This page is a collection of essays and online resources that might be helpful to my students or to other mathematicians. Or that I just think are cool.
See also my introduction to LaTeX elsewhere on this site.
General math:
- The Two Cultures of Mathematics by Timothy Gowers is a fun essay on two different ways working mathematicians think about their work.
- A Mathematician’s Apology by GH Hardy is famous, entertaining, and worth reading, even if I disagree with him more than I agree. It’s certainly a lovely defense of doing math for the pure beauty of doing math.
- Mathematics: What do grad students in math do all day? by Yasha Berchenko-Kogan. Originally an answer to a Quora question, this is one of the best explanations of what doing math research feels like that I’ve read.
Math Communication
- Gian-Carlo Rota has a short essay called Ten Lessons I Wish I had Been Taught which has excellent
- The MAA has an extensive online resource on how to communicate mathematics clearly. It’s mostly targeted at instructors, but can be very useful for students learning to communicate mathematics as well. I especially recommend checking out:
- Resources for writing and presenting
- Examples of good math writing and math presentations
- Proof-Writing Strategies
- Characteristics of Effective Student Talks and Papers
- Jordan Ellenberg has notes on giving a good talk, as does John E. McCarthy
- Bianca Viray has a nice set of slides on mathematical communication.
- Bjorn Poonen has a good roundup of practical advice on style in mathematical papers. This is mostly focused on users of $\LaTeX$ but may be useful for any mathematical writing.
Calculus and Analysis
- Calculus Made Easy by Silvanus P. Thompson is a calculus textbook from 1914 that has some of the clearest and most fun writing on the subject I’ve seen.
- Differential Forms and Integration by Terry Tao provides a different perspective on integration in multiple variables, and explains how to think about line integrals and surface integrals through the lens of differential forms.
Linear Algebra
- Down With Determinants! by Sheldon Axler is a famous essay arguing that linear algebra should be taught mostly avoiding the concept of determinants. It provides a perspective on linear algebra that is different from the standard introductory course approach, and one that I find clearer; it has influenced the way I teach the subject.
Group Theory
- Hermann Weyl has a lovely lecture series on the fundamentals of group theory, starting from a discussion of classical friezes and ornamental designs, which have been compiled in a book titled Symmetry. You can easily find free copies online, and it is also available in hard copy from Amazon.
- Timothy Gowers has an excellent series of posts on basic group theory on his blog, of which my favorite is his counterfactual history of normal subgroups. It explains normal subgroups nicely (in my view better than the standard treatment), and also models the way mathematicians work in the real world quite well.