Fun but serious mathematics books to gift advanced undergraduates.

Question

I am looking for fun, interesting mathematics textbooks which would make good studious holiday gifts for advanced mathematics undergraduates or beginning graduate students. They should be serious but also readable.

In particular, I am looking for readable books on more obscure topics not covered in a standard undergraduate curriculum which students may not have previously heard of or thought to study.

Some examples of suggestions I've liked so far:

• On Numbers and Games, by John Conway.
• Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups, by John Meier.
• Ramsey Theory on the Integers, by Bruce Landman.

I am not looking for pop math books, Godel, Escher, Bach, or anything of that nature.

I am also not looking for books on 'core' subjects unless the content is restricted to a subdiscipline which is not commonly studied by undergrads. (E.g. Finite Group Theory by Isaacs would be good, but Abstract Algebra by Dummit and Foote would not.)

Answer 1

Off the top, no particular order:

• Conceptual Mathematics - Lawvere and Schanuel
• Sets for Mathematics - Lawvere and Rosebrugh
• A Walk Through Combinatorics - Bona
• Combinatorial Species and Tree-Like Structures - Bergeron, Labelle & Leroux
• Ordinary Differential Equations - Arnold
• What Are and What's the Purpose of Numbers - Dedekind
• Collected Works of Karl Menger - Menger
• Algebraic Number Theory and Fermat's Last Theorem - Stewart

Just a couple if you're interested in applied areas:

• Theory of Gambling and Statistical Logic - Epstein
• Theoretical Introduction to Programming - Mills
• Elements of Statistical Learning - Hastie, Tibshirani & Friedman

Answer 2

The Book of Numbers by John H. Conway & Richard Guy. Although this does fall into the "popular mathematics" arena, it contains breadth and depth that will be of interest. Whilst I read many formal textbooks on mathematics, The Book of Numbers is a book I often return to when working on a problem. It contains many gems.

Answer 3

Here are several books that I have looked at frequently.

Proofs and Confirmations, David Bressoud

Winning Ways For your Mathematical Plays Vols. 1 to 4, Berlekamp, Conway, Guy

Integer Partitions, Andrews and Eriksson

Number Theory in Science and Communication, Schroeder

Fractals, Chaos, and Power Laws, Schroeder

The first part of The Road to Reality, Penrose contains a primer on the math required in modern physics.

Answer 4

Topics in Algebra by I.N.Herstein.

Answer 5

As someone in your target audience, I recommend A=B, by Marko Petkovsek, Herbert Wilf and Doron Zeilberger (with a foreword by Donald E. Knuth).

It's basically a book on generating combinatorial identities programmatically, inspired by Exercise 1.2.6.63 in Knuth's Art of Computer Programming, Volume 1:

[50] Develop computer programs for simplifying sums that involve binomial coeffcients.

The mathematics is not above an undergraduate's head, and a lot of the results are intuitive and attractive.

Best of all, the book is available in its entirety from the website (although paper copies can also be purchased)