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

This document contains a description of several of my papers, including remarks on history and connection with subsequent work. It also contains some new results and conjectures.

This survey article is an introduction to Diophantine Geometry at a basic undergraduate level. It focuses on Diophantine Equations and the qualitative description of their solutions rather than detailed proofs.

Personal recollections about Alexandre Grothendieck and early days of his theory of motives

These are the notes from my courses on the arithmetic of quadratic forms.

These are the notes for a set of lectures delivered by the two authors at the Les Houches Summer School on `Complex Systems' in July 2006. They provide an introduction to the basic concepts in modern (probabilistic) coding theory, highlighting connections with statistical mechanics. We also stress common concepts with other disciplines dealing with similar problems that can be generically referred to as `large graphical models'. While most of the lectures are devoted to the...

These lecture notes have been written for a course at the Algebraic Coding Theory (ACT) summer school 2022 that took place in the university of Zurich. The objective of the course propose an in-depth presentation of the proof of one of the most striking results of coding theory: Tsfasman Vl\u{a}du\c{t} Zink Theorem, which asserts that for some prime power $q$, there exist sequences of codes over $\mathbb{F}_q$ whose asymptotic parameters beat random codes.

The purpose of this article is to discuss recent advances in the growing field of phase retrieval, and to publicize open problems that we believe will be of interest to mathematicians in general, and algebraists in particular.

Geometric complexity theory (GCT) is an approach to the P vs. NP and related problems. This article gives its complexity theoretic overview without assuming any background in algebraic geometry or representation theory.

In this survey we recognize Enrique Arrondo's contributions over the whole of its career, recalling his professional history and collecting the results of his mathematical production.

A homage to the life and mathematics of John K. S. McKay. Obituary for the Bulletin of the London Mathematical Society.