This page has a bunch of links you might want to read. In addition, you might see the following sub-pages:
Please notify me of any broken links, suggestions, etc. by email.
An abridged version of this page for olympiad students can be found here.
I can’t help but link Paul Graham’s essays. Ones I felt hit closest to home: What You’ll Wish You’d Known, Undergraduation, The Age of the Essay, What You Can’t Say, Mean People Fail, The Lesson to Unlearn.
Undergraduate Math and Computer Science#
If you check Appendix A of Napkin, you can find listings of lecture notes or textbooks that I like for most undergraduate (or early graduate) topics. Here are some additional links.
MSci Category Theory notes by Tom Leinster. I highly enjoyed these notes; very carefully written and explains intuition. Some minimal knowledge of group theory and linear algebra is used in the examples. See also the corresponding print book.
Analytic NT notes by AJ Hildebrand. A set of lecture notes for analytic number theory, suitable for self-study. A light introduction where you get to prove versions of the Prime Number Theorem and Dirichlet’s Theorem.
Algebraic Geometry by Andreas Gathmann. My preferred introduction to algebraic geometry; short but complete. This was the source that finally got me to understand the concept of a ringed space.
Manifolds and Differential Forms by Reyer Sjamaar. My preferred introduction to differential geometry; very readable and works with minimal prerequisites. Also, beautifully drawn figures.
Harvard’s CS 125: Algorithms and Complexity has delightful lecture and section notes.
See also Geoff Smith’s page.
My own handouts (sorry, couldn’t resist linking them again).
Yufei Zhao’s site has several excellent handouts, especially in geometry. I consulted many of them when I was coming up with ideas for my geometry textbook. In particular, the Cyclic Quadrilaterals handout is especially worth reading.
Alexander Remorov, in particular the projective geometry handout, which the corresponding chapter in my textbook is based off of.
Po-Shen Loh, mostly combinatorics. See especially the handouts on the probabilistic method.
A Journey to the IMO, an overview of the IMO written from Nepal student Yuvraj Sarda.
Art and Craft of Problem Solving by Paul Zeitz, introduction to math olympiads in general.
An olympiad combinatorics book, by Pranav A. Sriram. The individual chapters are located in posts #1, #11, #49.
Olympiad NT through Challenging Problems, by Justin Stevens, is an introductory olympiad number theory text, at a level somewhat easier than what my own number theory handouts assume.
Modern Olympiad Number Theory, by Aditya Khurmi. More olympiad-oriented number theory textbook.
Each section is in alphabetical order. Obviously not an exhaustive list of good contests, there are too many; these are just ones I have seen recently.
- National olympiads:
- Japan Finals
- USA Junior
- Other olympiads:
- NICE Olympiad
- Team selection tests:
- International contests:
- European Girls Math Olympiad
- IMO Shortlist
- Romania Masters in Mathematics