Invited speakers:

Herbert Gangl (University of Durham)

Title: Harmonic polylogarithms and tensor calculus.

Abstract: We use a tensor calculus which captures the differential behaviour of multiple polylogarithms, motivated by a bar construction on algebraic cycles by Bloch and Kriz and developed in joint work with Levin and Goncharov, to reduce harmonic polylogarithms, as used in Feynman graph computations in particle physics, to better computable ones. As a result, many Feynman integrals for 2-loop calculations can be quickly evaluated to high precision, which has been formulated as a desirable goal, e.g. by particle physicists working on collider experiments. (joint with Claude Duhr and John Rhodes) 

Behrang Noohi (Kings College London)

Title: Galois cohomology of crossed-modules and cohomology of reductive groups.

Abstract: A 2-group (or a crossed-module) is a categorified version of a group. Line bundles over a scheme, for instance, form the Picard 2-group. Galois cohomology of 2-groups can be used to give information about Galois cohomology of ordinary groups (via, say, certain long exact sequences). We discuss the basics of the theory and give some simple examples involving Picard and Brauer groups. We then explain Borovoi's application of these ideas to the study of Galois cohomology of reductive groups.

Lorenzo Ramero (Universite Lille 1)

Title: Cohomological epsilon factors and p-adic analytic geometry.

Abstract: Following Grothendieck, one knows how to associate to every l-adic sheaf F on a curve C defined over a finite field, a function L(F,t), the L-function of the sheaf. It satisfies a functional equation, in which there appears a constant e(F) called the epsilon factor, that plays an important role in several arithmetical questions. About 25 years ago, in a series of lectures at IHES, Deligne proposed a program to prove a formula expressing e(F) as the product of local epsilon factors e_x(F) (where the index x varies over the closed points of C); each e_x(F) would be determined by the local monodromy representation of F around x. Deligne's idea was to reinterpret e(F) in terms of the determinant of the cohomology of F, and to give a canonical decomposition of this determinant as a tensor product of local modules. When formulated in this manner, the problem can be stated for a curve defined over an arbitrary field K. For the case of a finite field, such a decomposition has been found by Laumon. In my talk, I will discuss the case where K is a p-adic field of zero characteristic.

Andrei Yafaev (UCL)

Title: The Andre-Oort Conjecture.

Abstract: We will present the Andre-Oort conjecture and its proof under the generalised Riemann hypothesis. We will also present some recent developments, namely the ideas of Jonathan Pila involving model theory that allow, in some cases to prove the conjecture unconditionally.

Contributed talks:

Arsen Elkin (University of Warwick)

Title: Ekedahl-Oort strata of Artin-Schreier curves in characteristic 2.

Abstract: For a hyperelliptic curve X of genus g defined over an algebraically closed field k of characteristic p=2, we show that the de Rham cohomology of X decomposes into pieces corresponding to the branch points of the hyperelliptic cover and is completely determined by the ramification invariants associated with these branch points. This allows us to compute the isomorphism class of the group scheme J_X[2] of the 2-torsion of the Jacobian of X in terms of its Ekedahl-Oort type.

Sohail Iqbal (University of Warwick)

Title: Godeaux, Campedelli and Q-Gorenstein Smoothing.

Abstract: I will describe a link between Godeaux and Campedelli surfaces by using Q-Goresntein smoothing technique given by Lee and Park. But with explicit constructions based on writing out canonical rings by generators and relations. The rings are described as unprojections. The explicit construction of these surfaces are sections of higher dimensional key varieties.

Samir Silsek (University of Warwick)

Title: Generators for Rational Points on Cubic Surfaces.

Abstract: Let C be a smooth plane cubic curve over the rationals. The Mordell--Weil Theorem can be restated as follows: there is a finite subset B of rational points such that all rational points can be obtained from this subset by successive tangent and secant constructions. It is conjectured that a minimal such B can be arbitrarily large; this is indeed the well-known conjecture that there are elliptic curves with arbitrarily large ranks. This talk is concerned with the corresponding problem for cubic surfaces.