Exploring the toolkit of Jean Bourgain

(This text is a survey written for the Bourbaki seminar on the work of F. Manners.) Gowers uniformity norms are the central objects of higher order Fourier analysis, one of the cornerstones of additive combinatorics, and play an important role in both Gowers' proof of Szemer\'{e}di's theorem and the Green-Tao theorem. The inverse theorem states that if a function has a large uniformity norm, which is a robust combinatorial measure of structure, then it must correlate with a...

We summarize four different versions of our course notes on the limits of mathematics.

The following is a near complete set of notes of Bourgain's 1988 paper "Almost Sure Convergence and Bounded Entropy." Both entropy results are treated, as is one application. The proofs here are essentially those of Bourgain's.

In this note we generalize the Moroianu's fixed point theorem. We propose a very elegant common proof of the Bourbaki's fixed point theorem and our result. We apply our result to give a very elegant proof of the fact that, in the Zermelo-Fraenkel system, the axiom of choice is equivalent to each of the following statements: H. Kneser's Lemma, Zorn's Lemma, Zermelo's Lemma

This article presents some technical and pedagogical considerations about a series of videos made by the authors to teach elementary theory of differential equations. Intended to be a pedagogical support for undergraduate students of scientific curricula, these videos provide rigorous mathematical contents in a form that takes into account the specificities of the tool. Videos for teaching mathematics should be seen as a specific way for providing rigorous scientific content ...

Mathematical software systems are becoming more and more important in pure and applied mathematics in order to deal with the complexity and scalability issues inherent in mathematics. In the last decades we have seen a cambric explosion of increasingly powerful but also diverging systems. To give researchers a guide to this space of systems, we devise a novel conceptualization of mathematical software that focuses on five aspects: inference covers formal logic and reasoning a...

This book is the final version of a course on algorithmic information theory and the epistemology of mathematics and physics. This is camera-ready copy prepared for publication as a book, but at the last minute I decided to publish it electronically instead. This book discusses Einstein and Godel's views on the nature of mathematics in the light of information theory, and sustains the thesis that mathematics is quasi-empirical. There is a foreword by Cris Calude of the Univer...

Recent advances in computing have changed not only the nature of mathematical computation, but mathematical proof and inquiry itself. While artificial intelligence and formalized mathematics have been the major topics of this conversation, this paper explores another class of tools for advancing mathematics research: databases of mathematical objects that enable semantic search. In addition to defining and exploring examples of these tools, we illustrate a particular line of ...

This article is based on a talk given at a one-day meeting in NUI, Maynooth on the Fourth of April, 2008, held to honour David Walsh and Richard Watson. It surveys joint work with the author.

25th anniversary and new building dedication Centre de Recherches Math\'{e}matiques Montr\'{e}al, Canada, October 1994