writing this to recommend every textbook that's ever helped me understand something. I can, however, offer advice about how to find textbooks and how to read them.
Finding Textbooks​
I find textbooks either by asking a professor, or more commonly, by referring to the course I wish I had access to. Many schools like MIT, CMU, Princeton or what have you quietly list information about the textbooks they use. For example, if you want to learn about physics, you could google, "CalTech Physics Major" or "CalTech Physics Courses" or "CalTech Physics Course Offerings" or "CalTech Physics Course Registrar". Ultimately, you just want to find a list of course numbers and names. Sometimes the books are listed in the description, sometimes not. If not, your next step is to google "<School> <Course Number> <Course Name> syllabus". A course syllabus is your treasure map. It will tell you what textbook you need, and often what sections you should read. Occasionally it might even include problems you should complete and other helpful resources.
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Websites that have helped me find books:​
AoPS: https://artofproblemsolving.com/wiki/index.php/Math_books
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Books I've Enjoyed by Category
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Physics
Classical Mechanics by John Taylor
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Algebra
Any Linear Algebra book by Gilbert Strang
Abstract Algebra by Dummit and Foote
Visual Group Theory by Nathan Carter (don't use this as your main text; great supplemental text)
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Differential Equations
Differential Equations and Boundary Value Problems by Edwards and Penney
Nonlinear Dynamics and Chaos by Steven Strogatz
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Probability & Stats
Too many to list here. See my probability project. As I do more stats, I'll update, but you can also see my syllabi collection...when I finally get around to updating it...
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Machine Learning
Understanding Machine Learning: From Theory to Algorithms by Schwartz (note: not for beginners)
Machine Learning Refined by Jeremy Watt
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Graph Theory
Introduction to Graph Theory by Douglas West
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Econometrics
Econometric Analysis by William Greene
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Analysis
Understanding Analysis by Abbot is a decent introductory book.
Real and Complex Analysis by Rudin (much more advanced)
Real Analysis by Royden (also advanced)​​
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There are more, but I've exhausted this distraction enough (for now). At some point, I should really just create some dynamic document that I can link to.
How to read textbooks
The most important advice pertaining to reading textbooks is this: some books will work for you, some will not (and that's okay). In the subjects I really care about, I typically have 3-5 books. It's not uncommon that I come across a theorem that my favorite book explains in a way that I find unintuitive. When that happens, I check my next favorite book and so on until the idea clicks. Sometimes a slight change in word order or notation makes all the difference.
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Syllabi
The importance of syllabi cannot be overstated. Professors—good ones at least—put serious consideration into how to organize a course and will often include a textbook that closely (or sometimes exactly) follows their lectures. By following their recommended reading, you have a map that directs you where to go.
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Table of Contents
I bookmark the table of contents of every single textbook I ever use (even the books I don't like). The table of contents are the skeleton of the book and will help you navigate the material. Additionally, it's good to get a preview of what's to come and while some books address this idea directly, it's often sufficient to periodically skim the table of contents.
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Index
An index is sort of like a dictionary for your textbook. If you happen to have a PDF, you can usually do a key-word search when looking for a topic/theorem/definition etc. but the index is a good alternative. The index has a list of just about every major idea, key-word, theorem, or really anything you could need and the pages that those key words/ideas appear on.
Relationship Between Index and Table of Contents
Sometimes ideas are important enough to recur throughout the book. Say, for instance, you are using a calculus book and you need to look up some idea pertaining to derivatives. It's likely that there are dozens of listed pages (or even sections) in which the derivate appears in. If you know the table of contents, you don't need to check every single appearance listed in the index, you can quickly guess which pages will be most useful to you.
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Proofs
Proofs are the backbone of everything. If you don't know how to study some sort of mathematical subject, start with understanding the proofs.
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How to read proofs
Math is not a spectator sport. If you really want to understand a proof, at some point you should probably just derive the whole thing yourself. This does not mean you can't use the book. Quite the contrary, you should absolutely use the book, but reading something and writing it out in your own words are not equivalent. I'll not that mathematicians can be very lazy people and sometimes rely on ideas that are obvious to the initiated, but might be confusing to everyone else (which, to be clear, is fine; I do that myself, as do many of my peers and superiors). If you're working through a proof from a book and find yourself frustrated with an author that's skipping steps (or even if you just don't like their wording or notation), try another book. I usually have 3-5 books for a subject I really care about, and it's not uncommon for me to jump back and forth between books when I'm working on a proof.
Syllabi Collection
I already blabbed about syllabi, but they're so important that I decided they deserved their own section. In particular, I want to emphasize the idea of distributions. If you can find a syllabus for some course, say, mathematical statistics, across multiple different schools, you begin to see what works and what doesn't when it comes to learning a given topic. You can then begin to form a distribution of textbooks used and topics covered. More often than not, you'll find that a certain book (or a collection of books) begin to reappear over and over again. For mathematical statistics, one such book is Statistical Inference by Casella and Berger.
