Some of My Favorite Books



Some books that you might like to read on mathematics and more



Serge Lang was one of the great number theorists of the last century, who is perhaps best known for his influential conjectures and his many books. His conjectures include: the Mordell-Lang conjecture, the Bombieri-Lang conjecture, the Lang-Trotter conjecture, and many others. His books include all kinds: monographs, graduate textbooks, undergraduate textbooks, and more.

Today I want to talk about books. I love to read books of all genres; although my favorites are: mathematics, history of mathematics, and history in general. I did just read “Under the Dome” by Stephen King, so I am not limited to nonfiction—that is fiction—right?



Since it is the end of the year, I thought that some of us might have time to catch up on our reading. So I plan to share a few of my favorite books—I hope that you might let me know some of yours.

I knew Lang when I was faculty at Yale University in the mid-seventies. During our spring break one year, Dick Karp visited for a whole week. It was great having Karp visit the Computer Science department, and he and I had many fun discussions. On the last night of his visit, Karp, Lang, and I went out for dinner. I recall on the way to my car, Dick causally asked Serge what he had done during the spring break? Serge quickly replied:

I wrote a book.

Karp seemed at a loss for a reply. I had nothing to say, but I immediately thought, what did I do this last week? Besides enjoyable conversations and general stuff I had no answer—I certainly did not write a book. I did nothing. I do believe Lang, since he wrote many many books—as I said earlier he was extremely prolific. Oh well.

Let’s turn to reading books, not writing them.

My Book List

Here are some books that I have enjoyed, in no special order:

Math Talks for Undergraduates by Serge Lang. This book is a series of lectures for undergraduates. It is wonderful. He covers a wide range of topics, and demonstrates his great ability to explain things clearly. One example is the chapter on the famous ABC conjecture. This is due to Joseph Oesterlé and David Masser, and is:

Given , there exists a constant such that for all non-zero relatively prime integers with , we have the inequality

Here is defined as the product of the distinct prime factors of :

Note, where is a prime. Lang proves a version of the conjecture for polynomials, and explains why the conjecture is “the greatest conjecture of the century.”

The File by Serge Lang. This is one of the strangest books ever written, but it is hard to not find it compelling. It is a collection of letters that Lang wrote and were written to him. One letter after the other.

The letters all concern Lang’s campaign to keep a candidate out of the National Academy of Sciences. Lang was upset that the candidate, who worked in social science, was misusing mathematics to “prove” some point. Lang stated clearly in his letters against the candidate that this use of mathematics was not worthy of membership to the Academy. The defenders letters, of course, argued back the opposite point.

I was once given advice by a friend who is an attorney. He said:

Never throw out any letter you are sent, and never send anyone a letter.

This book is a tribute to this maxim. People wrote letters to Lang, and said things in those letters, that they never wanted to become public. But, they did become public.

Analytic Number Theory by Donald Newman. Analytic Number Theory is a deep and beautiful area. It is known for extremely technical proofs, which often require pages of complex formulas, careful estimates, and messy calculations. Yet, Newman takes you on a quick tour of some of the best results in 75 pages. He gives essentially full proofs of some of the major achievements of analytic number theory: Roth’s Theorem, the Waring Problem, and the Prime Number Theorem. This is not a book to read to become an expert, or even to become well versed in the area, but there is something magical about this book. All in 75 pages.

The Honors Class: Hilbert’s Problems and Their Solvers by Ben Yandell. This is a book on the famous list of 23 problems of David Hilbert. What I like so much about this book is the history behind the solutions to the problems. In some cases Hilbert problems were “solved” for decades, yet eventually it was discovered that the solutions were wrong or had gaps. Part of my “hidden” agenda is to remind us all that even the immortals make mistakes, have proofs with gaps, and are human.

The Poincaré Conjecture: In Search of the Shape of the Universe by Donal O’Shea.

Poincaré’s Prize by George Szpiro.

Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century by Masha Gessen. These three books are on the Poincaré conjecture and its resolution. The last one is the most recent, and it focuses mostly on Grigori Perelman as a person. The book discusses the question: Why did he turn down the Fields Medal? You will read that he had turned down awards before, but you will still be puzzled—I think—why he turned the Fields Medal down. The books approach the Poincaré problem and its history from different angles, and I enjoyed them all.

Prime Obsession by John Derbyshire.

The Riemann Hypothesis by Karl Sabbagh. These books are on the hunt for a solution to the Riemann Hypothesis. The second book highlights the work of Louis de Branges who solved the Bierberbach Conjecture of Ludwig Bierberbach. This was a major open question in analysis, and de Branges deserves a great deal of credit for solving this difficult problem.

Unfortunately, since his great success he has been claiming that he is “about” to solve the Riemann Hypothesis. Sabbagh spent many hours with de Branges and presents a case study in how a mathematical problem can almost take you over. The Riemann Hypothesis is definitely a mathematical disease for some.

The Mathematician’s Brain by David Ruelle. Ruelle is a first class mathematician and tells some wonderful stories about his views of everything—including mathematics.

Structure and Randomness by Terry Tao. Or anything else by Tao. This book is a collection of lectures from his blog posts of the year 2007. His ability to explain things is wonderful. For example, his section on ultrafilters is a classic. I thought I knew what they are, but I was wrong. Only after reading this did I really “get” what they are.

Kolmogorov’s Heritage in Mathematics edited by Eric Charpentier, Annick Lesne, and Nikolai Nikolski. This is a collection of articles about one of the great mathematicians of the twentieth century.

Lost Horizon by James Hilton. A classic novel—the cover claims that it was the first paperback ever published. I do not know if this is correct, and I decided not to check.

Open Problems

What are some of your favorites? I would love to hear.