Those of us who spend too much time reading math blogs periodically run across discussions about who the greatest living mathematicians are. Everyone could probably come up with their own list (as well as their own definition of what “great” even means), but I would guess that most people who work in combinatorics or number theory would place George Andrews high on that list. His name is among the first one would mention when discussing partitions, q-series, hypergeometric series, and several other topics in combinatorics, and his work has won him many honors and accolades. So it is not surprising that Imperial College Press has dedicated the third volume in their ICP Selected Papers series to collecting some of George Andrews’ works. (As of this writing, Andrews is also the only mathematician to be recognized in this series).
Whether or not he is on one’s personal list of greatest mathematicians, he is certainly on any list of the most prolific: according to the list in the book, Andrews is responsible for 292 papers, 15 books, and more than two dozen other pieces of writing; and this number has continued to grow, according to his website. Knowing that a complete collection of his papers would not be physically possible, Andrew Sills, one of Andrews’ doctoral students, devised a complicated system worthy of Nate Silver, and the two men narrowed down this list to a manageable number of papers from all areas and eras of his career to this point that appear in this volume. Well, “manageable” may be a bit of a stretch as the book is still more than 1000 pages and I’m pretty sure I will need an extra visit to the chiropractor after carrying it back and forth to the office a handful of times. But it is a beautifully put together volume filled with exciting mathematics and is well worth the size.
George Andrews was born in Oregon in 1938 and received his Bachelor’s and Master’s degrees from Oregon State University. He then went to the University of Pennsylvania where he was Hans Rademacher’s last doctoral student, studying the theory of mock theta functions. After graduating in 1964, he accepted a position at Pennsylvania State University, where he has spent the entirety of his career. Over the last six decades he has achieved many honors, including being President of the American Mathematical Society and being elected a Fellow of the National Academy of Sciences. Most of his work has been in combinatorics, and more specifically in analytic combinatorics. The 58 papers collected in the volume under review are broken into eight sections:
The Geometry of Numbers
Combinatorics, Fibonacci Numbers, and Computers
Education, History, etc.
The section on q-Series alone consists of 19 of the papers (and nearly 400 pages!), and is divided into subsections including Andrews’ Ph.D. Thesis, Heine’s Method, The Askey Influence, Well-poised Series, Bailey Chains, q-Trinomials and the Borwein Conjecture, Ramanujan’s “Lost” Notebook, Statistical Mechanics, The Pfaff Trilogy, and MacMahon’s Partition Analysis. It is this work, as well as his work on the theory of partitions, for which Andrews is most well known.
For the uninitiated, let me give a flavor of the types of mathematics that Andrews is interested in. A partition of an integer n is a way of writing n as the sum of nonnegative integers — for example, 1+1+1+1, 2+2, and 2+1+1 are all partitions of the number 4. The idea of partitions dates back to (at least) Leibniz. Much interesting work has been done to count the number of partitions, especially if you place various restrictions on the integers you are allowed to add. One example of such a result is the so-called Second Rogers-Ramanujan Identity, which says that the number of partitions of n into summands none of which are consecutive to each other is the same as the number of ways of partitioning n into parts that are all congruent to either 2 or 3 mod 5. As an example, if n=9 then the partitions of the first type are 9, 8+1, 7+2, 6+3, and 5+3+1 and the partitions of the second type are 9, 6+1, 4+4+1, 4+1+1+1+1+1, and 1+1+1+1+1+1+1+1+1 — there are five of each.
Andrews was intrigued by identities such as this one, and went on to prove many many such identities. (For much more detail, I would refer the reader to this writeup by Andrews himself or to the book Integer Partitions, by Andrews and Eriksson… or to the book under review)
Each section and subsection of The Selected Works comes with some new commentary by Andrews. In some cases, this gives some of the greater motivation for the works and how they fit into the bigger picture of his career. In other cases, it gives insight into why he chose to include the papers he did in this volume — and why others were left out. Some of the commentary relates anecdotes about his collaborators and his travel, and other gives updates onto what happened with the open questions posed in the paper. In a very small number of cases, errata are given about the works. These commentaries give some interesting insight into Andrews as a mathematician, and my only complaint about the collection is that I would have liked even more of these reflections.
As I mentioned earlier, the book is a very nice physical object, and even someone uninterested in the mathematics it contains will find it an interesting illustration of the history of typesetting and journal layouts over the last sixty years. Sills and Andrews have also included some biographical information about Andrews, a full (eighteen page long) curriculum vitae, a list of all of his doctoral students, and several pages of photographs from throughout his career. They have created a volume that will be extremely valuable for anyone who wants to work in the areas that Andrews has spent his career working in, and many of the papers in this volume should help lure new people into the area — or at least spend a few hours learning some combinatorial number theory.
I don’t know if ICP has any plans to release future volumes of selected works by mathematicians. The number of living mathematicians who have the depth and breadth of work that calls for this type of volume is somewhat small, and I imagine that even fewer would be willing to spend the energy to do this level of reflection and annotation of their own work. But we can be grateful that George Andrews was willing to and that he had someone like Andrew Sills to help edit the volume. We should all hope that there are more volumes like this in the future.
Darren Glass is an Associate Professor of Mathematics at Gettysburg College. While his research originally dealt with arithmetic geometry and Galois theory, he has found himself intrigued by a number of questions about integer partitions recently, and has found Andrews’ work indispensable in this area. He can be reached at email@example.com.