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This 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 cote 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.
List of contents
Ten abandoned gold mines.- The Fubini Lectures.- Foreword.- The adventures of measure theory.- What is invariant theory, really?.- Twelve problems in probability no one likes to bring up.- Recurrent Themes of Gian-Carlo Rota's Mathematical Thought.- Resolution of Weyl modules: the Rota touch.- Circulant recursive matrices.- Remarks on Invariant geometric calculus. Cayley-Grassmann algebras and geometric Clifford algebras.- Grassmann geometric calculus, invariant theory and superalgebras.- Rota-Metropolis cubic logic and Ulam-Rényi games.- Umbral nature of the Poisson random variables.- A formal theory of resultants (I): an algorithm in invariant theory.- A formal theory of resultants (II): a constructive definition of the resultant.- Focus on Catalan Numbers and Combinatorics on Words.- Foreword to the surveys by Aigner and Perrin.- Catalan and other numbers: a recurrent theme.- Enumerative combinatorics on words.- Algebraic Combinatorics and Theoretical Computer Science.- Alphabet splitting.- Some operations on the family of equivalence relations.- Solving linear recurrences using functionals.- Polynomiality of the q, t-Kostka revisited.- A combinatorial approach to the theory of Pi-algebras and exponential growth.- On the permanent of certain circulant matrices.- Episturmian words and morphisms (results and conjectures).- A curious characteristic property of standard Sturmian words.
Summary
This 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 cote 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.
