Resources

If you find yourself asking, "What book should I get to learn $X$?", then you have come to the right place! This is a question I personally get asked a lot, and one that seems to be posted almost daily on several online Math communities, so I put together this list to point out some resources I’ve personally used and found helpful for courses and various other topics.

Disclaimer: in the textbook section, a number of the outgoing Amazon links are affiliate links! These are meant to help cover the hosting cost of this site, but you should also be aware that many of these books are often available at university libraries, at cheaper resalers, or often even as PDFs floating around online.


Advice

For Incoming Math Majors

  • Start thinking about whether or not grad school as an option. You don’t have to decide within your first year, but you’ll want to pick courses accordingly.

  • Get an idea of what Math is all about - there are great things beyond Calculus!

    • To get an idea of what you’re in for as a Math major, take a look at the GRE Mathematics Subject Exam. It covers a broad array of pretty standardized topics.
    • Also take a look Garrity’s book (linked in the resources section) for a short, condensed survey of some of the major topics and theorems you’ll encounter.
    • One thing that was helpful to me was to make graphs and diagrams of different areas of Math I encountered. Take a look, knowing the landscape you’re traversing is valuable.
  • Learn Latex and Mathjax early, and then make efforts regularly typeset your written notes.

  • It may be intimidating, but try to find ways to talk to your professors outside of class! Don’t hesitate to ask for advice on just about anything related to Math or your academic career, in my experience they are always happy to help.

  • For those aiming for grad school, particularly in pure Math: here are what I would consider some of the most essential, core classes to take:

    • Calculus (of course)
    • Linear Algebra
    • Ordinary Differential Equations
    • Abstract Algebra
    • Real Analysis
    • Point-Set Topology

    Here are some “nice-to-haves” – still important, still beautiful, and great electives, but missing out on them in undergrad isn’t the end of the world:

    • Probability
    • Combinatorics
    • Complex Analysis
    • Number Theory
    • Discrete Math (e.g. Graph Theory)
    • More Linear Algebra (e.g. Numerical Analysis)

    Of course, I recommend taking as many Math courses as possible – explore the subject and discover what you like! But the ideas and concepts from these particular courses are relevant to just about any area you might go into, so having some exposure to them makes life much easier (regardless of what you pursue).

On Learning and Being a Student

  • Start everything early: Starting early gives you time to mull over questions and potentially visit professors/TAs for help.
  • Use a wide variety of resources: Books, lectures, notes, PDFs – use them all! The search modifiers inurl:edu filetype:pdf on Google are incredibly useful.
  • Extensively use good resources: There is also some value in working a single, particularly well-regarded book, front to back.
  • Spaced repetition: read up on it, and work it into your study habits. Cramming has diminishing returns in proof-based classes, you are much better served by visiting and revisiting material often enough to be familiar with it.
  • Sleep like it is your hobby. Two hours of extra sleep is almost always a more efficient use of your time than two hours of tired studying.

On Learning Mathematics Specifically

  • Vocabulary: for any new topic, it can be helpful to make a list of definitions and simply memorize them. Math is (at least partially) a language, so it is worth (partially) learning it like one!
  • Use multiple books: For most undergraduate topics, there are many “standard” references. Use them all! Cross-reference liberally as well.
  • Get the definitions down cold: Record every single definition you come across and just do whatever you have to do to memorize them! Some people like flash cards, I personally make a “dictionary” document for each subject. Meaning and understanding is often a gradual process, but knowing the full statement of a definition should just be automatic. I like to think of definitions as the analog “multiplication tables” within higher Mathematics – is it essential to memorize? Well no, but you also wouldn’t want to waste time stopping to multiply out $12^2$ when you’re trying to solve an integral.
  • Synthesize often: This includes things like revising class notes, making your own “cheat sheets”, writing up explanations of particular theorems or motivations for certain concepts, etc. Even better, find ways to give talks or teach other people! Present things as if you were trying to teach yourself the topic 6-12 months in the past.
  • Google for PDF notes: For just about any topic at any level of specificty, someone has probably written up notes or expository explanations and shared them online. So it can be extremely beneficial to look through the first page or two of Google results using filetype:pdf for any concept or proof you happen to be studying and see how other mathematicians think about and present them.
  • Work a lot of problems/proofs: Do more than you are assigned, but be judicious with your time! If you stare at a problem for more than an hour or two without making progress, pivot. Sometimes the best way to make progress on a problem is to take a break, work on something different, or sleep on it. A relevant quote from Alexander Grothendieck:

    I can illustrate the approach with the image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!