Careerism Stuffs
Some people seem to naturally understand the subtleties of writing a resume, answering interview questions, writing cover letters, etc. If that's you, these resources probably aren't necessary. If, like me, you hate all of that and just want to build things, you might find these resources as a decent starting place when you inevitably have to network, interview, apply etc.
Harvard's Resume/Cover Letter Guide.
The career center at Harvard has an entire website dedicated to details about how to look for a job, write a cover letter, etc. The website is designed for Harvard students, but there are tons of free, public resources available as well, including their resume guide.
Professors' Websites
Many professors have their own website, some of which contain advice and resources. For example, Dr. Steve J. Miller from Williams College has an excellent website.
MIT Admissions and Blogs
MIT hosts a blog page on their admissions website that is full of career and academic advice. Many of them are also quite funny. The posts are sometimes specific to MIT, but several of them contain advice that's just good for life. The "Applying Sideways" post gave me a lot of perspective as a teenager that I still appreciate today. If you're looking for advice on something specific, like imposter syndrome or studying physics, you can try to find it with "MIT Blogs Imposter Syndrome" or "MIT blogs physics" etc. The general admissions page is also quite good. Play around and see if anything helps you. I highly recommend the, "You should really read these" posts, which are found on the page I've linked below.
Contacting Professors about PhD Programs
I'm still learning how to email professors from other schools in an effective manner. As I find resources on this topic, I'll include them here. For now, below is some advice from someone at Georgia Tech:
Applying to Grad school or REUs
DISCLAIMER: These are my opinions and should not be treated as professional advice.
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There are many different links I've found that have helped me write my applications. I'll try to update them as I find them. As a summary, I'd point out a few key ideas:
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* REUs and Graduate school are funded experiences, and there isn't infinite money available. Your job is to convince professors that the work they do to achieve funding via grants won't be wasted on you should they choose you for their program.
* Use concrete examples of your successes. For instance, if you studied a subject on your own outside of class, don't say, "I studied subject X." Instead, use action words and tangible examples that professors will be familiar with. If you studied probability on your own, for example, you might say something like, "In my self study of probability I completed 10 out of 10 problem sets from MIT's OpenCourseWare platform and read every single section of their recommended reading. Additionally, I typed out 100 pages of notes in LaTeX by writing major ideas in my own words. Such ideas include (but aren't limited to) the proof of linearity of expectation and the proof of the central limit theorem."
* Suppose you think of application strategies as offensive and defensive. Offensive strategies are strategies in which you convince the reader why you're significant. Defensive strategies are proof reading, efforts of completeness, and really avoiding anything that might make you immediately rejected. I recommend prioritizing defense over offense. In today's world, there are usually more qualified applicants than there are spots, so even if you are perfectly qualified, you can be eliminated if you have avoidable errors (titling your essay to Harvard when you're applying to UCLA, for instance).
Here are links that have given me some advice. I hope they are useful to you. If you have links that you think I should include, please email me and I'll consider uploading them :)
LaTeX
Anyone studying anything even remotely mathematical [1] should, at some point, learn to typeset using LaTeX. If the reader is not familiar, LaTeX allows one to type beautiful documents with mathematical symbols. Most textbooks and assignments at the college level will be written using LaTeX.
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One problem with LaTeX is that it appears harder to use than it actually is. The best way to get good at using LaTeX is just to start. Many start with Overleaf, as it's an editor you can use from your browser. I personally use TexMaker, but you can use whatever editor you like. Some people use VS code, and some use VIM (which is probably what I'll switch to someday).
One useful thing to know about LaTeX is that it has been around for a long time (since 1984). Consequently, whatever problem you're having has almost certainly been someone else's problem, and likely has a solution somewhere on some forum. Whenever there's something I want to do but don't know how, I usually find an explanation with a single google search. Some of those searches might be, "How to align text in LaTeX", "How to draw graphs in LaTeX", "What is the TikZ package?" etc. Sometimes the answer is on a forum, sometimes it's in Overleaf documentation. Regardless, there will be a solution if you look hard enough.
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I'm working on a guide for TikZ (which is how you draw shapes, graphs, functions, and more) because we aren't required to learn TikZ at my school (though we really should be). It's incomplete, but it's a starting point one can use to get familiar with what TikZ can do (which is quite a lot).
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Notes:
1. Just about every subject remotely scientific will have some math at some point. Being able to present it cleanly using LaTeX will set you apart, especially in subjects where most people just use Word.
Professors' Websites (cont.)
Occasionally I find something that I think is worth saving, and I don't know where to put it. For now, this will do. Some of these resources might help people, but honestly, this is more for myself. It's convenient to have everything in one place that's accessible anywhere from any device with an internet connection.