Gian-Carlo Rota on Analysis and Probability

This volume consists of selected papers by Gian-Carlo Rota, a rare mathematician who made major contributions to several areas of mathematics. Presented in Part I are his papers in analysis, written at the beginning of his career and having a continuing and pervasive influence today. Part II is devoted to his papers on convexity and probability theory, written towards the end of his career and containing many ideas that have yet to be fully developed. Comprehensive commentaries by experts in the field are included in every chapter. Accessible to both experts and beginners.

"What makes this collection extraordinarily enjoyable to read are the commentaries that guide the reader between the gallery of results and are full of stories about Rota and of reflections about his mathematical style, about how he was led to areas and insights, about what he saw in them and what he thought they could do for other areas."

MATHEMATICAL REVIEWS

"This mixture of reprinted papers and articles about Rota tell the story of his mathematical journey. Apart from the opportunity to trace Rota's mathematical development through his papers, the book offers eighteen articles on Rota's work. These range from reminiscences and biographical details, to articles explaining the background to specific papers."

THE MATHEMATICAL GAZETTE "Rota's work contains many fresh ideas stimulating further research. If a PhD student asks for a book that might serve as a starting point to his/her research, the present volume is an excellent choice. Experts in analysis and probability can also find continual values in seeing Rota's writings in a single volume."

ACTA SCIENTIARUM MATHEMATICARUM

Résumé
Gian-Carlo Rota was born in Vigevano, Italy, in 1932. He died in Cambridge, Mas­ sachusetts, in 1999. He had several careers, most notably as a mathematician, but also as a philosopher and a consultant to the United States government. His mathe­ matical career was equally varied. His early mathematical studies were at Princeton (1950 to 1953) and Yale (1953 to 1956). In 1956, he completed his doctoral thesis under the direction of Jacob T. Schwartz. This thesis was published as the pa­ per "Extension theory of differential operators I", the first paper reprinted in this volume. Rota's early work was in analysis, more specifically, in operator theory, differ­ ential equations, ergodic theory, and probability theory. In the 1960's, Rota was motivated by problems in fluctuation theory to study some operator identities of Glen Baxter (see [7]). Together with other problems in probability theory, this led Rota to study combinatorics. His series of papers, "On the foundations of combi­ natorial theory", led to a fundamental re-evaluation of the subject. Later, in the 1990's, Rota returned to some of the problems in analysis and probability theory which motivated his work in combinatorics. This was his intention all along, and his early death robbed mathematics of his unique perspective on linkages between the discrete and the continuous. Glimpses of his new research programs can be found in [2,3,6,9,10].

Contenu

1. Differential Operators.- Reprinted papers.- Extension theory of differential operators I.- On the completeness of Sturm-Liouville expansions.- On the spectra of singular boundary value problems.- Articles.- Rota's work on differential operators.- Sturm-Liouville expansions.- 2. Theory of Linear Operators.- Reprinted papers.- Note on the invariant subspaces of linear operators.- On models for linear operators.- A note on the joint spectral radius.- Multiplicative extensions of positive linear operators.- Dilations of positive operators.- A method for the study of positive operators.- On the eigenvalues of positive operators.- Linear Operators. Part II. Spectral Theory.- Articles.- Invariant subspaces and models for linear operators.- Gian-Carlo Rota's work on positive operators.- The joint spectral radius.- 3. Reynolds Operators.- Reprinted papers.- On the representation of averaging operators.- Spectral theory of smoothing operators.- Endomorphismes de Reynolds et théorie ergodique.- Représentation des opérateurs de Reynolds.- Une généralisation de l'espérance mathématique conditionnelle qui se présente dans la théorie statistique de la turbulence.- On Reynolds operators in finite-dimensional algebras.- On the passage from the Navier-Stokes equations to the Reynolds equations.- Reynolds Operators.- Article.- Reynolds and averaging operators.- 4. Ergodic Theory.- Reprinted papers.- Une théorie unifiée des martingales et des moyennes ergodiques.- An alternierende Verfahren for general positive operators.- On the classification of periodic flows.- On the maximal ergodic theorem for Abel-limits.- A limit theorem for the time-dependent evolution equation.- Frobenius reciprocity in ergodic theory.- Articles.- Arlan Ramsay, Frobenius reciprocity.- Aunified theory for ergodic and martingale theorems.- Abel ergodic theory.- An alternierende Verfahren I.- An alternierende Verfahren II.- 5. Inequalities.- Reprinted paper.- On the inequality $${\left| {f'} \right|^2} \leqslant 4\left| f \right|.\left| {f''} \right|$$.- Article.- The Kallman-Rota inequality; a survey.- 6. Geometric Probability and Profinite Combinatorics.- Reprinted papers.- Totally invariant set functions of polynomial type.- A continuous analogue of Sperner's theorem.- A stochastic interpretation of the Riemann zeta function.- An example of profinite combinatorics.- Article.- Continuous and profinite combinatorics.- 7. Probability Theory.- Reprinted papers.- A measure theoretic approach to logical quantification.- Stochastic integrals: a combinatorial approach.- Probability set functions.- On the combinatorics of cumulants.- Articles.- Rota's foundations of probability theory.- Cumulants and probability set functions.