  • Work the exams, and if possible, do them in a timed setting.

Math Course Resources

General Notes on Resources

Anything lower-division: Khan Academy.

  • On Youtube videos: more and more lectures and talks are being posted online. These vary in quality, but this if offset by the fact that you can watch them at 2x speed.

  • On video lectures in general: they are a crutch! Use them to supplement and enrich your understanding, but they do not work well as primary resources. The best resources are always materials from your own instructors and courses. Passively watching videos is also not very helpful - take notes, write down timestamps and video titles for important topics, and don’t hesitate to rewatch several times.

  • MIT OCW: when video lectures are available, generally high quality. Covers a good number of areas, and a good source of practice problems and exams.

  • Look for book recommendations for your topic/class in places like Math StackExchange (MSE) or MathOverflow (MO), and don’t be afraid to consult multiple books on the same topic. I’ve had very good luck following the reference-request tags on these sites.

  • If you find an author/lecturer/general source that you particularly like, look up other things from that source! If they cover other topics, be open to learning those – Math is quite interrelated, and later topics will be easier to learn when you have a collection of concepts you can “hang your hat on”.

  • For lower division or introductory courses, try to obtain a textbook paired with a solutions manual. This is a great way to drill problems, check your answers, and identify your weaknesses. Beware typos!

  • Don’t bother with sites like Chegg.

  • For computational courses: Schaum’s Outlines can be useful, but the quality is very subject-dependent.

  • Learn how to use a Computer Algebra System (CAS). Don’t use it as a crutch - just learn enough syntax so that you can quickly run “sanity checks” on your computations. Wolfram Alpha is a good for quick things.

    • If you have the time, learn something like SageMath, which can do symbolic computations and can even be used for things like group theory and algebra.

Below are resources for specific courses/subjects; I’ve tried to roughly organize these by increasing complexity with respect to a typical undergraduate Math degree.

Legend

  • – Textbooks
  • – Videos or online lectures
  • – Websites or other online collections of resources
  • – Particularly excellent resources that I highly recommend
  • Details – Expand these sections for notes on the mathematical content or subject-specific advice.

Single Variable Calculus

Content

  • Todo Notes
  • Todo

Multiviarable / Vector Calculus

Content

  • Todo

Notes

  • Many Calculus books cover both single and multivariable, so most of the resources from the single variable section will be applicable here as well. However, the resources in this section focus almost entirely on multivariable cases.

Linear Algebra

  • Axler, Linear Algebra Done Right

    • This subject is usually taught as a bag of computational tricks and algorithms, which obscures the absolute beauty of the subject – this text motivates the theory nicely and shows how powerful it can be.
  • Goode and Annin, Differential Equations and Linear Algebra

    • Good balance of rigor vs. brevity and computation vs. theory. Very concise, gives you what you need to start calculating, but also takes time to list vector space axioms, mentions fields, a nice way of viewing the determinant formula, and (best of all) lists of many conditions that are equivalent to a matrix being singular or non-singular.
  • Anton, Elementary Linear Algebra

    • Has an entire chapter on many cool applications of Linear Algebra – things like graph theory, computer graphics, and Google’s Pagerank algorithm. Also has a lot of “historical note” blurbs that are pretty interesting.
  • Strang, Introduction to Linear Algebra

    • Strang is a giant in the world of linear algebra, so it’s worth seeing how he approaches the subject.
  • MIT OCW: Linear Algebra with Gilbert Strang

    • Again, it’s Strang, so worth checking out!

