Exploring the toolkit of Jean Bourgain

I present examples of mathematical objects that are of interest for public key cryptography. Text for the Journ\'ee Annuelle 2007 of the SMF.

This paper has been withdrawn by the author due to an error in the last paragraph of step 2 of the main proof, on page 6.

There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been sleep-walking in their infinitary mathematical paradise to take heed. Much of mathematics relies upon either (i) the "existence'" of objects that contain an infinite number of elements, (ii) our ability, "in theory", to compute with an arb...

A well known argument of James yields that if a Banach space $X$ contains $\ell_1^n$'s uniformly then $X$ contains $\ell_1^n$'s almost isometrically. In the first half of the paper we extend this idea to the ordinal $\ell_1$-indices of Bourgain. In the second half we use our results to calculate the $\ell_1$-index of certain Banach spaces. Furthermore we show that the $\ell_1$-index of a separable Banach space not containing $\ell_1$ must be of the form $\omega^{\alpha}$ for ...

The present work has been designed for students in secondary school and their teachers in mathematics. We will show how with the help of our knowledge of number systems we can solve problems from other fields of mathematics for example in combinatorial analysis and most of all when proving some combinatorial identities. To demonstrate discussed in this article method we have chosen several suitable mathematical tasks.

A theory graph is a network of axiomatic theories connected with meaning-preserving mappings called theory morphisms. Theory graphs are well suited for organizing large bodies of mathematical knowledge. Traditional and formal proofs do not adequately fulfill all the purposes that mathematical proofs have, and they do not exploit the structure inherent in a theory graph. We propose a new style of proof that fulfills the principal purposes of a mathematical proof as well as cap...

In this paper, I explore what mathematical research can tell us about ourselves, and our role in the world, using examples from my own experience. The paper is a sequel to my piece "Mathematics is a Quest for Truth", published in the Notices of the American Mathematical Society in 2014.

Beginning with the projectively invariant method for linear programming, interior point methods have led to powerful algorithms for many difficult computing problems, in combinatorial optimization, logic, number theory and non-convex optimization. Algorithms for convex optimization benefitted from many pre-established ideas from classical mathematics, but non-convex problems require new concepts. Lecture series I am presenting at the conference on Foundations of Computational...

The purpose of this short note is to demonstrate how some techniques from additive combinatorics recently developed by Peluse and Peluse-Prendiville can be applied to give an alternative proof for a trilinear smoothing inequality originally due to Bourgain.

We claim that human mathematics is only a limited part of the consequences of the chosen basic axioms. Properly human mathematics varies with time but appears to have universal features which we try to analyze. In particular the functioning of the human brain privileges concept naming and short formulations. This leads to organizing mathematical knowledge structurally. We consider briefly the problem of non-mathematical sciences.