Arithmetic, Geometry, and Coding Theory: Homage to Gilles Lachaud

The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be approached from a computational viewpoint.

In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One of the aims of this article is to show, by means of examples, the usefulness of computer algebra to mathematical research.

In the first part of this paper we try to explain to a general mathematical audience some of the remarkable web of conjectures linking representations of Galois groups with algebraic geometry, complex analysis and discrete subgroups of Lie groups. In the second part we briefly review some limited recent progress on these conjectures.

This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of t...

During the five days of this conference a very dense scientific program has enlighted our research fields, with the presentation of large number of interesting lectures. I will try to summarize the theoretical aspects of some of these new results.

These are the notes of my lectures at the 1996 European Congress of Mathematicians. {} Polynomials appear in mathematics frequently, and we all know from experience that low degree polynomials are easier to deal with than high degree ones. It is, however, not clear that there is a well defined class of "low degree" polynomials. For many questions, polynomials behave well if their degree is low enough, but the precise bound on the degree depends on the concrete problem. {} It ...

This note completes a talk given at the conference Curves over Finite Fields: past, present and future celebrating the publication the book {\em Rational Points on Curves over Finite Fields by J.-P. Serre and organised at Centro de ciencias de Benasque in june 2021. It discusses a part of the history of algebraic geometry codes together with some of their recent applications. A particular focus is done on the "multiplicative" structure of these codes, i.e. their behaviour wit...

This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area.

Geometric complexity theory (GCT) is an approach to the $P$ vs. $NP$ and related problems through algebraic geometry and representation theory. This article gives a high-level exposition of the basic plan of GCT based on the principle, called the flip, without assuming any background in algebraic geometry or representation theory.

- Synth\`ese des travaux pr\'esent\'es en vue d'une Habilitation \`a Diriger des Recherches - Synthesis of works presented towards the Habilitation degree This is a summary (in French) of my work in number theory, group theory and combinatorics in the last eight years.