Content

  • Todo Notes
  • Todo

Ordinary Differential Equations

Content

  • Todo Notes
  • Todo

Discrete Mathematics

Content

  • Basic logic (truth tables, quantifiers, implications, contrapositive and converse, induction)
  • Basic set theory (set-builder notation, venn diagrams, cartesian products)
  • Functions (injectivity/surjectivity, inverse images)
  • Relations (partial orders, equivalence relations)
  • Basic combinatorics (permutations and combinations, inclusion/exclusion, pigeonhole principle)
  • Recurrence relations
  • Graphs (Konigsberg problem, Hamiltonian/Eulerian cycles)
  • Number theory (divisibility, modular arithmetic, the Euclidean algorithm)
  • Generating functions
  • Probability (odds for dice rolls and cards, distributions, Stirling’s approximation)

Notes

  • Highlight equation aspect of proofs, ties between recurrence relations and ODEs TODO

Number Theory

Content

  • The prime counting function
  • Modular arithmetic, solving equations in rings, multiplicative functions (like the totient)
  • The Chinese remainder theorem, Euler’s theorem, Fermat’s Little Theorem
  • Quadratic reciprocity, the Legendre and Jacobi symbol

Notes

  • This is a subject that pairs very well with introductory abstract algebra (groups and rings).

Abstract Algebra

  • Dummit and Foote, Abstract Algebra

    • Essentially the de-facto standard, plus it also serves as an encylopedaic reference.
  • Beachy and Blair, Abstract Algebra

    • A good undergraduate-level reference.
  • Matthew Salomone Abstract Algebra Series

    • Fantastically well-motivated series, covers the equivalent of an entire year of material that naturally leads into Galois theory.
  • Benedict Gross, Lectures at Harvard

    • Very clear with lots of examples.
  • Math Dr. Bob, Abstract Algebra

    • I didn’t find these particularly useful, but I know people that have.

Content

  • Covers groups, rings, fields, and Galois theory.

Notes

  • Todo

Galois Theory

  • Matthew Salomone, Algebra Lectures
    • A portion of his Algebra series, the exposition is fantastic because the series follows a cohesive narrative that introduces some of the major results and benefits of Galois theory early on. Highly recommended.

Content

  • Todo

Notes

  • Todo

Real Analysis

Content

  • The reals as an ordered field, construction
  • Metric spaces, basic topology
  • The Riemann-Stieltjes Integral
  • Sequences and series, Cauchy sequences and completeness
  • Limits and continuity, pointwise and uniform convergence
  • The Mean Value Theorem
  • Measure theory and the Lebesgue integral

Notes

  • Todo

Complex Analysis

Content

  • The complex integral
  • Residue theorems
  • Analytic vs. Holomorphic vs. Complex Differentiable functions

Notes

  • Todo

Partial Differential Equations

Content

  • Heat/wave/Laplace equations. Notes
  • Todo

Numerical Analysis

Content

  • Fixed points and root-finding, Newton’s method, least squares.

Notes

  • Todo

Combinatorics

Content Notes

Probability

Content

  • Todo

Notes

  • It is useful to take (either beforehand or concurrently) introductory classes in both statistics and combinatorics.

Point-Set Topology

  • Munkres, Topology

    • A standard - the good stuff starts about 10 chapters in, everything before that is aimed at providing a solid grounding in set theory and proofs.
  • Lee, Introduction to Smooth Manifolds

    • The appendix has a great uber-compressed review of point-set.

Content Notes

  • Todo: Specify which chapters are useful. TODO
  • Common question: where is Topology useful?
    • See Brouwer fixed point theorem, Lie groups in physics, links to functional analysis.

Graduate Real Analysis

Content

  • Measure theory, sigma algebras
  • Monotone and dominated convergence theorems
  • $L^p$ spaces, Holder’s inequality

Notes

  • Todo

Algebra

Functional Analysis

Content

  • Measure theory, sigma algebra
  • Montone and dominated convergence theorems
  • $L^p$ spaces, Holder’s inequality

Notes

  • Todo!

