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

Talk at the International Conference ``G\'eom\'etrie au vingti\`eme ci\`ecle: 1930--2000'', Paris, Institut Henri Poincar\'e, Sept. 2001. The title is a homage to Hans Rademacher and Otto Toeplitz whose book fascinated the author many years ago.

These lecture notes where presented as a course of the CIMPA summer school in Manila, July 20-30, 2009, Semidefinite programming in algebraic combinatorics. This version is an update June 2010.

This is the text of a series of five lectures given by the author at the "Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras" held at Vanderbilt University in May 2004. It is meant as an overview of recent results illustrating the interplay between noncommutative geometry and arithmetic geometry/number theory.

This is an essay to accompany the author's lecture at the introductory workshop on `Nonabelian fundamental groups in arithmetic geometry' at the Newton Institute, Cambridge in July, 2009.

This is a brief exposition of the mathematical themes that motivate the special programme at the Newton Institute in 2009. It is mostly intended for the general public having mathematical training up to the level of secondary school.

This is the memoir of my habilitation thesis, defended on March 29 th, 2013 (Universit\'e Paris XI).

We describe how a systematic use the deep methods from $\ell$-adic cohomology pioneered by Grothendieck and Deligne and further developed by Katz, Laumon allow to make progress on various classical questions from analytic number theory. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016 Arizona Winter School.

This is an idiosyncratic account of the main results presented at the 31st Rencontres de Blois, which took place from June 2nd to June 7th, 2019 in the Castle of Blois, France.

We will present several examples in which ideas from ergodic theory can be useful to study some problems in arithmetic and algebraic geometry.

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