Related papers: Galois theory and Diophantine geometry
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.
A remark about the role of Galois theory in Diophantine geometry as reflected in the work of Serge Lang. An entry in `The mathematical contributions of Serge Lang.'
This paper develops a framework of algebra whereby every Diophantine equation is made quickly accessible by a study of the corresponding row entries in an array of numbers which we call the Newtonian triangles. We then apply this framework…
This paper is around the topics I discussed in the lecture I gave at the Isaac Newton Institute in Cambridge, July 2009, in the Introductory Workshop. This paper can be read as a companion to my paper [Sa\"i di], where detailed proofs can…
This is an expanded version of the 10 lectures given as the 2006 London Mathematical Society Invited Lecture Series at the Heriot-Watt University 31 July - 4 August 2006.
This text is the write-up of a talk at the Bellairs Workshop in Number Theory on Tropical and Non-Archimedean Geometry that took place at the Bellairs Research Institute, Barbados, in May 2011. The goal of this text is to present recent…
We introduce an abstract topos-theoretic framework for building Galois-type theories in a variety of different mathematical contexts; such theories are obtained from representations of certain atomic two-valued toposes as toposes of…
These are notes based on a series of talks that the author gave at the "Interactions between hyperbolic geometry and quantum groups" conference held at Columbia University in June of 2009.
An introduction to quantum groups and non-commutative differential calculus (Lecture at the III Workshop on Differential Geometry, Granada, September 1994)
This expository paper recounts the development and application of the concept of the diffeological groupoid, from its introduction in 1985 to its use in current research. We demonstrate how this single concept has served as a powerful and…
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…
We give a brief exposition on the uses of non-commutative fundamental groups for the study of Diophantine problems via a non-abelian Albanese map.
This is an expository article on the theory of formal group laws in homotopy theory, with the goal of leading to the connection with higher-dimensional abelian varieties and automorphic forms. These are roughly based on a talk at the…
This is an expository lecture on the subject of the title delivered at the Park-IAS mathematical institute in Princeton (July, 2000).
The lecture notes below correspond to the course given by the author in occasion of the VIASM school on Number Theory (18-24 June 2018, Hanoi). We have chosen to omit the proofs that are already presented in details in many references in…
We survey recent developments in the Birational Anabelian Geometry program aimed at the reconstruction of function fields of algebraic varieties over algebraically closed fields from pieces of their absolute Galois groups.
In the preprint we present an outline of the one dimensional version of topological Galois theory. The theory studies topological obstruction to solvability of equations "in finite terms" (i.e. to their solvabilty by radicals, by elementary…
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…
We explore connections between birational anabelian geometry and abstract projective geometry. One of the applications is a proof of a version of the birational section conjecture.
We present proofs of classical results in Poisson geometry using techniques from Dirac geometry. This article is based on mini-courses at the Poisson summer school in Geneva, June 2016, and at the workshop "Quantum Groups and Gravity" at…