Resources

Here is a list of books related to Math that I am or have been reading (mostly because I want the citations in Chicago style below). They are organized by topic alphabetically. I’ve also tried to make the categories as broad as possible so as to reduce the number of categories.

Algebra

Abstract Algebra

I have been reading Dummit and Foote’s Abstract Algebra.¹ The explanations are very clear with ample examples to motivate each definition or notion. However, it still is a graduate level book, and I have switched to Pinter’s A Book of Abstract Algebra, which covers topics more suited at my level.² Another book that has been recomended to me is Gallian’s Contemporary Algebra.³ An interesting feature is its focus on algebra applications such as cryptography and emphasis on documenting and teaching the relatively short history of abstract algebra.

Linear Algebra

Analysis

Following the suggestions of my friends and teachers in the US, I started out my journey into Real Analysis in highschool with, of course, Rudin’s Principles of Mathematical Analysis (PMA for short).⁴ It was the golden standard for elementary analysis, but I found its laconic, hand-wavy style of writing and lack of illustration to be difficult for developing intuition.⁵ Although quite strenuously, I was able to complete the first three chapters of PMA, and took the time to understand every definition and proof, sometimes drawing my own illustrations. Since I am reading and learning Analysis on my own, I thought it would be better to read a book with a more explanatory and detailed writing style, and of course, with diagrams. So I switched to Apostol’s Mathematical Analysis, which generally had the same content and structure as Rudin’s PMA.⁶ Apostol’s book has slightly more content than Rudin, which Rudin presumably leaves for his second book Real and Complex Analysis.⁷

Foundations

Category Theory

Set Theory

In my experience, Jech’s Set Theory is a popular book in advanced set theory.⁸ Personally, I read Cunningham’s introductory book on set theory, which was very pragmatic (showing and demonstrating common proof techniques) despite having a few publishing errors.⁹

Geometry

Of course, one cannot study geometry sans Euclid’s Elements.¹⁰ The particular publisher should not matter for Euclid’s Elements has been standard for over two millennia; Proposition numbers will be cited instead of page numbers from any book of Euclid’s Elements.

I am also reading Hartshorne’s Geometry: Euclid and Beyond concurrent to Euclid’s Elements.¹¹

Number Theory

I am currently trying to read Apostol’s Introduction to Analytic Number Theory.¹²

Topology

Knot Theory

Knot theory is knot easy to learn, it could be quite mind bending, so Adam’s Knot Book, which takes an elementary approach, is probably the best beginner resource.¹³ It is also available online for free (as of March 2021). A more rigorous approach would be Roberts’s “Knot Knotes” (pun intended).¹⁴

1. David Steven Dummit and Richard M. Foote, Abstract Algebra, 3rd ed. (Danvers, MA: John Wiley & Sons, 2004). ↩

2. Charles C. Pinter, A Book of Abstract Algebra, 2nd ed. (Mineola, NY: Dover Publications, 2013). ↩

3. Joseph A. Gallian, Contemporary Abstract Algebra, 9th ed. (Boston, MA: Cengage Learning, 2017). ↩

4. Walter Rudin, Principles of Mathematical Analysis (New York u.a., NY: McGraw-Hill u.a., 1964). ↩

5. I find the relationship between Rudin’s Principles of Mathematical Analysis with American students summed up best in this MAA book review. ↩

6. Tom Mike Apostol, Mathematical Analysis, 2nd ed. (Reading, MA: Addison-Wesley, 1974). ↩

7. Walter Rudin, Real and Complex Analysis, 3rd ed. (New York City, NY: McGraw-Hill, 1987). ↩

8. Thomas J. Jech, Set Theory: The Third Millennium Edition, Revised and Expanded, 3rd ed. (Berlin, Germany: Springer, 2003). ↩

9. Daniel W. Cunningham, Set Theory: A First Course (New York City, NY: Cambridge University Press, 2016). ↩

10. Euclid, Thomas Little Heath, and Dana Densmore, Euclid’s Elements: All Thirteen Books Complete in One Volume: the Thomas L. Heath Translation (Santa Fe, N.M.: Green Lion Press, 2002). ↩

11. Robin Hartshorne, Geometry: Euclid and Beyond, ed. K.A. Ribet, F. W. Gehring, and S. Axler (New York City, NY: Springer, 2000). ↩

12. Tom M. Apostol, Introduction to Analytic Number Theory (New York City, NY: Springer, 2011). ↩

13. Colin C. Adams, The Knot Book: An Elementary Introduction to the Theory of Knots (n.p.: American Mathematical Society, 2004). Online Access Link. ↩

14. Justin Roberts, “Knot Knotes,” 2015. Online Acess Link. ↩