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12th February

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Speaker: Spencer Bloch

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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).
- I. Iterated integrals; multiple zeta-numbers.
- II. Graphs; Feynman rules; Feynman amplitudes; Feynman parameters.
- III. External momenta and masses.
- IV. Renormalization; limiting mixed Hodge structures.
- V. Monodromy; Cutkowsky Rules.

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Speaker: Moulay-Tahar Benameur

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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.

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Speaker: Marius Crainic

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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.

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24th February

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Speaker: Matthias Flach

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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.

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Speaker: Don Zagier

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

Abstract:

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Speaker: Spencer Bloch

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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.

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26th February

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Speaker: Spencer Bloch

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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.

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Speaker: Hendrik Lenstra

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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.

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Speaker: Jean-Louis Colliot-Thélène

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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.

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Speaker: Spencer Bloch

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Title: Introduction to mixed Tate motives

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

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12th March

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Speaker: Spencer Bloch

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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.

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Speaker: Rob de Jeu

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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 L_{p}(s) associated to chi. We prove a general relation between
L_{p}(e) for any e in a finite extension of Q_{p} such that L_{p}(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.

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Speaker: Eduard Looijenga

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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.

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Speaker: Spencer Bloch

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Title: Cutkowsky rules and Monodromy of quadrics

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