Recommended Readings
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 on the beginner’s page.
Mathematician autobiographies#
Two of my past mentors have really moving pieces about their lives:
- Joe Gallian’s story: A Break for Mathematics (interview)
- Ken Ono’s story: My Search for Ramanujan (book)
(If you are an undergraduate math student, Joe and Ken run summer programs that I recommend, at Duluth and UVa respectively. I am a proud alum of both.)
PG essays#
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.
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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.
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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.
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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.
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Manifolds and Differential Forms by Reyer Sjamaar. My preferred introduction to differential geometry; very readable and works with minimal prerequisites. Also, beautifully drawn figures.
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Harvard’s CS 125: Algorithms and Complexity has delightful lecture and section notes.
Olympiad Resources#
See also Geoff Smith’s page.
Handouts#
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My own handouts (sorry, couldn’t resist linking them again).
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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.
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Alexander Remorov, in particular the projective geometry handout, which the corresponding chapter in my textbook is based off of.
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Po-Shen Loh, mostly combinatorics. See especially the handouts on the probabilistic method.
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A Journey to the IMO, an overview of the IMO written from Nepal student Yuvraj Sarda.
Books#
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My own: olympiad geometry book EGMO and OTIS Excerpts for non-geometry.
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Art and Craft of Problem-Solving by Paul Zeitz, introduction to math olympiads in general.
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The CMUMC POTD Book by Thomas Lam. This is among the best general problem collections I’ve seen. (CMUMC is Carnegie Mellon University Math Club, and POTD means Problem of the Day.)
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Combinatorics textbooks in increasing difficulty order:
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Counting Rocks! An Introduction to Combinatorics, by Henry Adams, Kelly Emmrich, Maria Gillespie, Shannon Golden, Rachel Pries. Freely available on arXiv. This is a beginner textbook that starts from pre-olympiad content (e.g. binomial coefficients) to some early olympiad-level material. It also has some linked video lectures.
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Olympiad Combinatorics, by Pranav A. Sriram. The individual chapters are located in posts #1, #11, #49. This is an intermediate-advanced textbook.
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Number theory textbooks in increasing difficulty order:
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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.
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Modern Olympiad Number Theory, by Aditya Khurmi. More olympiad-oriented number theory textbook.
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Problems from the Book by Titu Andreescu and Gabriel Dospinescu. Intermediate-advanced textbook covering topics in inequalities, algebra, analysis, combinatorics, and number theory.
Contests#
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:
- Other olympiads:
- Team selection tests:
- International contests: