September 14, 2020
Gian-Carlo Rota once asserted that "every mathematician only has a few tricks". The sheer breadth and ingenuity in the work of Jean Bourgain may at first glance appear to be a counterexample to this maxim. However, as we hope to illustrate in this article, even Bourgain relied frequently on a core set of tools, which formed the base from which problems in many disparate mathematical fields could then be attacked. We discuss a selected number of these tools here, and then perform a case study of how an argument in one of Bourgain's papers can be interpreted as a sequential application of several of these tools.
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January 4, 2023
We provide an exposition of the proofs of Bourgain's polynomial ergodic theorems. The focus is on the motivation and intuition behind his arguments.
July 7, 1994
This is a revised version of the course notes handed to each participant at the limits of mathematics short course, Orono, Maine, June 1994.
December 19, 2011
This note gives an informal overview of the proof in our paper "Borel Conjecture and Dual Borel Conjecture", see arXiv:1105.0823.
November 10, 2021
A variant of the Bourgain-Gamburd machine without using any girth bounds is obtained. Also, we find series of applications of %our result the Bourgain-Gamburd machine to problems of Additive Combinatorics, Number Theory and Probability.
November 28, 2018
We give a short proof of a theorem of J.-E. Pin (theorem 1.1 below), which can be found in his thesis. The part of the proof which is my own (not Pin's) is a complete replacement of the same part in an earlier version of this paper.
February 6, 2003
This is the second issue of the SPM Bulletin (SPM stands for "Selection Principles in Mathematics"). The first issue is math.GN/0301011 and contains some background and details.
April 7, 2020
We give an overview of several of the mathematical works of Gilles Lachaud and provide a historical context. This is interspersed with some personal anecdotes highlighting many facets of his personality.
March 31, 2011
These notes provide an opportunity to discover the beauty of Bourbaki set theory, and I hope that they will facilitate the task to those who find it difficult to read this book, one of the most critical elements of the mathematics of Bourbaki. ---- Ces notes constituent une occasion pour d\'ecouvrir la beaut\'e de la th\'eorie des ensembles de Bourbaki, et j'esp\`ere qu'elles faciliteront la t\^ache \`a ceux qui ont trouv\'e des difficult\'es en lisant ce livre qui est l'un...
January 14, 2019
In this diploma thesis (written in German) we present a detailed proof of Bourgain's Return Times Theorem due to Bourgain, Furstenberg, Katznelson and Ornstein following their paper as well as the book by Assani. Moreover, we generalize the result to return time sequences coming from systems with purely atomic invariant $\sigma$-algebra.
February 1, 2013
We have recently show that Poincare series of Hyperbolic Lie algebras have the form of a ratio between Poincare series of a chosen finite Lie algebra and a polynomial of finite degree. By the aid of some properly chosen examples, we now give some remarks on a related theorem of Bourbaki.