12th February

Speaker: Spencer Bloch

Title: Introduction to iterated integrals and multiple zeta-numbers

Abstract: (Relativistic) Quantum Field Theory is a rich source of fascinating problems in algebraic geometry. In this lecture I will outline some general questions involving physics and algebraic geometry (motives).

Speaker: Moulay-Tahar Benameur

Title: The foliated Cheeger-Gromov invariant.

Abstract: We shall review the classical rho invariant and introduce the foliated rho invariant associated with a holonomy invariant measure on a smooth closed odd foliation. Then we shall explain a foliated homotopy invariance theorem for the signature operator.

Speaker: Marius Crainic

Title: Local forms in Poisson geometry

Abstract: I will report on recent joint work with Ionut Marcut (PhD student, UU) on a Poisson geometric version of the slice theorem (in equivariant geometry) and of the local Reeb stability (in foliation theory). This theorem is a generalization of Conn's linearization theorem- a generalization which is possible due to recent geometric proof (joint with R.L. Fernandes) of Conn's theorem. I will spend most on the time on recalling the classical results, explaining the result and looking at examples. If time allows, some ideas of the proof will be given.

24th February

Speaker: Matthias Flach

Title: Weil-étale cohomology and values of zeta-functions

Abstract: We discuss Lichtenbaum's idea of a Weil-étale cohomology theory and its expected relation to zeta-functions of arithmetic schemes, with the analytic class number formula and the Birch Swinnerton-Dyer conjecture as starting points. If time permits we explain joint work with Baptiste Morin which constructs such a theory with some of the expected properties.

Speaker: Don Zagier

Title: q-series, modular forms, and the Bloch group


Speaker: Spencer Bloch

Title: An algebraic geometer looks at renormalization in physics (colloquium talk)

Abstract: Algebraic geometry has some powerful tools to deal with divergent integrals. I will outline one approach in elementary terms and sketch how it can be applied to integrals arising in physics.

26th February

Speaker: Spencer Bloch

Title: Some algebraic geometry associated to graphs and configurations

Abstract: Introduction to configuration polynomials for rational and quaternionic vector spaces. The quaternionic pfaffian of Eliakim Hastings Moore. Configuration polynomials arising from Feynman rules. Geometric and motivic properties of hypersurfaces defined by configuration polynomials.

Speaker: Hendrik Lenstra

Title: Anisotropic groups and integral closures

Abstract: A vector space over a field, equipped with an inner product, is called anisotropic if no non-zero vector has zero inner product with itself. There is a similar but more subtly defined notion in the context of finite abelian groups. It has an unexpected application to the description of rings of integers of algebraic number fields.

Speaker: Jean-Louis Colliot-Thélène

Title: Brauer-Manin obstruction and integral points

Abstract: For a smooth, projective variety X over a number field, one wonders what is the closure of the set of rational points in the adelic points of X. It is contained in the Brauer-Manin set, a set defined in terms of the Brauer group of X by means of the reciprocity laws of class field theory. For some classes of varieties, this is known or conjectured to be the actual closure of the set of rational points. There are parallel investigations for integral points of affine varieties. Here one raises such questions as strong approximation, a generalization of the chinese remainder theorem. The Brauer group of schemes has also been brought to bear on such matters. I shall describe the situation for homogeneous spaces, including the problem of representing an integer by an integral quadratic form. I will end with the question of representation of an integer as a sum of three cubes.

Speaker: Spencer Bloch

Title: Introduction to mixed Tate motives

Abstract: Dilogarithm motives and Feynman amplitudes associated to 1-loop graphs with masses and external momenta.

12th March

Speaker: Spencer Bloch

Title: The algebraic geometry of graphs and configurations II; Hodge structures

Abstract: The graph motive. Dodgson polynomials; singularities and Hodge structure. Time permitting, I will also discuss sums of graph motives.

Speaker: Rob de Jeu

Title: Valuations of p-adic L-functions and multiplicative Euler characteristics in étale cohomology

Abstract: If k is a totally real field and chi an even character of its Galois group, then Coates and Lichtenbaum conjectured a relation between the p-adic valuation of the L-function associated to chi at a negative odd integer n, and a certain multiplicative Euler characteristic associated to chi with Tate twist 1-n. This was proved by Bayer-Neukirch for the trivial character assuming the main conjecture of Iwasawa theory (since proved by Wiles) by using a p-adic L-function Lp(s) associated to chi. We prove a general relation between Lp(e) for any e in a finite extension of Qp such that Lp(e) is defined, and such an Euler characteristic involving a modified Tate twist 1-e. As an application, we determine some very specific extensions of certain number fields. All of this is joint work with Tejaswi Navilarekallu.

Speaker: Eduard Looijenga

Title: Connectivity properties of decomposability loci

Abstract: In the moduli space of principally polarized abelian varieties of given dimension g, the locus parameterizing the decomposable ones is a closed subvariety. Similarly, in the moduli space of stable genus g>1 curves that have compact jacobian, the locus parameterizing singular ones is a closed subvariety. In either case, we find that this pair is highly connected. We discuss some applications and intermediate results of interest long the way. Most of this talk represents joint work with Wilberd van der Kallen.

Speaker: Spencer Bloch

Title: Cutkowsky rules and Monodromy of quadrics

Abstract: Cutkowsky rules. Propagators, intersection of quadrics, and monodromy.