Category Theory

Content

  • Todo!

Notes Expected in some classes, but often assumed. Used in Algebraic Topology and Algebraic Geometry heavily. You can also find many full lectures online by people like Steve Awodey, Bartosz Milewski, and Eugenia Cheng.

Algebraic Topology

Content

  • Todo! Notes
  • Todo!

Algebraic Geometry

Content
Notes

Differential Geometry / Manifolds

Content

  • Todo

Notes

  • Todo

I’m continually expanding this list – if you have any recommendations, please feel free to let me know in the comments so I can possibly add them!


General Resources

Extracurricular Math

  • Garrity, All the Mathematics You Missed: But Need to Know for Graduate School

    • If you’re thinking about grad school at all, read this! Even if you’re not, it’s a pretty good collection of mathematics that it’s good to at least be familiar with. You can also use this to get an idea of some of the major theorems and results in a variety of subfields.
  • Introduction to Higher Mathematics

    • A great survey that highlights many different areas of advanced mathematics.
  • Abbott, Flatland

    • As the professor who recommended this to me said, reading this book is an absolutely essential step in the professional development of every mathematician. It’s a short fiction that explores a world of 1-dimensional beings that live in a 2-dimensional universe, which provides a powerful analogy for thinking about higher dimensional objects.

Computer Science

The Math Subject GRE

If you are thinking about applying to graduate school in Mathematics (pure or applied), I would recommend that you start looking at the material for this exam within your first few years, and take it as soon as your Junior year.

Note that the Math Subject GRE is vastly different than the Math portion of the General GRE. The latter covers primarily high-school level mathematics and requires no courses beyond Calculus (although the questions can still be tricky). The former is over 50% Calculus, and often includes a wide variety of upper-division topics as well. Many sources confuse these two exams!

Topics

  • Single Variable Calculus
  • Multivariable Calculus
  • Ordinary Differential Equations
  • Linear Algebra
  • Complex Analysis
  • Abstract Algebra
    • Primarily groups and rings, virtually no module or Galois theory.
  • Probability/Statistics
  • Real Analysis
    • Convergence of sums/sequences, topological properties of $\RR$.
  • Combinatorics
    • Graph theory, counting problems
  • Point-Set Topology
  • Numerical Analysis
  • Set Theory and Logic

Resources

Advice

  • Most problems don’t require extensive computations; there is usually a trick that solves it very quickly.
  • Time is the most difficult factor, be sure to take timed practice tests. You have right around two minutes per problem.
  • Sign up early, as these exams are often only held a few times per year.

Fun Stuff

Don’t forget to take some time to get excited about what you’re studying!

  • Hyper Rouge

    • A cool little dungeon crawler in which the in-game world is a hpyerbolic space.
  • Eversion of Spheres: Part 1 and Part 2

    • A great way to get some visual intuition for what homeomorphisms and/or homotopy equivalences might look like
  • Understanding Tensors

  • Hopf Fibration Toy

  • Reviews of some of the big Math-related Youtubers:

    • VSauce - 2/10
      • Often hints at neat concepts, but usually focused on “pop-math”. The cotton candy of the Math video world.
    • Mathologer - 9/10
      • Rigorous without drowning in technical details, and good topic selection!
    • PBS Infinite Series - 9/10
      • Topics are much closer to “serious” math, and are often accompanied with proofs.
    • Numberphile - 7/10
      • Often eschews rigorous explanations, which leads to a very misinformed public! Many good interviews, though.
    • 3blue1brow - 8/10
      • Amazing visualization, topics are interesting but are less serious. Good for general conceptual understanding.
  • Some interesting Math-related movies:

  • Proof

    • Good Will Hunting
    • A Beautiful Mind (John Nash)
    • The Imitation Game (Alan Turing)
    • The Theory of Everything (Stephen Hawking)
    • The Man Who Knew Infinity (Srinivasa Ramanujan)

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