Crapo H., Senato D. (eds.) - Algebraic combinatorics and computer science. A tribute to Gian-Carlo Rota

Milano: Springer-Verlag Italia, 2001. — x+546 p. — ISBN: 978-88-470-2159-4This book, dedicated to the memory of Gian-Carlo Rota, is the result of a collaborative effort by his friends, students and admirers. Rota was one of the great thinkers of our times, innovator in both mathematics and phenomenology. I feel moved, yet touched by a sense of sadness, in presenting this volume of work, despite the fear that I may be unworthy of the task that befalls me. Rota, both the scientist and the man, was marked by a generosity that knew no bounds. His ideas opened wide the horizons of fields of research, permitting an astonishing number of students from all over the globe to become enthusiastically involved. The contagious energy with which he demonstrated his tremendous mental capacity always proved fresh and inspiring.
Beyond his renown as gifted scientist, what was particularly striking in Gian-Carlo Rota was his ability to appreciate the diverse intellectual capacities of those before him and to adapt his communications accordingly. This human sense, complemented by his acute appreciation of the importance of the individual, acted as a catalyst in bringing forth the very best in each one of his students. Whosoever was fortunate enough to enjoy Gian-Carlo Rota's longstanding friendship was most enriched by the experience, both mathematically and philosophically, and had occasion to appreciate "son cotè de bon vivant".
The book opens with a heartfelt piece by Henry Crapo in which he meticulously pieces together what Gian-Carlo Rota's untimely demise has bequeathed to science. Mathematics is a discipline which does not follow straight lines of progression, but which undergoes sudden interruptions, expansions and contractions. Mathematicians are lured by contemporary trends, prolifically writing articles each of which unveils some aspect of the truth. The more the subject grows, the more complex the language becomes. As a consequence, it is increasingly difficult, even for experts in the field, to determine the importance of new results and the potential avenues which they open before them. Gian-Carlo Rota's most outstanding talent, was his all-encompassing vision of the manifold aspects of mathematics, his ability to distil the 'underlying ideas' of each domain and precisely to state the formulation which would most naturally lend itself to limpidity of results. This is borne witness to by David Buchsbaum's salient article, which introduces the second chapter which deals with a number of themes of research which were particularly dear to Rota himself or which formed
part of his work in progress.
Rota would take no "accepted truth" for granted and reconsidered and discussed issues unremittingly. In one of his most recent essays, "Twelve problems in probability no one likes to bring up", Rota bravely reconsidered the question of Kolmogorov' s elegant and neat axiomatization, and pointed to the possibility of a deeper level of understanding. The essay in question forms part of the highly regarded Fubini lectures, which he gave in 1998 at the Institute for Scientific Interchange and which are published here as the first chapter of this book, with a refined preface to the lectures by Edoardo Vesentini.
The third chapter contains two thought-provoking expository articles, by Martin Aigner and Dominique Perrin, presented by Marilena Barnabei and Flavio Bonetti. The first exposition takes an in-depth look at the properties of one of the most profoundly studied sequences of combinatorial counting numbers, the Catalan Numbers. In the second exposition, Perrin presents an overview of the field of generating series for formal languages and rational series. Chapter 4, which opens with a piece by Alain Lascoux (in which he describes the impact of "On the Foundation of Combinatorial Theory. VIII. Finite Operator Calculus"), provides the reader with a selection of texts, representative of some of the most important lines of research in the fields of Algebraic Combinatorics and Theoretical Computer Science, fields profoundly influenced by Rota's thinking. Over and above the fact that many of these essays contain new results, they also furnish us with an up-to- date, self-contained overview of these areas of ongoing research. Gian-Carlo Rota has left us an invaluable wealth of ideas, which we shall do our utmost to keep alive. This volume is the first step.