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

We present detailed summaries of the talks that were given during a week-long workshop on Arithmetic Groups at the Banff International Research Station in April 2013. The vast majority of these reports are based on abstracts that were kindly provided by the speakers. Video recordings of many of the lectures are available online.

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 article surveys the development of the theory of algebraic geometry codes since their discovery in the late 70's. We summarize the major results on various problems such as: asymptotic parameters, improved estimates on the minimum distance, and decoding algorithms. In addition, we present various modern applications of these codes such as public-key cryptography, algebraic complexity theory, multiparty computation or distributed storage.

This article is a survey of recent developments in, and a tutorial on, the approach to P v. NP and related questions called Geometric Complexity Theory (GCT). It is written to be accessible to graduate students. Numerous open questions in algebraic geometry and representation theory relevant for GCT are presented.

Lecture notes on an introductory course on arithmetic lattices (EPFL 2014).

This is a report on the work of Robert Langlands, following his award of the Abel Prize in 2018. It includes his contributions to the general areas of Representation Theory, Automorphic Forms, Number Theory and Arithmetic Geometry. We have tried to communicate the remarkable continuity that runs throughout all of his work, with its roots in several fundamental areas of mathematics. What is now known as the Langlands Program represents a unification of some of the deepest part...

This work is a collection of old and new aplications of Galois cohomology to the clasification of algebraic and arithmetical objects.

Notes for a course at the H.-C. R. I., Allahabad, 15 August 2008 -- 26 January 2009

We summarize the major results in number theory of the last decade.

A method of constructing algebraic-geometric codes with many automorphisms arising from Galois points for algebraic curves is presented.