Grothendieck memorial conference

(Department of Mathematics, Imperial College, 180 Queen's Gate, room 140)

**4 June **

10-11 Luc Illusie

*Grothendieck
and algebraic geometry*

Abstract : Between 1957 and 1970 Grothendieck deeply and durably transformed algebraic geometry. I will discuss some of his revolutionary contributions.

11.30-12.30 Yves André

*The
evolution of Grothendieck's idea of a motivic Galois theory *

15-16 John Coates

*Quadratic
twists of elliptic curves*

Abstract: In this lecture, I will discuss some recent work with Y. Li, Y. Tian, and S. Zhai proving, for a large class of elliptic curves defined over the rationals, the existence of many quadratic twists, which have either no zero, or a simple zero at the point s=1. We use a curious arithmetic induction argument, discovered initially for the congruent number elliptic curve by Y. Tian. We establish particularly strong results for the quadratic twists of X_0(49), and I will also discuss some extensive numerical data obtained by A. Dabrowski, T. Jedrzejak, and L. Szymaszkiewicz based on our work.

16.30-17.30 Mohamed Saïdi

*Grothendieck anabelian section conjecture*

Abstract: In the first half of my talk I will explain the anabelian philosophy of Grothendieck, as formulated in his famous letter to Faltings, and the two main anabelian type conjectures he formulated, including the section conjecture. I will then explain a possible approach to tackling the section conjecture: the local-global approach, via the theory of cuspidalization of arithmetic fundamental groups, and explain how this approach relates to the adelic Mordell-Lang conjecture of Michael Stoll and the Brauer-Manin obstruction. Finally I will present two new results concerning this conjecture.

**5 June**

10-11 Leila Schneps

*Grothendieck-Teichmüller
theory *

Abstract: In this talk we will investigate the ideas that occupied Grothendieck in the early 80's, as explored at some length in his text "The Long March through Galois Theory" and summarized in his "Sketch of a Programme", concentrating essentially on his attempt to find a new description of the absolute Galois group by considering its action on geometric objects; first finite covers of the thrice-punctured sphere, giving rise to the theory of dessins d'enfants, and then, more generally, covers of moduli spaces of curves. We will discuss the relations that emerged between Grothendieck's ideas and seminal work by Drinfeld in the early 90's, and progress that has been made in the direction of his programme since then.

11.30-12.30 Nick Shepherd-Barron

*Del Pezzo surfaces as Springer
fibres*

Abstract: For a long time it has been known that deformations of elliptic singularities possess simultaneous log resolutions, in which del Pezzo surfaces appear. In this talk I will explain how this construction arises in connexion with exceptional simple groups, in a way that extends the construction by Brieskorn, Grothendieck and Springer for simple singularities.

15-16 René Pannekoek

*Rational
points on Kummer varieties and the Brauer-Manin obstruction *

Abstract: Let B be an abelian variety over a number field k and let X be the Kummer variety Km(B) of B, which is a smooth projective model of the quotient B/<-1>. Roughly speaking, rational points on X correspond to rational points on quadratic twists of B. We use this to show that if the Brauer–Manin obstruction controls the failure of weak approximation on all Kummer varieties, then for every positive-dimensional abelian variety A over a number field, the ranks of quadratic twists of A are unbounded. This is joint work with David Holmes (Leiden University).

16.30-17.30 Jan Nekovář

*Generalised
modular parametrisations of elliptic curves *