Galois Groups in Enumerative Geometry and Applications

These notes are an exposition of Galois Theory from the original Lagrangian and Galoisian point of view. A particular effort was made here to better understand the connection between Lagrange's purely combinatorial approach and Galois algebraic extensions of the latter. Moreover, stimulated by the necessities of present day computer explorations, the algorithmic approach has been given priority here over every other aspect of presentation. In particular, you may not find here...

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.

A Fano problem consists of enumerating linear spaces of a fixed dimension on a variety, generalizing the classical problem of 27 lines on a cubic surface. Those Fano problems with finitely many linear spaces have an associated Galois group that acts on these linear spaces and controls the complexity of computing them in coordinates via radicals. Galois groups of Fano problems were first studied by Jordan, who considered the Galois group of the problem of 27 lines on a cubic s...

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.

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.

In this article we present a new method to obtain polynomial lower bounds for Galois orbits of torsion points of one dimensional group varieties.

This is a survey paper discussing the moduli problem for varieties of general type.

This paper surveys some applications of moduli theory to issues concerning the distribution of rational points on algebraic varieties. It will appear on the proceedings of the Fano Conference.

These notes are a self-contained introduction to Galois theory, designed for the student who has done a first course in abstract algebra.

A strategy to address the inverse Galois problem over Q consists of exploiting the knowledge of Galois representations attached to certain automorphic forms. More precisely, if such forms are carefully chosen, they provide compatible systems of Galois representations satisfying some desired properties, e.g. properties that reflect on the image of the members of the system. In this article we survey some results obtained using this strategy.