If publications like the American Mathematical Monthly or Notices of the American Mathematical Society had to worry about their circulation numbers in the way that magazines like Rolling Stone or Sports Illustrated do, we would probably be subject to lots more 'special issues' consisting of lists. What are the Ten Best Theorems in Topology? Who are the Algebraists To Watch In 2010? And who were the most influential mathematicians of the last thirty years?
I'm not sure about the other lists, but I know that Joe Gallian would get my vote for one of the top spots in the "most influential" list. Gallian wouldn't earn his spot on the list based on his own mathematics, although his research in the theory of finite groups is nothing to sneeze at, but instead he would make the list based on the influence he has had over a generation of mathematicians, through his expository writing, his REU program, and his involvement with the MAA in general and Project NExT in particular. (Full disclosure: It was Gallian's famous "Just Say Yes!" pep talk at a Project NExT meeting which first inspired me to start volunteering to write book reviews for MAA Online, so you have him to blame for that among all of the great things he has inspired.)
Because his influence is so widespread, it is no surprise that a conference was held in order to celebrate his 65th birthday. This conference, held at his home institution of the University of Minnesota in Duluth in the summer of 2007, focussed on 'Communicating Mathematics', and also commemorated the 30th anniversary of the REU which Gallian runs.
The proceedings of this conference has been published in the AMS's Contemporary Mathematics series. It collects a number of wonderful articles inspired by Gallian and his work in different areas. Aparna Higgings, one of Gallian's partners-in-crime in encouraging student research and in running Project NExT, has written a nice biographical sketch of Joe and his many contributions to the mathematical community. The other articles are all mathematical, with topics in graph theory, group theory, voting theory, and a number of different areas of mathematics. They range from highly expository to true research articles, although, true to the spirit of the conference, the level of exposition is significantly higher on average than one finds in most conference proceedings. Many of the articles would be accessible to advanced undergraduate students (and would provide excellent examples to show what a paper which is high in both mathematical and expository quality looks like), and a number of them even suggest further research projects which the reader might be interested in pursuing.
The articles collected in this volume are the following:
- A journey of discovery: Orthogonal matrices and wireless communications — Sarah Spence Adams
- Probabilistic expectations on unstructured spaces — John Beam
- A beginner’s guide to forcing — Timothy Y. Chow
- Higher order necessary conditions in smooth constrained optimization — Elena Constantin
- Hamiltonian paths and hyperbolic patterns — Douglas Dunham
- When graph theory meets knot theory — Joel S. Foisy and Lewis D. Ludwig
- Can an asymmetric power structure always be achieved? — Jane Friedman and Cameron Parker
- McKay’s canonical graph labeling algorithm — Stephen G. Hartke and A. J. Radcliffe
- A multiplicative deformation of the Mobius function for the poset of partitions of a multiset — Patricia Hersh and Robert Kleinberg
- Communicating, mathematics, communicating mathematics: Joe Gallian style — Aparna Higgins
- Fair allocation methods for coalition games — David Housman
- Sums-of-squares formulas — Daniel C. Isaksen
- Product-free subsets of groups, then and now — Kiran S. Kedlaya
- Generalizations of product-free subsets — Kiran S. Kedlaya and Xuancheng Shao
- What is a superrigid subgroup? — Dave Witte Morris
- Averaging points two at a time — David Petrie Moulton
- Vertex algebras as twisted bialgebras: On a theorem of Borcherds — Manish M. Patnaik
I had a lot of fun reading the articles in this volume, and I also learned quite a bit of interesting mathematics. To stand on a soapbox momentarily, I wish there were more venues for these types of articles: well-written mathematics, which blurs the line between expository and research, broadly accessible without watering things down too much. I hope we don't have to wait for another mathematician like Joe Gallian to come along to get another volume of this nature, because we would likely be waiting a very long time.
Darren Glass is assistant professor of mathematics at Gettysburg College.