Galois Groups in Enumerative Geometry and Applications

Hopf Galois theory expands the classical Galois theory by considering the Galois property in terms of the action of the group algebra k[G] on K/k and then replacing it by the action of a Hopf algebra. We review the case of separable extensions where the Hopf Galois property admits a group-theoretical formulation suitable for counting and classifying, and also to perform explicit computations and explicit descriptions of all the ingredients involved in a Hopf Galois structure....

This is a renovated list of open problems, to appear in: "Affine Algebraic Geometry" conference Proceedings volume in Contemporary Mathematics series of the Amer. Math. Soc. Ed. by Jaime Gutierrez, Vladimir Shpilrain, and Jie-Tai Yu.

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.

We provide an expository introduction to $\mathbb{A}^1$-enumerative geometry, which uses the machinery of $\mathbb{A}^1$-homotopy theory to enrich classical enumerative geometry questions over a broader range of fields. Included is a discussion of enriched local degrees of morphisms of smooth schemes, following Morel, $\mathbb{A}^1$-Milnor numbers, as well as various computational tools and recent examples. Based off lectures delivered by Kirsten Wickelgren at the 2018 LMS-...

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

The author surveys Galois theory of function fields with non-zero caracteristic and its relation to the structure of finite permutation groups and matrix groups.

This is a survey of select recent results by a number of authors, inspired by the classical configuration theorems of projective geometry.

The inverse problem of Galois Theory was developed in the early 1800 s as an approach to understand polynomials and their roots. The inverse Galois problem states whether any finite group can be realized as a Galois group over Q (field of rational numbers). There has been considerable progress in this as yet unsolved problem. Here, we shall discuss some of the most significant results on this problem. This paper also presents a nice variety of significant methods in connectio...

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

Computational Galois theory, in particular the problem of computing the Galois group of a given polynomial is a very old problem. Currently, the best algorithmic solution is Stauduhar's method. Computationally, one of the key challenges in the application of Stauduhar's method is to find, for a given pair of groups H<G a G-relative H-invariant, that is a multivariate polynomial F that is H-invariant, but not G-invariant. While generic, theoretical methods are known to find su...