February 27, 2023
The inverse Galois problem asks whether any finite group can be realised as the Galois group of a Galois extension of the rationals. This problem and its refinements have stimulated a large amount of research in number theory and algebraic geometry in the past century, ranging from Noether's problem (letting X denote the quotient of the affine space by a finite group acting linearly, when is X rational?) to the rigidity method (if X is not rational, does it at least contain i...
January 30, 2018
The subjects in the title are interwoven in many different and very deep ways. I recently wrote several expository accounts [64-66] that reflect a certain range of developments, but even in their totality they cannot be taken as a comprehensive survey. In the format of a 30-page contribution aimed at a general mathematical audience, I have decided to illustrate some of the basic ideas in one very interesting example - that of HilbpC2, nq, hoping to spark the curiosity of coll...
April 13, 2010
This survey is about Galois theory of curves in characteristic p, a topic which has inspired major research in algebraic geometry and number theory and which contains many open questions. We illustrate important phenomena which occur for covers of curves in characteristic p. We explain key results on the structure of fundamental groups. We end by describing areas of active research and giving two new results about the genus and p-rank of certain covers of the affine line.
July 31, 2008
This text is an introduction to algebraic enumerative geometry and to applications of tropical geometry to classical geometry, based on a course given during the X-UPS mathematical days, 2008 May 14th and 15th. The aim of this text is to be understandable by a first year master student.
April 25, 2005
Below are the problems that I formulated at Open Problems Session of {\it Workshop on Group Actions on Rational Varieties}, McGill University and University of Montreal, Canada, March 2002. 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.
September 23, 2004
This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups defined over the fields which admit arbitrary cyclic extensions.
January 8, 2014
In this paper we improve our previous results on classification of groups of points on abelian varieties over finite fields. The classification is given in terms of the Weil polynomial of abelian varieties in a given $k$-isogeny class.
December 16, 2015
This is a survey paper on the theory of scattered spaces in Galois geometry and its applications.
September 26, 2023
In view of the recent proofs of the $P=W$ conjecture, the present paper reviews and relate the latest results in the non abelian Hodge theory of curves, with a view on how $P=W$ phenomena appear in multiple areas of algebraic geometry. Finally, we retrace the history of results on the conjecture up to the three new proofs of the statement in full generality.
November 29, 2023
We introduce equivariant L-genera associated to actions of Galois and related groups on algebraic varieties. We explain the role which these L-genera play in the spectral analysis of Eisenstein series. We also discuss the natural categorification of L-genera and its potential role in enumerative geometry.